Discussion: Please discuss, elaborate and give example on the topic below. Please use the Module/reference I provided. Professor will not allow outside sources.
Author: Jackson, S. L. (2017). Statistics plain and simple, (4th ed.). Boston, MA: Cengage Learning
Topic:
Using the sample provided, address the following:
· How would you interpret the results of the two-way ANOVA?
· What does the p value tell you?
· The results mention df. What does that term represent? How is it calculated? Write a plainly stated sentence that explains what these results tell you about the groups.
Sample
Sum of Squares df Mean Square F Sig.
SCORES Between Groups 351.520 4 87.880 9.085 .000
Within Groups 435.300 45 9.673
Total 7 86.820 49
Module/reference
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in which different subjects are used in each group) or a correlated-groups design. (Remember, correlated-groups designs are of two types: within-subjects designs, in which the same subjects are used repeatedly in each group, and matched-subjects designs, in which different subjects are matched between conditions on variables that the researcher believes are relevant to the study.)
We will look at the typical inferential statistics used to analyze interval-ratio data for between-subjects designs. In Module 13 we discuss the advantages and rati ...
Discussion Please use the Referencemodule I provided. Professor widdowsonerica
Discussion: Please use the Reference/module I provided. Professor will not consider outside source. Please Discuss, elaborate and give example. Be careful with spelling and grammar.
Author: Jackson, S. L. (2017). Statistics plain and simple, (4th ed.). Boston, MA: Cengage Learning
Topic:
Review this week’s course materials and learning activities, and reflect on your learning so far this week. Respond to one or more of the following prompts in one to two paragraphs:
1. Provide citation and reference to the material(s) you discuss. Describe what you found interesting regarding this topic, and why.
2. Describe how you will apply that learning in your daily life, including your work life.
3. Describe what may be unclear to you, and what you would like to learn.
Reference/Module
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in which different subjects are used in each group) or a correlated-groups design. (Remember, correlated-groups designs are of two types: within-subjects designs, in which the same subjects are used repeatedly in each group, and matched-subjects designs, in which different subjects are matched between conditions on variables that the researcher believes are relevant to the study.)
We will look at the typical inferential statistics used to analyze interval-ratio data for between-subjects designs. In Mo ...
Discussion Discuss, elaborate and give example on the topic belowwiddowsonerica
Discussion: Discuss, elaborate and give example on the topic below. Please use the Reference/Module I provided below. Professor wont consider outside sources. Please be careful with spelling and Grammar.
Author: Jackson, S.L. (2017) Statistics plain and simple. (4th ed.). Boston, MA: Cengage Learning
Topic:
When there is a between-subjects design, use a one-way between-subjects ANOVA, which uses only one independent variable. 275 words.
Reference/Module
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in which different subjects are used in each group) or a correlated-groups design. (Remember, correlated-groups designs are of two types: within-subjects designs, in which the same subjects are used repeatedly in each group, and matched-subjects designs, in which different subjects are matched between conditions on variables that the researcher believes are relevant to the study.)
We will look at the typical inferential statistics used to analyze interval-ratio data for between-subjects designs. In Module 13 we discuss the advantages and rationale for studying more than two groups; in Module 14 we discuss the statistics appropriate for use with between-subjects designs involving more than two groups.
MODULE 13
Comparing More Than Two Groups
Learning Objectives
•Explain what additional information can be gained by using des ...
#35921 Topic Discussion2Number of Pages 1 (Double Spaced)Ntroutmanboris
#35921 Topic: Discussion2
Number of Pages: 1 (Double Spaced)
Number of sources: 1
Writing Style: APA
Type of document: Essay
Academic Level:Master
Category: Psychology
VIP Support: N/A
Language Style: English (U.S.)
Order Instructions:
I will upload the instruction
Discussion: Discuss, elaborate and give example on the topic below. Please use the Reference/Module I provided below. Professor wont consider outside sources. Please be careful with spelling and Grammar.
Author: Jackson, S.L. (2017) Statistics plain and simple. (4th ed.). Boston, MA: Cengage Learning
Topic:
When there is a between-subjects design, use a one-way between-subjects ANOVA, which uses only one independent variable. 275 words.
Reference/Module
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in which different subjects are used in each group) or a correlated-groups design. (Remember, correlated-groups designs are of two types: within-subjects designs, in which the same subjects are used repeatedly in each group, and matched-subjects designs, in which different subjects are matched between conditions on variables that the researcher believes are relevant to the study.)
We will look at the typical inferential statistics used to analyze interval-ratio data for between-subjects designs. In Module 13 we discuss the advantages and rational ...
Happiness Data SetAuthor Jackson, S.L. (2017) Statistics plain ShainaBoling829
Happiness Data Set
Author: Jackson, S.L. (2017) Statistics plain and simple. (4th ed.). Boston, MA: Cengage Learning.
I attach the previous essay so you have idea on how to do this assignment. It is similar to the assignment last week.
Assignment Content
1.
Top of Form
As you get closer to the final project in Week 6, you should have a better idea of the role of statistics in research. This week, you will calculate a one-way ANOVA for the independent groups. Reading and interpreting the output correctly is highly important. Most people who read research articles never see the actual output or data; they read the results statements by the researcher, which is why your summary must be accurate.
Consider your hypothesis statements you created in Part 2.
Calculate a one-way ANOVA, including a Tukey's HSD for the data from the Happiness and Engagement Dataset.
Write a 125- to 175-word summary of your interpretation of the results of the ANOVA, and describe how using an ANOVA was more advantageous than using multiple t tests to compare your independent variable on the outcome. Copy and paste your Microsoft® Excel® output below the summary.
Format your summary according to APA format.
Submit your summary, including the Microsoft® Excel® output to the assignment.
Reference/Module:
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check AnswersModule 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check AnswersChapter 7 Summary and ReviewChapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in ...
BUS 308 Week 3 Lecture 1 Examining Differences - Continued.docxcurwenmichaela
BUS 308 Week 3 Lecture 1
Examining Differences - Continued
Expected Outcomes
After reading this lecture, the student should be familiar with:
1. Issues around multiple testing
2. The basics of the Analysis of Variance test
3. Determining significant differences between group means
4. The basics of the Chi Square Distribution.
Overview
Last week, we found out ways to examine differences between a measure taken on two
groups (two-sample test situation) as well as comparing that measure to a standard (a one-sample
test situation). We looked at the F test which let us test for variance equality. We also looked at
the t-test which focused on testing for mean equality. We noted that the t-test had three distinct
versions, one for groups that had equal variances, one for groups that had unequal variances, and
one for data that was paired (two measures on the same subject, such as salary and midpoint for
each employee). We also looked at how the 2-sample unequal t-test could be used to use Excel
to perform a one-sample mean test against a standard or constant value. This week we expand
our tool kit to let us compare multiple groups for similar mean values.
A second tool will let us look at how data values are distributed – if graphed, would they
look the same? Different shapes or patterns often means the data sets differ in significant ways
that can help explain results.
Multiple Groups
As interesting as comparing two groups is, often it is a bit limiting as to what it tells us.
One obvious issue that we are missing in the comparisons made last week was equal work. This
idea is still somewhat hard to get a clear handle on. Typically, as we look at this issue, questions
arise about things such as performance appraisal ratings, education distribution, seniority impact,
etc.
Some of these can be tested with the tools introduced last week. We can see, for
example, if the performance rating average is the same for each gender. What we couldn’t do, at
this point however, is see if performance ratings differ by grade, do the more senior workers
perform relatively better? Is there a difference between ratings for each gender by grade level?
The same questions can be asked about seniority impact. This week will give us tools to expand
how we look at the clues hidden within the data set about equal pay for equal work.
ANOVA
So, let’s start taking a look at these questions. The first tool for this week is the Analysis
of Variance – ANOVA for short. ANOVA is often confusing for students; it says it analyzes
variance (which it does) but the purpose of an ANOVA test is to determine if the means of
different groups are the same! Now, so far, we have considered means and variance to be two
distinct characteristics of data sets; characteristics that are not related, yet here we are saying that
looking at one will give us insight into the other.
The reason is due to the way the variance is an.
Assessment 4 ContextRecall that null hypothesis tests are of.docxfestockton
Assessment 4 Context
Recall that null hypothesis tests are of two types: (1) differences between group means and (2) association between variables. In both cases there is a null hypothesis and an alternative hypothesis. In the group means test, the null hypothesis is that the two groups have equal means, and the alternative hypothesis is that the two groups do not have equal means. In the association between variables type of test, the null hypothesis is that the correlation coefficient between the two variables is zero, and the alternative hypothesis is that the correlation coefficient is not zero.
Notice in each case that the hypotheses are mutually exclusive. If the null is false, the alternative must be true. The purpose of null hypothesis statistical tests is generally to show that the null has a low probability of being true (the p value is less than .05) – low enough that the researcher can legitimately claim it is false. The reason this is done is to support the allegation that the alternative hypothesis is true.
In this context you will be studying the details of the first type of test again, with the added capability of comparing the means among more than two group at a time. This is the same type of test of difference between group means. In variations on this model, the groups can actually be the same people under different conditions. The main idea is that several group mean values are being compared. The groups each have an average score or mean on some variable. The null hypothesis is that the difference between all the group means is zero. The alternative hypothesis is that the difference between the means is not zero. Notice that if the null is false, the alternative must be true. It is first instructive to consider some of the details of groups.
One might ask why we would not use multiple t tests in this situation. For instance, with three groups, why would I not compare groups one and two with a t test, then compare groups one and three, and then compare groups two and three?
The answer can be found in our basic probability review. We are concerned with the probability of a TYPE I error (rejecting a true null hypothesis). We generally set an alpha level of .05, which is the probability of making a TYPE I error. Now consider what happens when we do three t tests. There is .05 probability of making a TYPE I error on the first test, .05 probability of the same error on the second test, and .05 probability on the third test. What happens is that these errors are essentially additive, in that the chances of at least one TYPE I error among the three tests much greater than .05. It is like the increased probability of drawing an ace from a deck of cards when we can make multiple draws.
ANOVA allows us do an "overall" test of multiple groups to determine if there are any differences among groups within the set. Notice that ANOVA does not tell us which groups among the three groups are different from each other. The primary test ...
Assessment 4 ContextRecall that null hypothesis tests are of.docxgalerussel59292
Assessment 4 Context
Recall that null hypothesis tests are of two types: (1) differences between group means and (2) association between variables. In both cases there is a null hypothesis and an alternative hypothesis. In the group means test, the null hypothesis is that the two groups have equal means, and the alternative hypothesis is that the two groups do not have equal means. In the association between variables type of test, the null hypothesis is that the correlation coefficient between the two variables is zero, and the alternative hypothesis is that the correlation coefficient is not zero.
Notice in each case that the hypotheses are mutually exclusive. If the null is false, the alternative must be true. The purpose of null hypothesis statistical tests is generally to show that the null has a low probability of being true (the p value is less than .05) – low enough that the researcher can legitimately claim it is false. The reason this is done is to support the allegation that the alternative hypothesis is true.
In this context you will be studying the details of the first type of test again, with the added capability of comparing the means among more than two group at a time. This is the same type of test of difference between group means. In variations on this model, the groups can actually be the same people under different conditions. The main idea is that several group mean values are being compared. The groups each have an average score or mean on some variable. The null hypothesis is that the difference between all the group means is zero. The alternative hypothesis is that the difference between the means is not zero. Notice that if the null is false, the alternative must be true. It is first instructive to consider some of the details of groups.
One might ask why we would not use multiple t tests in this situation. For instance, with three groups, why would I not compare groups one and two with a t test, then compare groups one and three, and then compare groups two and three?
The answer can be found in our basic probability review. We are concerned with the probability of a TYPE I error (rejecting a true null hypothesis). We generally set an alpha level of .05, which is the probability of making a TYPE I error. Now consider what happens when we do three t tests. There is .05 probability of making a TYPE I error on the first test, .05 probability of the same error on the second test, and .05 probability on the third test. What happens is that these errors are essentially additive, in that the chances of at least one TYPE I error among the three tests much greater than .05. It is like the increased probability of drawing an ace from a deck of cards when we can make multiple draws.
ANOVA allows us do an "overall" test of multiple groups to determine if there are any differences among groups within the set. Notice that ANOVA does not tell us which groups among the three groups are different from each other. The primary test.
Discussion Please use the Referencemodule I provided. Professor widdowsonerica
Discussion: Please use the Reference/module I provided. Professor will not consider outside source. Please Discuss, elaborate and give example. Be careful with spelling and grammar.
Author: Jackson, S. L. (2017). Statistics plain and simple, (4th ed.). Boston, MA: Cengage Learning
Topic:
Review this week’s course materials and learning activities, and reflect on your learning so far this week. Respond to one or more of the following prompts in one to two paragraphs:
1. Provide citation and reference to the material(s) you discuss. Describe what you found interesting regarding this topic, and why.
2. Describe how you will apply that learning in your daily life, including your work life.
3. Describe what may be unclear to you, and what you would like to learn.
Reference/Module
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in which different subjects are used in each group) or a correlated-groups design. (Remember, correlated-groups designs are of two types: within-subjects designs, in which the same subjects are used repeatedly in each group, and matched-subjects designs, in which different subjects are matched between conditions on variables that the researcher believes are relevant to the study.)
We will look at the typical inferential statistics used to analyze interval-ratio data for between-subjects designs. In Mo ...
Discussion Discuss, elaborate and give example on the topic belowwiddowsonerica
Discussion: Discuss, elaborate and give example on the topic below. Please use the Reference/Module I provided below. Professor wont consider outside sources. Please be careful with spelling and Grammar.
Author: Jackson, S.L. (2017) Statistics plain and simple. (4th ed.). Boston, MA: Cengage Learning
Topic:
When there is a between-subjects design, use a one-way between-subjects ANOVA, which uses only one independent variable. 275 words.
Reference/Module
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in which different subjects are used in each group) or a correlated-groups design. (Remember, correlated-groups designs are of two types: within-subjects designs, in which the same subjects are used repeatedly in each group, and matched-subjects designs, in which different subjects are matched between conditions on variables that the researcher believes are relevant to the study.)
We will look at the typical inferential statistics used to analyze interval-ratio data for between-subjects designs. In Module 13 we discuss the advantages and rationale for studying more than two groups; in Module 14 we discuss the statistics appropriate for use with between-subjects designs involving more than two groups.
MODULE 13
Comparing More Than Two Groups
Learning Objectives
•Explain what additional information can be gained by using des ...
#35921 Topic Discussion2Number of Pages 1 (Double Spaced)Ntroutmanboris
#35921 Topic: Discussion2
Number of Pages: 1 (Double Spaced)
Number of sources: 1
Writing Style: APA
Type of document: Essay
Academic Level:Master
Category: Psychology
VIP Support: N/A
Language Style: English (U.S.)
Order Instructions:
I will upload the instruction
Discussion: Discuss, elaborate and give example on the topic below. Please use the Reference/Module I provided below. Professor wont consider outside sources. Please be careful with spelling and Grammar.
Author: Jackson, S.L. (2017) Statistics plain and simple. (4th ed.). Boston, MA: Cengage Learning
Topic:
When there is a between-subjects design, use a one-way between-subjects ANOVA, which uses only one independent variable. 275 words.
Reference/Module
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in which different subjects are used in each group) or a correlated-groups design. (Remember, correlated-groups designs are of two types: within-subjects designs, in which the same subjects are used repeatedly in each group, and matched-subjects designs, in which different subjects are matched between conditions on variables that the researcher believes are relevant to the study.)
We will look at the typical inferential statistics used to analyze interval-ratio data for between-subjects designs. In Module 13 we discuss the advantages and rational ...
Happiness Data SetAuthor Jackson, S.L. (2017) Statistics plain ShainaBoling829
Happiness Data Set
Author: Jackson, S.L. (2017) Statistics plain and simple. (4th ed.). Boston, MA: Cengage Learning.
I attach the previous essay so you have idea on how to do this assignment. It is similar to the assignment last week.
Assignment Content
1.
Top of Form
As you get closer to the final project in Week 6, you should have a better idea of the role of statistics in research. This week, you will calculate a one-way ANOVA for the independent groups. Reading and interpreting the output correctly is highly important. Most people who read research articles never see the actual output or data; they read the results statements by the researcher, which is why your summary must be accurate.
Consider your hypothesis statements you created in Part 2.
Calculate a one-way ANOVA, including a Tukey's HSD for the data from the Happiness and Engagement Dataset.
Write a 125- to 175-word summary of your interpretation of the results of the ANOVA, and describe how using an ANOVA was more advantageous than using multiple t tests to compare your independent variable on the outcome. Copy and paste your Microsoft® Excel® output below the summary.
Format your summary according to APA format.
Submit your summary, including the Microsoft® Excel® output to the assignment.
Reference/Module:
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control Group
Comparing a Placebo Group to the Control and Experimental Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It Does
Review of Key Terms
Module Exercises
Critical Thinking Check AnswersModule 14: One-Way Between-Subjects Analysis of Variance (ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check AnswersChapter 7 Summary and ReviewChapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical analyses used with designs involving more than two groups. The inferential statistics discussed in this chapter differ from those presented in the previous two chapters. In Chapter 5, single samples were being compared to populations (z test and t test), and in Chapter 6, two independent or correlated samples were being compared. In this chapter, the statistics are designed to test differences between more than two equivalent groups of subjects.
Several factors influence which statistic should be used to analyze the data collected. For example, the type of data collected and the number of groups being compared must be considered. Moreover, the statistic used to analyze the data will vary depending on whether the study involves a between-subjects design (designs in ...
BUS 308 Week 3 Lecture 1 Examining Differences - Continued.docxcurwenmichaela
BUS 308 Week 3 Lecture 1
Examining Differences - Continued
Expected Outcomes
After reading this lecture, the student should be familiar with:
1. Issues around multiple testing
2. The basics of the Analysis of Variance test
3. Determining significant differences between group means
4. The basics of the Chi Square Distribution.
Overview
Last week, we found out ways to examine differences between a measure taken on two
groups (two-sample test situation) as well as comparing that measure to a standard (a one-sample
test situation). We looked at the F test which let us test for variance equality. We also looked at
the t-test which focused on testing for mean equality. We noted that the t-test had three distinct
versions, one for groups that had equal variances, one for groups that had unequal variances, and
one for data that was paired (two measures on the same subject, such as salary and midpoint for
each employee). We also looked at how the 2-sample unequal t-test could be used to use Excel
to perform a one-sample mean test against a standard or constant value. This week we expand
our tool kit to let us compare multiple groups for similar mean values.
A second tool will let us look at how data values are distributed – if graphed, would they
look the same? Different shapes or patterns often means the data sets differ in significant ways
that can help explain results.
Multiple Groups
As interesting as comparing two groups is, often it is a bit limiting as to what it tells us.
One obvious issue that we are missing in the comparisons made last week was equal work. This
idea is still somewhat hard to get a clear handle on. Typically, as we look at this issue, questions
arise about things such as performance appraisal ratings, education distribution, seniority impact,
etc.
Some of these can be tested with the tools introduced last week. We can see, for
example, if the performance rating average is the same for each gender. What we couldn’t do, at
this point however, is see if performance ratings differ by grade, do the more senior workers
perform relatively better? Is there a difference between ratings for each gender by grade level?
The same questions can be asked about seniority impact. This week will give us tools to expand
how we look at the clues hidden within the data set about equal pay for equal work.
ANOVA
So, let’s start taking a look at these questions. The first tool for this week is the Analysis
of Variance – ANOVA for short. ANOVA is often confusing for students; it says it analyzes
variance (which it does) but the purpose of an ANOVA test is to determine if the means of
different groups are the same! Now, so far, we have considered means and variance to be two
distinct characteristics of data sets; characteristics that are not related, yet here we are saying that
looking at one will give us insight into the other.
The reason is due to the way the variance is an.
Assessment 4 ContextRecall that null hypothesis tests are of.docxfestockton
Assessment 4 Context
Recall that null hypothesis tests are of two types: (1) differences between group means and (2) association between variables. In both cases there is a null hypothesis and an alternative hypothesis. In the group means test, the null hypothesis is that the two groups have equal means, and the alternative hypothesis is that the two groups do not have equal means. In the association between variables type of test, the null hypothesis is that the correlation coefficient between the two variables is zero, and the alternative hypothesis is that the correlation coefficient is not zero.
Notice in each case that the hypotheses are mutually exclusive. If the null is false, the alternative must be true. The purpose of null hypothesis statistical tests is generally to show that the null has a low probability of being true (the p value is less than .05) – low enough that the researcher can legitimately claim it is false. The reason this is done is to support the allegation that the alternative hypothesis is true.
In this context you will be studying the details of the first type of test again, with the added capability of comparing the means among more than two group at a time. This is the same type of test of difference between group means. In variations on this model, the groups can actually be the same people under different conditions. The main idea is that several group mean values are being compared. The groups each have an average score or mean on some variable. The null hypothesis is that the difference between all the group means is zero. The alternative hypothesis is that the difference between the means is not zero. Notice that if the null is false, the alternative must be true. It is first instructive to consider some of the details of groups.
One might ask why we would not use multiple t tests in this situation. For instance, with three groups, why would I not compare groups one and two with a t test, then compare groups one and three, and then compare groups two and three?
The answer can be found in our basic probability review. We are concerned with the probability of a TYPE I error (rejecting a true null hypothesis). We generally set an alpha level of .05, which is the probability of making a TYPE I error. Now consider what happens when we do three t tests. There is .05 probability of making a TYPE I error on the first test, .05 probability of the same error on the second test, and .05 probability on the third test. What happens is that these errors are essentially additive, in that the chances of at least one TYPE I error among the three tests much greater than .05. It is like the increased probability of drawing an ace from a deck of cards when we can make multiple draws.
ANOVA allows us do an "overall" test of multiple groups to determine if there are any differences among groups within the set. Notice that ANOVA does not tell us which groups among the three groups are different from each other. The primary test ...
Assessment 4 ContextRecall that null hypothesis tests are of.docxgalerussel59292
Assessment 4 Context
Recall that null hypothesis tests are of two types: (1) differences between group means and (2) association between variables. In both cases there is a null hypothesis and an alternative hypothesis. In the group means test, the null hypothesis is that the two groups have equal means, and the alternative hypothesis is that the two groups do not have equal means. In the association between variables type of test, the null hypothesis is that the correlation coefficient between the two variables is zero, and the alternative hypothesis is that the correlation coefficient is not zero.
Notice in each case that the hypotheses are mutually exclusive. If the null is false, the alternative must be true. The purpose of null hypothesis statistical tests is generally to show that the null has a low probability of being true (the p value is less than .05) – low enough that the researcher can legitimately claim it is false. The reason this is done is to support the allegation that the alternative hypothesis is true.
In this context you will be studying the details of the first type of test again, with the added capability of comparing the means among more than two group at a time. This is the same type of test of difference between group means. In variations on this model, the groups can actually be the same people under different conditions. The main idea is that several group mean values are being compared. The groups each have an average score or mean on some variable. The null hypothesis is that the difference between all the group means is zero. The alternative hypothesis is that the difference between the means is not zero. Notice that if the null is false, the alternative must be true. It is first instructive to consider some of the details of groups.
One might ask why we would not use multiple t tests in this situation. For instance, with three groups, why would I not compare groups one and two with a t test, then compare groups one and three, and then compare groups two and three?
The answer can be found in our basic probability review. We are concerned with the probability of a TYPE I error (rejecting a true null hypothesis). We generally set an alpha level of .05, which is the probability of making a TYPE I error. Now consider what happens when we do three t tests. There is .05 probability of making a TYPE I error on the first test, .05 probability of the same error on the second test, and .05 probability on the third test. What happens is that these errors are essentially additive, in that the chances of at least one TYPE I error among the three tests much greater than .05. It is like the increased probability of drawing an ace from a deck of cards when we can make multiple draws.
ANOVA allows us do an "overall" test of multiple groups to determine if there are any differences among groups within the set. Notice that ANOVA does not tell us which groups among the three groups are different from each other. The primary test.
5
ANOVA: Analyzing Differences
in Multiple Groups
Learning Objectives
After reading this chapter, you should be able to:
• Describe the similarities and differences between t-tests and ANOVA.
• Explain how ANOVA can help address some of the problems and limitations associ-
ated with t-tests.
• Use ANOVA to analyze multiple group differences.
• Use post hoc tests to pinpoint group differences.
• Determine the practical importance of statistically significant findings using effect
sizes with eta-squared.
iStockphoto/Thinkstock
tan81004_05_c05_103-134.indd 103 2/22/13 4:28 PM
CHAPTER 5Section 5.1 From t-Test to ANOVA
Chapter Overview
5.1 From t-Test to ANOVA
The ANOVA Advantage
Repeated Testing and Type I Error
5.2 One-Way ANOVA
Variance Between and Within
The Statistical Hypotheses
Measuring Data Variability in the ANOVA
Calculating Sums of Squares
Interpreting the Sums of Squares
The F Ratio
The ANOVA Table
Interpreting the F Ratio
Locating Significant Differences
Determining Practical Importance
5.3 Requirements for the One-Way ANOVA
Comparing ANOVA and the Independent t
One-Way ANOVA on Excel
5.4 Another One-Way ANOVA
Chapter Summary
Introduction
During the early part of the 20th century R. A. Fisher worked at an agricultural research station in rural southern England. In his work analyzing the effect of pesticides and
fertilizers on results like crop yield, he was stymied by the limitations in Gosset’s indepen-
dent samples t-test, which allowed him to compare just two samples at a time. In the effort
to develop a more comprehensive approach, Fisher created a statistical method he called
analysis of variance, often referred to by its acronym, ANOVA, which allows for making
multiple comparisons at the same time using relatively small samples.
5.1 From t-Test to ANOVA
The process for completing an independent samples t-test in Chapter 4 illustrated a number of things. The calculated t value, for example, is a score based on a ratio, one
determined by dividing the variability between the two groups (M1 2 M2) by the vari-
ability within the two groups, which is what the standard error of the difference (SEd)
measures. So both the numerator and the denominator of the t-ratio are measures of data
variability, albeit from different sources. The difference between the means is variability
attributed primarily to the independent variable, which is the group to which individual
subjects belong. The variability in the denominator is variability for reasons that are unex-
plained—error variance in the language of statistics.
tan81004_05_c05_103-134.indd 104 2/22/13 4:28 PM
CHAPTER 5Section 5.1 From t-Test to ANOVA
In his method, ANOVA, Fisher also embraced this
pattern of comparing between-groups variance to
within-groups variance. He calculated the variance
statistics differently, as we shall see, but he followed
Gosset’s pattern of a ratio of between-groups vari-
ance compared to within.
The ANOVA .
STATISTICS : Changing the way we do: Hypothesis testing, effect size, power, ...Musfera Nara Vadia
STATISTICS : Changing the way we do: Hypothesis testing, effect size, power, confidence interval, two-tailed and one tailed test, and other misunderstood issues.
3Type your name hereType your three-letter and -number cours.docxlorainedeserre
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Using citations to give credit to others whose ideas or words you have used is an essential requirement to avoid issues of plagiarism. Just as you would never steal someone else’s car, you should not steal his or her words either. To avoid potential problems, always be sure to cite your sources. Cite by referring to the author’s last name, the year of publication in parentheses at the end of the sentence, such as (George & Mallery, 2016), and page numbers if you are using word-for-word materials. For example, “The developments of the World War II years firmly established the probability sample survey as a tool for describing population characteristics, beliefs, and attitudes” (Heeringa, West, & Berglund, 2017, p. 3).
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References
George, D., & Mallery, P. (2016). IBM SPSS statistics 23 step by step: A simple guide and reference. New York, NY: Routledge.
Heeringa, S. G., West, B. T., & Berglund, P. A. (2017). Applied survey data analysis (2nd ed.). New York, NY: Chapman & Hall/CRC Press.
Smith, P. D., Martin, B., Chewning, B., ...
Experimental design is a way to carefully plan experiments in advance so that results are both objective and valid. Ideally, an experimental design should:
• Describe how participants are allocated to experimental groups. A common method is completely randomized design, where participants are assigned to groups at random. A second method is randomized block design, where participants are divided into homogeneous blocks (for example, age groups) before being randomly assigned to groups.
• Minimize or eliminate confounding variables, which can offer alternative explanations for the experimental results.
• Allows making inferences about the relationship between independent variables and dependent variables.
• Reduce variability, to make it easier to find differences in treatment outcomes.
Types of Experimental Design
1. Between Subjects Design.
2. Completely Randomized Design.
3. Factorial Design.
4. Matched-Pairs Design.
5. Observational Study
• Longitudinal Research
• Cross Sectional Research
6. Pretest-Posttest Design.
7. Quasi-Experimental Design.
8. Randomized Block Design.
9. Randomized Controlled Trial
10. Within subjects Design.
ANOVA Interpretation Set 1 Study this scenario and ANOVA.docxfestockton
ANOVA Interpretation Set 1
Study this scenario and ANOVA table, then answer the questions in the assignment instructions.
A researcher wants to compare the efficacy of three different techniques for memorizing
information. They are repetition, imagery, and mnemonics. The researcher randomly assigns
participants to one of the techniques. Each group is instructed in their assigned memory
technique and given a document to memorize within a set time period. Later, a test about the
document is given to all participants. The scores are collected and analyzed using a one-way
ANOVA. Here is the ANOVA table with the results:
Source SS df MS F p
Between 114.3111 2 57.1556 19.74 <.0001
Within 121.6 42 2.8952
Total 235.9111 44
9/10/2019 Print
https://content.ashford.edu/print/AUPSY325.16.1?sections=ch6,ch6sec1,ch6sec2,ch6sec3,ch6sec4,ch6sec5,ch6sec6,ch6sec7,ch6sec8,ch6summary,ch7,ch7sec1,ch7s… 1/76
Chapter Learning Objectives
After reading this chapter, you should be able to do the following:
1. Explain why it is a mistake to analyze the differences between more than two groups with
multiple t tests.
2. Relate sum of squares to other measures of data variability.
3. Compare and contrast t test with analysis of variance (ANOVA).
4. Demonstrate how to determine significant differences among groups in an ANOVA with more
than two groups.
5. Explain the use of eta squared in ANOVA.
6Analysis of Variance
Peter Ginter/Science Faction/Corbis
9/10/2019 Print
https://content.ashford.edu/print/AUPSY325.16.1?sections=ch6,ch6sec1,ch6sec2,ch6sec3,ch6sec4,ch6sec5,ch6sec6,ch6sec7,ch6sec8,ch6summary,ch7,ch7sec1,ch7s… 2/76
Introduction
From one point of view at least, R. A. Fisher was present at the creation of modern statistical analysis. During
the early part of the 20th century, Fisher worked at an agricultural research station in rural southern England.
Analyzing the effect of pesticides and fertilizers on crop yields, he was stymied by independent t tests that
allowed him to compare only two samples at a time. In the effort to accommodate more comparisons, Fisher
created analysis of variance (ANOVA).
Like William Gosset, Fisher felt that his work was important enough to publish, and like Gosset, he met
opposition. Fisher’s came in the form of a fellow statistician, Karl Pearson. Pearson founded the first department
of statistical analysis in the world at University College, London. He also began publication of what is—for
statisticians at least—perhaps the most influential journal in the field, Biometrika. The crux of the initial conflict
between Fisher and Pearson was the latter’s commitment to making one comparison at a time, with the largest
groups possible.
When Fisher submitted his work to Pearson’s journal, suggesting that samples can be small and many
comparisons can be made in the same analysis, Pearson rejected the manuscript. So began a long and
increasingly acrimonious relationship between two men w ...
This presentation deals with the basics of design of experiments and discusses all the three basic statistical designs i.e. CRD, RBD and LSD. Further it explains the guidelines for developing experimental research.
7
Repeated Measures Designs
for Interval Data
Learning Objectives
After reading this chapter, you should be able to:
• Explain the advantages and drawbacks of using data from non-independent groups.
• Complete a paired-samples t-test.
• Complete a within-subjects F.
• Describe “power” as it relates to statistical testing.
iStockphoto/Thinkstock
tan81004_07_c07_163-192.indd 163 2/22/13 3:41 PM
CHAPTER 7Introduction
Chapter Outline
7.1 Dependent Groups Designs
Reconsidering the t and F ratios
An Example
A Matched Pairs Example
Comparing the Paired-Samples t-Test to the Independent Samples t-Test
The Power of the Dependent Groups Test
The Dependent Groups t-Test on Excel
The Alternate Approaches to Dependent t-Tests
7.2 The Within-Subjects F
Managing Error Variance in the Within-Subjects F
A Within-Subjects F Example
Calculating the Within-Subjects F
Understanding the Result
Comparing the Within-Subjects F and the One-Way ANOVA
Another Within-Subjects F Example
A Within-Subjects F in Excel
Chapter Summary
Introduction
Some of the most critical questions in management relate to change over time. For exam-ple, managers are deeply interested in assessing sales growth, shifts in shopping trends,
improvements in employee attitudes, increases in employee performance, and decreases in
absenteeism or turnover. They are also often keen to find out the influence of various
managerial decisions and business strategies on these and many other change-oriented
outcomes. However, none of the analyses completed to this point address these change-
related questions, because these analyses do not accommodate repeated measures of the
same variables within the same group of subjects over time. For instance, the t-tests and
ANOVAs discussed so far compared independent groups, groups that have completely
separate subjects. Each subject was only measured once on each variable of interest. The
same group of subjects was not measured repeatedly on the same variables to assess
change over time.
Another important issue is that independent samples t-tests and ANOVAs assume that
the groups being compared are equivalent on most aspects to begin with, except for the
independent (grouping or treatment) variable being investigated. When groups are large
and individuals are randomly selected, this is usually a reasonable assumption, because
any differences between groups tend to be relatively unimportant. The logic behind ran-
dom selection is that when groups are randomly drawn from the same population they
will differ only by chance—the larger the random sample, the lower the probability of
a substantial pre-existing difference. However, when groups are relatively small it can
be difficult to determine whether a difference in the measures of the dependent variable
occurred because the independent variable had a different impact on the different groups
or because there were differences between the groups to begin with.
tan81004_07.
WEEK 6 – EXERCISES Enter your answers in the spaces pr.docxwendolynhalbert
WEEK 6 – EXERCISES
Enter your answers in the spaces provided. Save the file using your last name as the beginning of the file name (e.g., ruf_week6_exercises) and submit via “Assignments.” When appropriate,
show your work
. You can do the work by hand, scan/take a digital picture, and attach that file with your work.
1
.
A psychotherapist studied whether his clients self-disclosed more while sitting in an easy chair or lying down on a couch. All clients had previously agreed to allow the sessions to be videotaped for research purposes. The therapist randomly assigned 10 clients to each condition. The third session for each client was videotaped and an independent observer counted the clients’ disclosures. The therapist reported that “clients made more disclosures when sitting in easy chairs (
M
= 18.20) than when lying down on a couch (
M
= 14.31),
t
(18) = 2.84,
p
< .05, two-tailed.” Explain these results to a person who understands the
t
test for a single sample but knows nothing about the
t
test for independent means.
2.
A researcher compared the adjustment of adolescents who had been raised in homes that were either very structured or unstructured. Thirty adolescents from each type of family completed an adjustment inventory. The results are reported in the table below. Explain these results to a person who understands the
t
test for a single sample but knows nothing about the
t
test for independent means.
Means on Four Adjustment Scales for
Adolescents from Structured versus Unstructured Homes
Scale
Structured Homes
Unstructured Homes
t
Social Maturity
106.82
113.94
–1.07
School Adjustment
116.31
107.22
2.03*
Identity Development
89.48
94.32
1.93*
Intimacy Development
102.25
104.33
.32
______________________
*
p
< .05
3.
Do men with higher levels of a particular hormone show higher levels of assertiveness? Levels of this hormone were tested in 100 men. The top 10 and the bottom 10 were selected for the study. All participants took part in a laboratory simulation in which they were asked to role-play a person picking his car up from a mechanic’s shop. The simulation was videotaped and later judged by independent raters on each of four types of assertive statements made by the participant. The results are shown in the table below. Explain these results to a person who fully understands the
t
test for a single sample but knows nothing about the
t
test for independent means.
Mean Number of Assertive Statements
Type of Assertive Statement
Group
1
2
3
4
Men with High Levels
2.14
1.16
3.83
0.14
Men with Low Levels
1.21
1.32
2.33
0.38
t
3.81**
0.89
2.03*
0.58
______________________
*
p
< .05;
**
p
< 0.1
4.
A manager of a small store wanted to discourage shoplifters by putting signs around the store saying “Shoplifting is a crime!” However, he wanted to make sure this would not result in customers buying less. To test this, he displayed the signs every other W.
Using the data in the file named Ch. 11 Data Set 2, test the resea.docxdaniahendric
Using the data in the file named Ch. 11 Data Set 2, test the research hypothesis at the .05 level of significance that boys raise their hands in class more often than girls. Do this practice problem by hand using a calculator. What is your conclusion regarding the research hypothesis? Remember to first decide whether this is a one- or two-tailed test.
Using the same data set (Ch. 11 Data Set 2), test the research hypothesis at the .01 level of significance that there is a difference between boys and girls in the number of times they raise their hands in class. Do this practice problem by hand using a calculator. What is your conclusion regarding the research hypothesis? You used the same data for this problem as for Question 1, but you have a different hypothesis (one is directional and the other is nondirectional). How do the results differ and why?
Practice the following problems by hand just to see if you can get the numbers right. Using the following information, calculate the
t
test statistic.
Using the results you got from Question 3 and a level of significance at .05, what are the two-tailed critical values associated with each? Would the null hypothesis be rejected?
Using the data in the file named Ch. 11 Data Set 3, test the null hypothesis that urban and rural residents both have the same attitude toward gun control. Use IBM
®
SPSS
®
software to complete the analysis for this problem.
A public health researcher tested the hypothesis that providing new car buyers with child safety seats will also act as an incentive for parents to take other measures to protect their children (such as driving more safely, child-proofing the home, and so on). Dr. L counted all the occurrences of safe behaviors in the cars and homes of the parents who accepted the seats versus those who did not. The findings: a significant difference at the .013 level. Another researcher did exactly the same study; everything was the same—same type of sample, same outcome measures, same car seats, and so on. Dr. R’s results were marginally significant (recall Ch. 9) at the .051 level. Which result do you trust more and why?
In the following examples, indicate whether you would perform a
t
test of independent means or dependent means.
Two groups were exposed to different treatment levels for ankle sprains. Which treatment was most effective?
A researcher in nursing wanted to know if the recovery of patients was quicker when some received additional in-home care whereas when others received the standard amount.
A group of adolescent boys was offered interpersonal skills counseling and then tested in September and May to see if there was any impact on family harmony.
One group of adult men was given instructions in reducing their high blood pressure whereas another was not given any instructions.
One group of men was provided access to an exercise program and tested two times over a 6-month period for heart health.
For Ch. 12 Data Set 3, comput.
Information Technology, Computer Networks and CyberspaceKizza (201.docxwiddowsonerica
Information Technology, Computer Networks and Cyberspace
Kizza (2014) largely treated cyber-attacks as “cybercrime,” conveying the assumption that these attacks are by definition illegitimate and criminal. However, he occasionally seems to recognize that there might be motives that are more ambiguous and may have a social, moral, or political basis. Lessig (2006) also referred to Internet hacker ethic as rebellious and libertarian and not always criminal in a moral sense. One example cited by Kizza (2014) is the Seattle WTO protest and mass computer attacks that are “increasingly being used to avenge what the attackers consider to be injustices” (p. 97). Another example we discussed was the Stratfor attack. Also, recently we see many examples of cyber security invasions by governments and corporations.
Search the Internet to find an example of a hacking activity or situation that represents a morally, ethically, or criminally ambiguous situation but is different from any examples you used in other assignments for this course.
Write a critical essay that addresses the following items:
Cite and briefly describe your example.
Apply what you have learned from the course to this point to identify arguments both in support of and critical of the behavior of the attackers.
Describe and explain the relationships among morality, ethics, law, and crime as they intersect in the case example that you have found.
Cite Winner’s “Do Artifacts Have Politics?” to identify and explain the political, moral, and ethical choices and consequences that may be embedded in technical choices and artifacts described in your example.
Discuss and cite the article “Do Artifacts Have Politics?”, the source of your example, and at least one additional credible or scholarly source other than the course textbook to support your analysis and positions. You may cite the course textbooks as well if you wish. Use standards and APA style guidelines, citing references as appropriate. Your paper should be two to three pages in length. Before submitting your final version, be sure to submit a draft version to the TurnItIn Checker.
I will support you with
Kizza (2014) and Lessig (2006)
Textbook
.
Information to use for paperStep 1 Select your topic and focus.docxwiddowsonerica
Information to use for paper:
Step 1: Select your topic and focus question! Read the topics from the list on page 2 of this document and choose the topic and focus that interests you. Fill out the box below.
What’s your topic and focus question?
Topic:
This Land is My Land
Focus Question:
Evaluate the impact of the Pequot War on either the Europeans or Natives.
Step 2: In 50 words or more, state why you chose the topic and focus question that you chose. It could be how the topic is of interest to you and that you have studied it previously, or it could be a subject that you want to learn more about something of which you do not have knowledge.
I opted for “This Land is My Land” topic due to my insightful interest to explore historical events related to patriotism and balancing power between Europeans and Natives in situations such as trade negotiations. The trade aspect drove me to be specific on the focus question that need to address the impacts of Pequot War to Europeans and Natives. In this case, the contact of both had profound effect where balance of power shifted overnight from natives to Europeans. My choice is therefore grounded on the enthusiasm to discover more knowledge into this particular aspect.
Step 3: List the two primary source and two secondary sources that you have chosen in the boxes below.
Source Type
Source Name
Primary Source #1
Indian Complaints about English Settlers, 1675
Primary Source #2
Edward Randolph’s Report of King Philip’s War, 1675
Secondary Source #1
Adam J. Hirsch, “The Collision of Military Cultures in Seventeenth-Century New England.”
Secondary Source #2
Michal L. Fickes, “’They Could Not Endure that Yoke’: The Capitivity of Pequot Women and Children after the War of 1637”
Step 4: In 50 words or more, describe your initial thoughts about how your sources relate to your chosen topic and focus. Make sure to provide specific examples from each of the four sources that illustrate how they will help you answer your focus question. This will help you begin to think about the form of your paper!
In primary #1 the natives who were the Indians, revolted against the Europeans led by King Phillip which meant the next combinations of Indians to threaten European dominance after
Pequot War
. Primary #2 contribute to this research since
Randolph, sent by King James II, compiled a report to account for the war involving Indians and colonies led by King Phillip.
On the other hand, secondary #1 is relevant to this paper since it analyze military conflicts in 17th-century, warfare practice in Europe; facets of native warfare of Indians and differences between European and native military prowess. Finally, Secondary #2 inputs to my compilation as it addresses how natives of New England were enslaved as it document how Europeans eventually thought of
Pequot War
as the “civilized” English versus the “savage” natives. The colonialists killed and enslaved Pequot women and children.
Topic instructions:
.
More Related Content
Similar to Discussion Please discuss, elaborate and give example on the topi
5
ANOVA: Analyzing Differences
in Multiple Groups
Learning Objectives
After reading this chapter, you should be able to:
• Describe the similarities and differences between t-tests and ANOVA.
• Explain how ANOVA can help address some of the problems and limitations associ-
ated with t-tests.
• Use ANOVA to analyze multiple group differences.
• Use post hoc tests to pinpoint group differences.
• Determine the practical importance of statistically significant findings using effect
sizes with eta-squared.
iStockphoto/Thinkstock
tan81004_05_c05_103-134.indd 103 2/22/13 4:28 PM
CHAPTER 5Section 5.1 From t-Test to ANOVA
Chapter Overview
5.1 From t-Test to ANOVA
The ANOVA Advantage
Repeated Testing and Type I Error
5.2 One-Way ANOVA
Variance Between and Within
The Statistical Hypotheses
Measuring Data Variability in the ANOVA
Calculating Sums of Squares
Interpreting the Sums of Squares
The F Ratio
The ANOVA Table
Interpreting the F Ratio
Locating Significant Differences
Determining Practical Importance
5.3 Requirements for the One-Way ANOVA
Comparing ANOVA and the Independent t
One-Way ANOVA on Excel
5.4 Another One-Way ANOVA
Chapter Summary
Introduction
During the early part of the 20th century R. A. Fisher worked at an agricultural research station in rural southern England. In his work analyzing the effect of pesticides and
fertilizers on results like crop yield, he was stymied by the limitations in Gosset’s indepen-
dent samples t-test, which allowed him to compare just two samples at a time. In the effort
to develop a more comprehensive approach, Fisher created a statistical method he called
analysis of variance, often referred to by its acronym, ANOVA, which allows for making
multiple comparisons at the same time using relatively small samples.
5.1 From t-Test to ANOVA
The process for completing an independent samples t-test in Chapter 4 illustrated a number of things. The calculated t value, for example, is a score based on a ratio, one
determined by dividing the variability between the two groups (M1 2 M2) by the vari-
ability within the two groups, which is what the standard error of the difference (SEd)
measures. So both the numerator and the denominator of the t-ratio are measures of data
variability, albeit from different sources. The difference between the means is variability
attributed primarily to the independent variable, which is the group to which individual
subjects belong. The variability in the denominator is variability for reasons that are unex-
plained—error variance in the language of statistics.
tan81004_05_c05_103-134.indd 104 2/22/13 4:28 PM
CHAPTER 5Section 5.1 From t-Test to ANOVA
In his method, ANOVA, Fisher also embraced this
pattern of comparing between-groups variance to
within-groups variance. He calculated the variance
statistics differently, as we shall see, but he followed
Gosset’s pattern of a ratio of between-groups vari-
ance compared to within.
The ANOVA .
STATISTICS : Changing the way we do: Hypothesis testing, effect size, power, ...Musfera Nara Vadia
STATISTICS : Changing the way we do: Hypothesis testing, effect size, power, confidence interval, two-tailed and one tailed test, and other misunderstood issues.
3Type your name hereType your three-letter and -number cours.docxlorainedeserre
3
Type your name here
Type your three-letter and -number course code here
The date goes here
Type instructor’s name here
Your Title Goes Here
This is an electronic template for papers written in GCU style. The purpose of the template is to help you follow the basic writing expectations for beginning your coursework at GCU. Margins are set at 1 inch for top, bottom, left, and right. The first line of each paragraph is indented a half inch (0.5"). The line spacing is double throughout the paper, even on the reference page. One space after punctuation is used at the end of a sentence. The font style used in this template is Times New Roman. The font size is 12 point. When you are ready to write, and after having read these instructions completely, you can delete these directions and start typing. The formatting should stay the same. If you have any questions, please consult with your instructor.
Citations are used to reference material from another source. When paraphrasing material from another source (such as a book, journal, website), include the author’s last name and the publication year in parentheses.When directly quoting material word-for-word from another source, use quotation marks and include the page number after the author’s last name and year.
Using citations to give credit to others whose ideas or words you have used is an essential requirement to avoid issues of plagiarism. Just as you would never steal someone else’s car, you should not steal his or her words either. To avoid potential problems, always be sure to cite your sources. Cite by referring to the author’s last name, the year of publication in parentheses at the end of the sentence, such as (George & Mallery, 2016), and page numbers if you are using word-for-word materials. For example, “The developments of the World War II years firmly established the probability sample survey as a tool for describing population characteristics, beliefs, and attitudes” (Heeringa, West, & Berglund, 2017, p. 3).
The reference list should appear at the end of a paper (see the next page). It provides the information necessary for a reader to locate and retrieve any source you cite in the body of the paper. Each source you cite in the paper must appear in your reference list; likewise, each entry in the reference list must be cited in your text. A sample reference page is included below; this page includes examples (George & Mallery, 2016; Heeringa et al., 2017; Smith et al., 2018; “USA swimming,” 2018; Yu, Johnson, Deutsch, & Varga, 2018) of how to format different reference types (e.g., books, journal articles, and a website). For additional examples, see the GCU Style Guide.
References
George, D., & Mallery, P. (2016). IBM SPSS statistics 23 step by step: A simple guide and reference. New York, NY: Routledge.
Heeringa, S. G., West, B. T., & Berglund, P. A. (2017). Applied survey data analysis (2nd ed.). New York, NY: Chapman & Hall/CRC Press.
Smith, P. D., Martin, B., Chewning, B., ...
Experimental design is a way to carefully plan experiments in advance so that results are both objective and valid. Ideally, an experimental design should:
• Describe how participants are allocated to experimental groups. A common method is completely randomized design, where participants are assigned to groups at random. A second method is randomized block design, where participants are divided into homogeneous blocks (for example, age groups) before being randomly assigned to groups.
• Minimize or eliminate confounding variables, which can offer alternative explanations for the experimental results.
• Allows making inferences about the relationship between independent variables and dependent variables.
• Reduce variability, to make it easier to find differences in treatment outcomes.
Types of Experimental Design
1. Between Subjects Design.
2. Completely Randomized Design.
3. Factorial Design.
4. Matched-Pairs Design.
5. Observational Study
• Longitudinal Research
• Cross Sectional Research
6. Pretest-Posttest Design.
7. Quasi-Experimental Design.
8. Randomized Block Design.
9. Randomized Controlled Trial
10. Within subjects Design.
ANOVA Interpretation Set 1 Study this scenario and ANOVA.docxfestockton
ANOVA Interpretation Set 1
Study this scenario and ANOVA table, then answer the questions in the assignment instructions.
A researcher wants to compare the efficacy of three different techniques for memorizing
information. They are repetition, imagery, and mnemonics. The researcher randomly assigns
participants to one of the techniques. Each group is instructed in their assigned memory
technique and given a document to memorize within a set time period. Later, a test about the
document is given to all participants. The scores are collected and analyzed using a one-way
ANOVA. Here is the ANOVA table with the results:
Source SS df MS F p
Between 114.3111 2 57.1556 19.74 <.0001
Within 121.6 42 2.8952
Total 235.9111 44
9/10/2019 Print
https://content.ashford.edu/print/AUPSY325.16.1?sections=ch6,ch6sec1,ch6sec2,ch6sec3,ch6sec4,ch6sec5,ch6sec6,ch6sec7,ch6sec8,ch6summary,ch7,ch7sec1,ch7s… 1/76
Chapter Learning Objectives
After reading this chapter, you should be able to do the following:
1. Explain why it is a mistake to analyze the differences between more than two groups with
multiple t tests.
2. Relate sum of squares to other measures of data variability.
3. Compare and contrast t test with analysis of variance (ANOVA).
4. Demonstrate how to determine significant differences among groups in an ANOVA with more
than two groups.
5. Explain the use of eta squared in ANOVA.
6Analysis of Variance
Peter Ginter/Science Faction/Corbis
9/10/2019 Print
https://content.ashford.edu/print/AUPSY325.16.1?sections=ch6,ch6sec1,ch6sec2,ch6sec3,ch6sec4,ch6sec5,ch6sec6,ch6sec7,ch6sec8,ch6summary,ch7,ch7sec1,ch7s… 2/76
Introduction
From one point of view at least, R. A. Fisher was present at the creation of modern statistical analysis. During
the early part of the 20th century, Fisher worked at an agricultural research station in rural southern England.
Analyzing the effect of pesticides and fertilizers on crop yields, he was stymied by independent t tests that
allowed him to compare only two samples at a time. In the effort to accommodate more comparisons, Fisher
created analysis of variance (ANOVA).
Like William Gosset, Fisher felt that his work was important enough to publish, and like Gosset, he met
opposition. Fisher’s came in the form of a fellow statistician, Karl Pearson. Pearson founded the first department
of statistical analysis in the world at University College, London. He also began publication of what is—for
statisticians at least—perhaps the most influential journal in the field, Biometrika. The crux of the initial conflict
between Fisher and Pearson was the latter’s commitment to making one comparison at a time, with the largest
groups possible.
When Fisher submitted his work to Pearson’s journal, suggesting that samples can be small and many
comparisons can be made in the same analysis, Pearson rejected the manuscript. So began a long and
increasingly acrimonious relationship between two men w ...
This presentation deals with the basics of design of experiments and discusses all the three basic statistical designs i.e. CRD, RBD and LSD. Further it explains the guidelines for developing experimental research.
7
Repeated Measures Designs
for Interval Data
Learning Objectives
After reading this chapter, you should be able to:
• Explain the advantages and drawbacks of using data from non-independent groups.
• Complete a paired-samples t-test.
• Complete a within-subjects F.
• Describe “power” as it relates to statistical testing.
iStockphoto/Thinkstock
tan81004_07_c07_163-192.indd 163 2/22/13 3:41 PM
CHAPTER 7Introduction
Chapter Outline
7.1 Dependent Groups Designs
Reconsidering the t and F ratios
An Example
A Matched Pairs Example
Comparing the Paired-Samples t-Test to the Independent Samples t-Test
The Power of the Dependent Groups Test
The Dependent Groups t-Test on Excel
The Alternate Approaches to Dependent t-Tests
7.2 The Within-Subjects F
Managing Error Variance in the Within-Subjects F
A Within-Subjects F Example
Calculating the Within-Subjects F
Understanding the Result
Comparing the Within-Subjects F and the One-Way ANOVA
Another Within-Subjects F Example
A Within-Subjects F in Excel
Chapter Summary
Introduction
Some of the most critical questions in management relate to change over time. For exam-ple, managers are deeply interested in assessing sales growth, shifts in shopping trends,
improvements in employee attitudes, increases in employee performance, and decreases in
absenteeism or turnover. They are also often keen to find out the influence of various
managerial decisions and business strategies on these and many other change-oriented
outcomes. However, none of the analyses completed to this point address these change-
related questions, because these analyses do not accommodate repeated measures of the
same variables within the same group of subjects over time. For instance, the t-tests and
ANOVAs discussed so far compared independent groups, groups that have completely
separate subjects. Each subject was only measured once on each variable of interest. The
same group of subjects was not measured repeatedly on the same variables to assess
change over time.
Another important issue is that independent samples t-tests and ANOVAs assume that
the groups being compared are equivalent on most aspects to begin with, except for the
independent (grouping or treatment) variable being investigated. When groups are large
and individuals are randomly selected, this is usually a reasonable assumption, because
any differences between groups tend to be relatively unimportant. The logic behind ran-
dom selection is that when groups are randomly drawn from the same population they
will differ only by chance—the larger the random sample, the lower the probability of
a substantial pre-existing difference. However, when groups are relatively small it can
be difficult to determine whether a difference in the measures of the dependent variable
occurred because the independent variable had a different impact on the different groups
or because there were differences between the groups to begin with.
tan81004_07.
WEEK 6 – EXERCISES Enter your answers in the spaces pr.docxwendolynhalbert
WEEK 6 – EXERCISES
Enter your answers in the spaces provided. Save the file using your last name as the beginning of the file name (e.g., ruf_week6_exercises) and submit via “Assignments.” When appropriate,
show your work
. You can do the work by hand, scan/take a digital picture, and attach that file with your work.
1
.
A psychotherapist studied whether his clients self-disclosed more while sitting in an easy chair or lying down on a couch. All clients had previously agreed to allow the sessions to be videotaped for research purposes. The therapist randomly assigned 10 clients to each condition. The third session for each client was videotaped and an independent observer counted the clients’ disclosures. The therapist reported that “clients made more disclosures when sitting in easy chairs (
M
= 18.20) than when lying down on a couch (
M
= 14.31),
t
(18) = 2.84,
p
< .05, two-tailed.” Explain these results to a person who understands the
t
test for a single sample but knows nothing about the
t
test for independent means.
2.
A researcher compared the adjustment of adolescents who had been raised in homes that were either very structured or unstructured. Thirty adolescents from each type of family completed an adjustment inventory. The results are reported in the table below. Explain these results to a person who understands the
t
test for a single sample but knows nothing about the
t
test for independent means.
Means on Four Adjustment Scales for
Adolescents from Structured versus Unstructured Homes
Scale
Structured Homes
Unstructured Homes
t
Social Maturity
106.82
113.94
–1.07
School Adjustment
116.31
107.22
2.03*
Identity Development
89.48
94.32
1.93*
Intimacy Development
102.25
104.33
.32
______________________
*
p
< .05
3.
Do men with higher levels of a particular hormone show higher levels of assertiveness? Levels of this hormone were tested in 100 men. The top 10 and the bottom 10 were selected for the study. All participants took part in a laboratory simulation in which they were asked to role-play a person picking his car up from a mechanic’s shop. The simulation was videotaped and later judged by independent raters on each of four types of assertive statements made by the participant. The results are shown in the table below. Explain these results to a person who fully understands the
t
test for a single sample but knows nothing about the
t
test for independent means.
Mean Number of Assertive Statements
Type of Assertive Statement
Group
1
2
3
4
Men with High Levels
2.14
1.16
3.83
0.14
Men with Low Levels
1.21
1.32
2.33
0.38
t
3.81**
0.89
2.03*
0.58
______________________
*
p
< .05;
**
p
< 0.1
4.
A manager of a small store wanted to discourage shoplifters by putting signs around the store saying “Shoplifting is a crime!” However, he wanted to make sure this would not result in customers buying less. To test this, he displayed the signs every other W.
Using the data in the file named Ch. 11 Data Set 2, test the resea.docxdaniahendric
Using the data in the file named Ch. 11 Data Set 2, test the research hypothesis at the .05 level of significance that boys raise their hands in class more often than girls. Do this practice problem by hand using a calculator. What is your conclusion regarding the research hypothesis? Remember to first decide whether this is a one- or two-tailed test.
Using the same data set (Ch. 11 Data Set 2), test the research hypothesis at the .01 level of significance that there is a difference between boys and girls in the number of times they raise their hands in class. Do this practice problem by hand using a calculator. What is your conclusion regarding the research hypothesis? You used the same data for this problem as for Question 1, but you have a different hypothesis (one is directional and the other is nondirectional). How do the results differ and why?
Practice the following problems by hand just to see if you can get the numbers right. Using the following information, calculate the
t
test statistic.
Using the results you got from Question 3 and a level of significance at .05, what are the two-tailed critical values associated with each? Would the null hypothesis be rejected?
Using the data in the file named Ch. 11 Data Set 3, test the null hypothesis that urban and rural residents both have the same attitude toward gun control. Use IBM
®
SPSS
®
software to complete the analysis for this problem.
A public health researcher tested the hypothesis that providing new car buyers with child safety seats will also act as an incentive for parents to take other measures to protect their children (such as driving more safely, child-proofing the home, and so on). Dr. L counted all the occurrences of safe behaviors in the cars and homes of the parents who accepted the seats versus those who did not. The findings: a significant difference at the .013 level. Another researcher did exactly the same study; everything was the same—same type of sample, same outcome measures, same car seats, and so on. Dr. R’s results were marginally significant (recall Ch. 9) at the .051 level. Which result do you trust more and why?
In the following examples, indicate whether you would perform a
t
test of independent means or dependent means.
Two groups were exposed to different treatment levels for ankle sprains. Which treatment was most effective?
A researcher in nursing wanted to know if the recovery of patients was quicker when some received additional in-home care whereas when others received the standard amount.
A group of adolescent boys was offered interpersonal skills counseling and then tested in September and May to see if there was any impact on family harmony.
One group of adult men was given instructions in reducing their high blood pressure whereas another was not given any instructions.
One group of men was provided access to an exercise program and tested two times over a 6-month period for heart health.
For Ch. 12 Data Set 3, comput.
Information Technology, Computer Networks and CyberspaceKizza (201.docxwiddowsonerica
Information Technology, Computer Networks and Cyberspace
Kizza (2014) largely treated cyber-attacks as “cybercrime,” conveying the assumption that these attacks are by definition illegitimate and criminal. However, he occasionally seems to recognize that there might be motives that are more ambiguous and may have a social, moral, or political basis. Lessig (2006) also referred to Internet hacker ethic as rebellious and libertarian and not always criminal in a moral sense. One example cited by Kizza (2014) is the Seattle WTO protest and mass computer attacks that are “increasingly being used to avenge what the attackers consider to be injustices” (p. 97). Another example we discussed was the Stratfor attack. Also, recently we see many examples of cyber security invasions by governments and corporations.
Search the Internet to find an example of a hacking activity or situation that represents a morally, ethically, or criminally ambiguous situation but is different from any examples you used in other assignments for this course.
Write a critical essay that addresses the following items:
Cite and briefly describe your example.
Apply what you have learned from the course to this point to identify arguments both in support of and critical of the behavior of the attackers.
Describe and explain the relationships among morality, ethics, law, and crime as they intersect in the case example that you have found.
Cite Winner’s “Do Artifacts Have Politics?” to identify and explain the political, moral, and ethical choices and consequences that may be embedded in technical choices and artifacts described in your example.
Discuss and cite the article “Do Artifacts Have Politics?”, the source of your example, and at least one additional credible or scholarly source other than the course textbook to support your analysis and positions. You may cite the course textbooks as well if you wish. Use standards and APA style guidelines, citing references as appropriate. Your paper should be two to three pages in length. Before submitting your final version, be sure to submit a draft version to the TurnItIn Checker.
I will support you with
Kizza (2014) and Lessig (2006)
Textbook
.
Information to use for paperStep 1 Select your topic and focus.docxwiddowsonerica
Information to use for paper:
Step 1: Select your topic and focus question! Read the topics from the list on page 2 of this document and choose the topic and focus that interests you. Fill out the box below.
What’s your topic and focus question?
Topic:
This Land is My Land
Focus Question:
Evaluate the impact of the Pequot War on either the Europeans or Natives.
Step 2: In 50 words or more, state why you chose the topic and focus question that you chose. It could be how the topic is of interest to you and that you have studied it previously, or it could be a subject that you want to learn more about something of which you do not have knowledge.
I opted for “This Land is My Land” topic due to my insightful interest to explore historical events related to patriotism and balancing power between Europeans and Natives in situations such as trade negotiations. The trade aspect drove me to be specific on the focus question that need to address the impacts of Pequot War to Europeans and Natives. In this case, the contact of both had profound effect where balance of power shifted overnight from natives to Europeans. My choice is therefore grounded on the enthusiasm to discover more knowledge into this particular aspect.
Step 3: List the two primary source and two secondary sources that you have chosen in the boxes below.
Source Type
Source Name
Primary Source #1
Indian Complaints about English Settlers, 1675
Primary Source #2
Edward Randolph’s Report of King Philip’s War, 1675
Secondary Source #1
Adam J. Hirsch, “The Collision of Military Cultures in Seventeenth-Century New England.”
Secondary Source #2
Michal L. Fickes, “’They Could Not Endure that Yoke’: The Capitivity of Pequot Women and Children after the War of 1637”
Step 4: In 50 words or more, describe your initial thoughts about how your sources relate to your chosen topic and focus. Make sure to provide specific examples from each of the four sources that illustrate how they will help you answer your focus question. This will help you begin to think about the form of your paper!
In primary #1 the natives who were the Indians, revolted against the Europeans led by King Phillip which meant the next combinations of Indians to threaten European dominance after
Pequot War
. Primary #2 contribute to this research since
Randolph, sent by King James II, compiled a report to account for the war involving Indians and colonies led by King Phillip.
On the other hand, secondary #1 is relevant to this paper since it analyze military conflicts in 17th-century, warfare practice in Europe; facets of native warfare of Indians and differences between European and native military prowess. Finally, Secondary #2 inputs to my compilation as it addresses how natives of New England were enslaved as it document how Europeans eventually thought of
Pequot War
as the “civilized” English versus the “savage” natives. The colonialists killed and enslaved Pequot women and children.
Topic instructions:
.
Information and Knowledge Needs of Nurses in the 21st CenturyIn th.docxwiddowsonerica
Information and Knowledge Needs of Nurses in the 21st Century
In the year 2025, nursing practice enabled by technology has created a professional culture of reflection, critical inquiry, and inter-professional collaboration. Examples of current technology that could change in the future include:
Nurses use technology at the point of care in all clinical settings (e.g., primary care, acute care, community, and long-term care) to inform their clinical decisions and effect the best possible outcomes for their clients.
Information is gathered and retrieved via human–technology biometric interfaces including voice, visual, sensory, gustatory, and auditory interfaces, continuously monitoring physiologic parameters for potentially harmful imbalances.
Longitudinal records are maintained for all citizens from their initial prenatal assessment to death; all life-long records are aggregated into the knowledge bases of expert systems. These systems are providing the basis of the artificial intelligence being embedded in emerging technologies.
Smart technologies and invisible computing are ubiquitous in all sectors where care is delivered.
Clients and families are empowered to review and contribute actively to their record of health and wellness.
Invasive diagnostic techniques are obsolete, nanotechnology therapeutics are the norm, and robotics supplement or replace much of the traditional work of all health professions.
Nurses provide expertise to citizens to help them effectively manage their health and wellness life plans, and navigate access to appropriate information and services.
Scholarly writing includes clarity and flow of thoughts. You may want to include headings in your paper to ensure you are meeting all elements of the rubric.
Rubric
APA Paper
APA Paper
Criteria
Ratings
Pts
Choose at least two of the technology enabled changes listed above and discuss clinical practice workflow impacts comparing current technology and the new technology.
10
pts
Based on the two technology enabled changes you chose, reflect on these advances and looking to the future, how do you envision nursing care will change?
5
pts
Find a current scholarly article that discusses nursing job satisfaction and the factors that influence such. Discuss the potential impacts on job satisfaction of the technologies outlined in the case scenario.
10
pts
Identify at least 3 ways that nurses could use the new technologies to partner with patients to manage their health.
10
pts
APA and Grammar- Title page/Running head- Body of paper/Header/Title: appropriate flow, syntax and context/3-5 pages- In-text citations- References-
10
pts
Total Points:
45
.
Infectious and Noninfectious Prevention and Control TechniquesIn.docxwiddowsonerica
Infectious and Noninfectious Prevention and Control Techniques
In this assignment, you will review, compare, and contrast the prevention and control techniques for an infectious and a noninfectious disease.
Visit the diseases and conditions index of the Centers for Disease Control and Prevention (CDC) Web site. Select an infectious and a noninfectious disease and complete a one-page review of the prevention and control techniques utilized for each of the two diseases you have selected to compare and contrast. Outline the following for the selected diseases:
Signs, symptoms, and transmission of the diseaseDiagnosis and treatment for the diseaseHow the disease is prevented and controlledThe public health impact of the disease
Submit your focus area review in a Word document to the W2: Assignment 2 Dropbox by Saturday, July 11, 2015. On a separate page, cite all sources using the APA format.
Name your document: SU_HSC4010_W2_A2_LastName_FirstInitial.doc.
Assignment 2 Grading Criteria
Maximum Points
Included a 1-page synopsis of a selected infectious and noninfectious disease and outlined the public health importance of the selected diseases.
15
Described signs, symptoms, and transmission of the selected diseases.
10
Described prevention, control, and treatment methods for the selected diseases.
10
Justified ideas and responses by using appropriate examples and references from texts, Web sites, and other references and cited the sources in the APA format.
10
Used correct spelling, grammar, and professional vocabulary.
5
Total:
50
.
Information Management and Software Development Please respo.docxwiddowsonerica
"
Information Management and Software Development
"
Please respond to the following:
According to your point of view, which of the defined criteria of measurement is considered as most important for evaluation and monitoring of information systems? How does computer-aided software engineering (CASE) aid in analyzing and evaluating characteristics of a project?
Discuss how
IT is impacting the business processes and applications of e-Business? How does the system development life cycle (SDLC) assist in accomplishing essential tasks and objectives of an organization?
.
Information Security Please respond to the followingIdentify t.docxwiddowsonerica
"Information Security" Please respond to the following:
Identify two events you believe are important to progression of Information Security. Research the events and develop a PowerPoint presentation to provide the class. You should find research from Online to support the events you believe are significant to the development of Information Security.
.
Information PaperThe Chief of Staff is preparing preliminary studi.docxwiddowsonerica
Information Paper
The Chief of Staff is preparing preliminary studies for the DHS secretary. He has directed you to prepare an information paper that identifies possible mission areas or responsibility overlaps within the Department of Homeland Security (DHS) operating agencies and to provide recommendations for possible consolidation. You are required to conduct library research to locate scholarly studies on the topic.
.
INF 410 week 4 assign onAssume you are the project manager for a s.docxwiddowsonerica
INF 410 week 4 assign on
Assume you are the project manager for a software company and your team is in charge of delivering an update to tax preparation software for the next tax season. The project team has ten software developers working together, but located across the country. You have the following information about the project, and need to provide a status to the sponsors.
New items you learned this month about the project:
There is a three day weekend in the coming month.
Nine of the developers are on track to finish their tasks two weeks early.
One software developer took the recently announced “early retirement” package, and is leaving in one week. He is confident he will be able to finish his piece of the work before he leaves.
It is hurricane season in Florida, where three of the developers are located.
The developer who is retiring is the only person who knows how to get approval for the final product from the IRS.
The project costs are under budget and there has been no need for overtime.
Write a one page status report (excluding the title page) to the sponsors providing a clear indication of status of the project (suggestion below into their assignment.
Green
Yellow
Red
The sponsors also like to only know the two most important risks, and what could be done to mitigate the risk if it happens. The conclusion of the status report should include a paragraph that describes any requests for additional resources, if they are necessary.
.
Information Technology in Public HealthPublic health surveillance .docxwiddowsonerica
Information Technology in Public Health
Public health surveillance and prevention efforts in the U.S. reflect the involvement of numerous health care providers and organizations. Government agencies such as the CDC often function as a hub for the collection, coordination, and communication of public health-related data.
Common applications of public health IT to promote and improve the health of populations include (but are not limited to) biosurveillance, disease registries, and immunization tracking. The timely and appropriate implementation of these applications can have a significant impact on the ability of providers to deliver health care to the their patients, which, in turn, can have a profound effect on a community's overall state of health and wellness. Consider the dire impact a SARS outbreak might have, for example, if patient diagnoses were not reported and shared regionally, nationally, and internationally.
Prepare for this Application as follows:
Conduct a literature review of a specific application of health information technology in public health and locate an example in which the application was used to address a public health-related concern. (You might wish to begin by researching a particular surveillance program, disease outbreak, disease registry, prevention program, or other public health-related event, and then identify the technology involved in this activity.)
Find out how this application has impacted the public's health. Learn also about any ways in which health care providers and organizations were involved in the use of this information technology.
Then write a 2- to 3-page paper that addresses the following:
Describe the specific application you selected and the ways in which it addresses a public health-related concern. Summarize the potential benefits to the public, as well as any possible drawbacks, arising from the use of this application.
Explain the role of hospitals and other health care organizations in contributing to or making use of this public health-related IT application.
Discuss other implications of this application for health care providers and organizations. How has it impacted (or might it impact) clinical or administrative practices in hospitals or other health care institutions? Provide a rationale for your response.
.
Information System and Enterprise SystemsIdentify the key factors.docxwiddowsonerica
Information System and Enterprise Systems"
Identify the key factors that companies should consider when deciding whether to buy or to build their own information systems. Explain your rationale.
Select one (1) type of the four (4) enterprise systems (i.e., supply chain management, customer relationship management, knowledge management systems, and enterprise resource planning), and speculate on the main issues that companies may encounter when implementing the type of enterprise system that you have selected. Next, suggest two or three (2 or 3) general techniques that companies could use in order to mitigate the issues in question.
50 + word answer
.
Individuals react to a variety of social cues to construct their per.docxwiddowsonerica
Individuals react to a variety of social cues to construct their personal realities. These cues include social categorization. Are social categories a valid basis for the construction of personal reality? Why or why not? How might the misinterpretation of social categorization cues lead to errant self-categorization and falsely shape the personal reality of the individual?
Note: Just need 150 - 200 words explaining the personal relaity of an individual
.
Information Technology and Decision MakingPatient safety and care .docxwiddowsonerica
Information Technology and Decision Making
Patient safety and care is reliant upon the quality of nurses' daily decision making. Nurses are constantly tending to the needs of multiple patients by providing health care information, administering drug treatments, managing patient family concerns, as well as informing other nurses and staff on the patient's condition.
Review your reading from this week and then address the following:
• Explain how well your current work setting supports you and your colleagues in decision making.
• Share your thoughts on two ways that electronic or computerized information technology could help you and your nursing colleagues improve decision making in your work setting. You can include decision support applications or not, as you wish.
Support your ideas or those of others with references from the professional nursing literature.
Cite the references used in APA style format.
.
Individuals use social movements when they feel a social institution.docxwiddowsonerica
Individuals use social movements when they feel a social institution is unjust. Current theoretical perspectives on social movements provide an awareness of the nature of social movements and their role in defining human behavior. In terms of human services, it is important to be able to recognize and assess the costs and benefits to individuals who participate in social movement activities (for example, healthcare reform, charity organizations, and disability movements) as well as the effects of social movements on human behavior.
Tasks:
Using the vignette provided in this module, your textbook, the Argosy University online library resources, and the Internet, conduct research to support your response.
In a 3- to 4-page essay, address the following:
Provide a summary of the vignette's key points as related to the social movements it represents. Identify and describe the concepts from this module that can be applied to the vignette to describe human behavior (i.e., cultural framing).
Identify and discuss the effects of the identified social movement on the individual described in the vignette.
Provide a summary of service methods or options that could be used to support this person. You can use examples you have identified in your own community as well.
Submission Details:
By
Wednesday, October 14, 2015
, prepare a 3- to 4-page essay. Your response should rely upon at least three sources from professional literature. This may include the Argosy University online library resources, relevant textbooks, peer-reviewed journal articles, and websites created by professional organizations, agencies, or institutions (.edu, .org, or .gov). Write in a clear, concise, and organized manner; demonstrate ethical scholarship in accurate representation and attribution of sources (i.e., APA format); and use accurate spelling, grammar, and punctuation.
.
Industry Averages and Financial Ratios PaperCalculate the 14 rat.docxwiddowsonerica
Industry Averages and Financial Ratios Paper
Calculate
the 14 ratios (show your calculations) for the company using the two most recent annual financial statements found on the financial information website you used earlier. Be careful not to use quarterly information, and include ratios for both years.
.
Information is provided for you below that will help you construct a.docxwiddowsonerica
Information is provided for you below that will help you construct an answer for your interpretation of the figures on page 2-3.
Background:
Dendrobates pumilo
is a poison dart frog with a distributional range from Nicaragua, Costa Rica, and Panama. The bright colors from the skin of these frogs are considered to be a warning sign of toxicity to predators. Although
D. pumilo
is one species, there are 10 color polymorphic populations within this range. The warning signs as displayed on
D. pumilo
are favored by natural selection in terms of predator avoidance.
Purpose of the study (what scientists wanted to understand):
1) How visual conspicuousness and toxicity is correlated.
2) Differences between visual backgrounds and predator visual systems on detection of
D.
pumilo
.
Glossary:
Dorsal
: upper side or back
Ventral
: underside
1
Visual background
: Poison dart frogs live on different natural substrates; live dead plant parts,
so this would bring about different visual backgrounds.
Visual systems
: In this study they are referring to how predators “view” D. pumilo. Remember, not all predators will see their prey the same way.
Color polymorphisms
:
alternative phenotypes, and in this case alternative phenotypes of frog skin color.
NAME: ______________________
Figure 1.
Sampled populations of
Dendrobates pumilio
in the Bocas del Toro Archipelago, Panama. From Isla Bastimentos, two
D. pumilio
color morphs were collected (green and orange), as well as four individuals of the closely related but nontoxic control species
Allobates talamancae
.
Bio 112 Olave Spring 2016
2
Figure 2.
Toxicity scores and coloration brightness in
Dendrobates pumilio
.
A
, Open bars indicate
toxicity scores with standard errors. Different letters above the bars indicate statistically significant differences; numbers are population numbers referred to in
C
and in subsequent figures.
Allobates talamancae
(a closely related Dendrobatid frog) and saline solution served as toxicity measure controls.
B
, Gray bars indicate the overall brightness of dorsal coloration (total reflectance flux, S
R
, in arbitrary units) for the 11 frog taxa.
C
, The inset gives the correlation between toxicity and dorsal brightness for the 10
D. pumilio
populations. Numbers refer to the population labels in
A
.
Bio 112 Olave Spring 2016
3
NAME: ______________________
Figure 3.
Illustration of the perceptual differences between potential predator taxa in species-specific color space. Plots show the estimated brightness and color contrast generated by the dorsal coloration of individuals from three
Dendrobates pumilio
populations (Solarte [orange], Aguacate [blue], and Pastores [green]), viewed against a
Heliconia
background, in the visual systems of birds, crabs, and diurnal snakes. Each dot represents reflectance spectra measured from an individual frog. The relative conspicuousness of the different frog morphs differs by viewer. For exa.
Individuals who are frequent voters share common characteristics, re.docxwiddowsonerica
Individuals who are frequent voters share common characteristics, regardless of the nation in which they may reside. Identify and describe at least three of these common characteristics. Explain why each characteristic is influential in motivating a person to vote.
Your response should be at least 200 words in length. You are required to use at least your textbook as source material for your response. All sources used, including the textbook, must be referenced; paraphrased and quoted material must have accompanying citations.
.
Infectious diseases come with extremely tough challenges to mitigate.docxwiddowsonerica
Infectious diseases come with extremely tough challenges to mitigate them and then finally get them under control. Bringing any such infectious disease under control involves a lot of decisions and cooperation between various branches of government and the health services.
Using the Internet, choose an infectious disease that was prevalent in the United States and had lasting consequences or select a disease (
Yellow Fever is the chosen disease)
Based on your research and understanding, create a 3 to 4-page report in a Microsoft Word document that includes the following:
A brief description of your chosen infectious disease along with your reasons for choosing the disease.
Information on the work conducted by government departments to mitigate the impact of your chosen infectious disease.
Investigations, research studies, and other surveillance data analyses regarding your chosen infectious disease.
Instances of the emergence and re-emergence of your chosen disease.
A brief summary of the government's findings and investigations about your chosen disease.
Past, current, and ongoing research pertaining to your chosen disease.
Support your writing with relevant facts or figures and indicate your current knowledge of the infectious disease.
Support your responses with examples.
Cite any sources in APA format.
.
Individual Reflection in which you describe the plan that you would .docxwiddowsonerica
Individual Reflection in which you describe the plan that you would like to implement, including:
Your list of the stakeholders whose approval/support will be necessary for your success. Take the time to ensure your list of stakeholders is as complete as possible.
Considering the various stakeholder perspectives, identify what is important to each of the stakeholders on your list.
Your assessment of the risks to your plan and impact of stakeholders with different goals and interests. Explain how you might manage the impact and address any risks that emerge.
The specific details of the strategy, or strategies, you will use to align stakeholder interests in your plan, with your rationale as to why you think these will be successful based on your analysis of stakeholder perceptions and interests. (
Hint:
Strategies can include team-building exercises, conflict resolution methods, partnering, change management, best practices, etc. You will need to provide details to support your selected strategy or strategies.)
.
Individual ProjectGroups in the WorkplaceSun, 10816.docxwiddowsonerica
Individual Project
Groups in the Workplace
Sun, 1/08/16
3-6 paragraphs
Assignment Objectives
Explain the effect of diversity on group performance
Assignment Description
You are preparing to write an article for a professional magazine regarding the value of diversity in teams. Diversity can include any of the following:
Gender
Age
National origin
Race
Sexual orientation
Marital status
Parental status
Height
Weight
Profession
Education
Geographic location of residence
Financial status
Artistic endeavors
Research the subject using the library and Internet, and prepare an outline or a draft of the article you are to write.
.
Individual ProjectRecruitment Strategies and Tools Mon, 83.docxwiddowsonerica
Individual Project
Recruitment Strategies and Tools
Mon
, 8/3/15
Numeric
3-4 paragraphs
Assignment Objectives
Describe validity and reliability with regard to personnel selection tools
Assignment Details
Complete the following for this assignment:
Research different types of prescreening tests, and construct a report for the president of the company.
Discuss the validity and reliability of each test.
Recommend which tests should be used in your company's selection process.
Use the library, Internet, and other resources to research your response.
.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Discussion Please discuss, elaborate and give example on the topi
1. Discussion: Please discuss, elaborate and give example on the
topic below. Please use the Module/reference I provided.
Professor will not allow outside sources.
Author: Jackson, S. L. (2017). Statistics plain and simple, (4th
ed.). Boston, MA: Cengage Learning
Topic:
Using the sample provided, address the following:
· How would you interpret the results of the two-way ANOVA?
· What does the p value tell you?
· The results mention df. What does that term represent? How is
it calculated? Write a plainly stated sentence that explains what
these results tell you about the groups.
Sample
Sum of Squares df Mean Square F Sig.
SCORES Between Groups 351.520 4 87.880 9.085 .000
Within Groups 435.300 45 9.673
Total 7 86.820 49
Module/reference
Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent
Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control
Group
Comparing a Placebo Group to the Control and Experimental
Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It
Does
2. Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance
(ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical
analyses used with designs involving more than two groups. The
inferential statistics discussed in this chapter differ from those
presented in the previous two chapters. In Chapter 5, single
samples were being compared to populations (z test and t test),
and in Chapter 6, two independent or correlated samples were
being compared. In this chapter, the statistics are designed to
test differences between more than two equivalent groups of
subjects.
Several factors influence which statistic should be used to
analyze the data collected. For example, the type of data
collected and the number of groups being compared must be
considered. Moreover, the statistic used to analyze the data will
vary depending on whether the study involves a between-
subjects design (designs in which different subjects are used in
each group) or a correlated-groups design. (Remember,
correlated-groups designs are of two types: within-subjects
designs, in which the same subjects are used repeatedly in each
group, and matched-subjects designs, in which different
subjects are matched between conditions on variables that the
researcher believes are relevant to the study.)
3. We will look at the typical inferential statistics used to analyze
interval-ratio data for between-subjects designs. In Module
13 we discuss the advantages and rationale for studying more
than two groups; in Module 14 we discuss the statistics
appropriate for use with between-subjects designs involving
more than two groups.
MODULE 13
Comparing More Than Two Groups
Learning Objectives
•Explain what additional information can be gained by using
designs with more than two levels of an independent variable.
•Explain and be able to use the Bonferroni adjustment.
•Identify what a one-way between-subjects ANOVA is and what
it does.
•Describe what between-groups variance is.
•Describe what within-groups variance is.
•Understand conceptually how an F-ratio is derived.
The experiments described so far have involved manipulating
one independent variable with only two levels—either a control
group and an experimental group or two experimental groups. In
this module, we discuss experimental designs involving one
independent variable with more than two levels. Examining
more levels of an independent variable allows us to address
more complicated and interesting questions. Often, experiments
begin as two-group designs and then develop into more complex
designs as the questions asked become more elaborate and
sophisticated.
Using Designs with three or More Levels of an Independent
Variable
Researchers may decide to use a design with more than two
levels of an independent variable for several reasons. First, it
allows them to compare multiple treatments. Second, it allows
them to compare multiple treatments to no treatment (the
control group). Third, more complex designs allow researchers
4. to compare a placebo group to control and experimental groups
(Mitchell & Jolley, 2001).
Comparing More than Two Kinds of Treatment in One Study
To illustrate this advantage of more complex experimental
designs, imagine that we want to compare the effects of various
types of rehearsal on memory. We have participants study a list
of 10 words using either rote rehearsal (repetition) or some
form of elaborative rehearsal. In addition, we specify the type
of elaborative rehearsal to be used in the different experimental
groups. Group 1 (the control group) uses rote rehearsal, Group 2
uses an imagery mnemonic technique, and Group 3 uses a story
mnemonic device. You may be wondering why we do not simply
conduct three studies or comparisons. Why don't we compare
Group 1 to Group 2, Group 2 to Group 3, and Group 1 to Group
3 in three different experiments? There are several reasons why
this is not recommended.
You may remember from Module 11 that a t test is used to
compare performance between two groups. If we do three
experiments, we need to use three t tests to determine any
differences. The problem is that using multiple tests inflates the
Type I error rate. Remember, a Type I error means that we
reject the null hypothesis when we should have failed to reject
it; that is, we claim that the independent variable has an effect
when it does not. For most statistical tests, we use the .05 alpha
level, meaning that we are willing to accept a 5% risk of making
a Type I error. Although the chance of making a Type I error on
one t test is .05, the overall chance of making a Type I error
increases as more tests are conducted.
Imagine that we conducted three t tests or comparisons among
the three groups in the memory experiment. The probability of a
Type I error on any single comparison is .05. The probability of
a Type I error on at least one of the three tests, however, is
considerably greater. To determine the chance of a Type I error
when making multiple comparisons, we use the formula 1 − (1 −
α)c, where c equals the number of comparisons performed.
Using this formula for the present example, we get
5. 1 − (1 − .05)3 = 1 − (.95 )3 = 1 − .86 = .14
Thus, the probability of a Type I error on at least one of the
three tests is .14, or 14%.
Bonferroni adjustment A means of setting a more stringent
alpha level in order to minimize Type | errors.
One way of counteracting the increased chance of a Type I error
is to use a more stringent alpha level. The Bonferroni
adjustment, in which the desired alpha level is divided by the
number of tests or comparisons, is typically used to accomplish
this. For example, if we were using the t test to make the three
comparisons just described, we would divide .05 by 3 and get
.017. By not accepting the result as significant unless the alpha
level is .017 or less, we minimize the chance of a Type I error
when making multiple comparisons. We know from discussions
in previous modules, however, that although using a more
stringent alpha level decreases the chance of a Type I error, it
increases the chance of a Type II error (failing to reject the null
hypothesis when it should have been rejected—missing an effect
of an independent variable). Thus, the Bonferroni adjustment is
not the best method of handling the problem. A better method is
to use a single statistical test that compares all groups rather
than using multiple comparisons and statistical tests. Luckily
for us, there is a statistical technique that will do this—the
analysis of variance (ANOVA), which will be discussed shortly.
Another advantage of comparing more than two kinds of
treatment in one experiment is that it reduces both the number
of experiments conducted and the number of subjects needed.
Once again, refer back to the three-group memory experiment.
If we do one comparison with three groups, we can conduct only
one experiment, and we need subjects for only three groups. If,
however, we conduct three comparisons, each with two groups,
we need to perform three experiments, and we need participants
for six groups or conditions.
Comparing Two or More Kinds of Treatment with a Control
Group
Using more than two groups in an experiment also allows
6. researchers to determine whether each treatment is more or less
effective than no treatment (the control group). To illustrate
this, imagine that we are interested in the effects of aerobic
exercise on anxiety. We hypothesize that the more aerobic
activity engaged in, the more anxiety will be reduced. We use a
control group that does not engage in any aerobic activity and a
high aerobic activity group that engages in 50 minutes per day
of aerobic activity—a simple two-group design. Assume,
however, that when using this design, we find that both those in
the control group and those in the experimental group have high
levels of anxiety at the end of the study—not what we expected
to find. How could a design with more than two groups provide
more information? Suppose we add another group to this
study—a moderate aerobic activity group (25 minutes per
day)—and get the following results:
Control Group
High Anxiety
Moderate Aerobic Activity
Low Anxiety
High Aerobic Activity
High Anxiety
Based on these data, we have a V-shaped function. Up to a
certain point, aerobic activity reduces anxiety. However, when
the aerobic activity exceeds a certain level, anxiety increases
again. If we had conducted only the original study with two
groups, we would have missed this relationship and erroneously
concluded that there was no relationship between aerobic
activity and anxiety. Using a design with multiple groups allows
us to see more of the relationship between the variables.
Figure 13.1 illustrates the difference between the results
obtained with the three-group versus the two-group design in
this hypothetical study. It also shows the other two-group
comparisons—control compared to moderate aerobic activity,
and moderate aerobic activity compared to high aerobic activity.
This set of graphs allows you to see how two-group designs
limit our ability to see the full relationship between variables.
7. FIGURE 13.1FIGURE>Determining relationships with three-
group versus two-group designs: (a) three-group design; (b)
two-group comparison of control to high aerobic activity; (c)
two-group comparison of control to moderate aerobic activity;
(d) two-group comparison of moderate aerobic activity to high
aerobic activity
Figure 13.1a shows clearly how the three-group design allows
us to assess more fully the relationship between the variables. If
we had only conducted a two-group study, such as those
illustrated in Figure 13.1b, c, or d, we would have drawn a
much different conclusion from that drawn from the three-group
design. Comparing only the control to the high aerobic activity
group (Figure 13.1b) would have led us to conclude that aerobic
activity does not affect anxiety. Comparing only the control to
the moderate aerobic activity group (Figure 13.1c) would have
led to the conclusion that increasing aerobic activity reduces
anxiety. Comparing only the moderate aerobic activity group to
the high aerobic activity group (Figure 13.1d) would have led to
the conclusion that increasing aerobic activity increases
anxiety.
Being able to assess the relationship between the variables
means that we can determine the type of relationship that exists.
In the previous example, the variables produced a V-shaped
function. Other variables may be related in a straight linear
manner or in an alternative curvilinear manner (for example, a
J-shaped or S-shaped function). In summary, adding levels to
the independent variable allows us to determine more accurately
the type of relationship that exists between the variables.
Comparing a Placebo Group to the Control and Experimental
Groups
A final advantage of designs with more than two groups is that
they allow for the use of a placebo group—a group of subjects
who believe they are receiving treatment but in reality are not.
A placebo is an inert substance that participants believe is a
treatment. How can adding a placebo group improve an
experiment? Consider an often-cited study by Paul (1966, 1967)
8. involving children who suffered from maladaptive anxiety in
public-speaking situations. Paul used a control group, which
received no treatment; a placebo group, which received a
placebo that they were told was a potent tranquilizer; and an
experimental group, which received desensitization therapy. Of
the participants in the experimental group, 85% showed
improvement, compared with only 22% in the control condition.
If the placebo group had not been included, the difference
between the therapy and control groups (85% − 22% = 63%)
would overestimate the effectiveness of the desensitization
program. The placebo group showed 50% improvement,
meaning that the therapy's true effectiveness is much less (85%
− 50% = 35%). Thus, a placebo group allows for a more
accurate assessment of a therapy's effectiveness because, in
addition to spontaneous remission, it controls for participant
expectation effects.
DESIGNS WITH MORE THAN TWO LEVELS OF AN
INDEPENDENT VARIABLE
Advantages
Considerations
Allows comparisons of more than two types of treatment
Type of statistical analysis (e.g., multiple t tests or ANOVA)
Fewer subjects are needed
Multiple t tests increase chance of Type I error; Bonferroni
adjustment increases chance of Type II error
Allows comparisons of all treatments to control condition
Allows for use of a placebo group with control and experimental
groups
Imagine that a researcher wants to compare four different types
of treatment. The researcher decides to conduct six individual
studies to make these comparisons. What is the probability of a
Type I error, with α = .05, across these six comparisons? Use
9. the Bonferroni adjustment to determine the suggested alpha
level for these six tests.
Analyzing the Multiple-Group Design
ANOVA (analysis of variance) An inferential parametric
statistical test for comparing the means of three or more groups.
As noted previously, t tests are not recommended for comparing
performance across groups in a multiple-group design because
of the increased probability of a Type I error. For multiple-
group designs in which interval-ratio data are collected, the
recommended parametric statistical analysis is the ANOVA
(analysis of variance). As its name indicates, this procedure
allows us to analyze the variance in a study. You should be
familiar with variance from Chapter 3 on descriptive statistics.
Nonparametric analyses are also available for designs in which
ordinal data are collected (the Kruskal-Wallis analysis of
variance) and for designs in which nominal data are collected
(chi-square test). We will discuss some of these tests in later
modules.
We will begin our coverage of statistics appropriate for
multiple-group designs by discussing those used with data
collected from a between-subjects design. Recall that a
between-subjects design is one in which different participants
serve in each condition. Imagine that we conducted the study
mentioned at the beginning of the module in which subjects are
asked to study a list of 10 words using rote rehearsal or one of
two forms of elaborative rehearsal. A total of 24 participants are
randomly assigned, 8 to each condition. Table 13.1 lists the
number of words correctly recalled by each participant.
one-way between-subjects ANOVA An inferential statistical test
for comparing the means of three or more groups using a
between-subjects design.
Because these data represent an interval-ratio scale of
measurement and because there are more than two groups, an
ANOVA is the appropriate statistical test to analyze the data. In
addition, because this is a between-subjects design, we use
a one-way between-subjects ANOVA. As discussed earlier, the
10. term between-subjects indicates that participants have been
randomly assigned to conditions in a between-subjects design.
The term one-way indicates that the design uses only one
independent variable—in this case, type of rehearsal. We will
discuss statistical tests appropriate for correlated-groups
designs and tests appropriate for designs with more than one
independent variable in Chapter 8. Please note that although the
study used to illustrate the ANOVA procedure in this section
has an equal number of subjects in each condition, this is not
necessary to the procedure.
TABLE 13.1 Number of words recalled correctly in rote,
imagery, and story conditions
ROTE REHEARSAL
IMAGERY
STORY
2
4
6
4
5
5
3
7
9
5
6
10
2
5
8
11. 7
4
7
6
8
10
3
5
9
¯¯¯X=4X¯=4
¯¯¯X=5.5X¯=5.5
¯¯¯X=8X¯=8
Grand Mean = 5.833
One-Way Between-Subjects ANOVA: What It Is and What It
Does
The analysis of variance (ANOVA) is an inferential statistical
test for comparing the means of three or more groups. In
addition to helping maintain an acceptable Type I error rate, the
ANOVA has the added advantage over using multiple t tests of
being more powerful and thus less susceptible to a Type II
error. In this section, we will discuss the simplest use of
ANOVA—a design with one independent variable with three
levels.
Let's continue to use the experiment and data presented in Table
13.1. Remember that we are interested in the effects of
rehearsal type on memory. The null hypothesis (H0) for an
ANOVA is that the sample means represent the same population
(H0: μ1 = μ2 = μ3). The alternative hypothesis (Ha) is that they
represent different populations (Ha: at least one μ ≠ another μ).
When a researcher rejects H0 using an ANOVA, it means that
the independent variable affected the dependent variable to the
extent that at least one group mean differs from the others by
more than would be expected based on chance. Failing to
12. reject H0 indicates that the means do not differ from each other
more than would be expected based on chance. In other words,
there is not enough evidence to suggest that the sample means
represent at least two different populations.
grand mean The mean performance across all participants in a
study.
error variance The amount of variability among the scores
caused by chance or uncontrolled variables.
In our example, the mean number of words recalled in the rote
rehearsal condition is 4, for the imagery condition it is 5.5, and
in the story condition it is 8. If you look at the data from each
condition, you will notice that most subjects in each condition
did not score exactly at the mean for that condition. In other
words, there is variability within each condition. The overall
mean, or grand mean, across all participants in all conditions is
5.83. Because none of the participants in any condition recalled
exactly 5.83 words, there is also variability between conditions.
What we are interested in is whether this variability is due
primarily to the independent variable (differences in rehearsal
type) or to error variance—the amount of variability among the
scores caused by chance or uncontrolled variables (such as
individual differences between subjects).
within-groups variance The variance within each condition; an
estimate of the population error variance.
The amount of error variance can be estimated by looking at the
amount of variability within each condition. How will this give
us an estimate of error variance? Each subject in each condition
was treated similarly—they were each instructed to rehearse the
words in the same manner. Because the subjects in each
condition were treated in the same manner, any differences
observed in the number of words recalled are attributable only
to error variance. In other words, some participants may have
been more motivated, or more distracted, or better at memory
tasks—all factors that would contribute to error variance in this
case. Therefore, the within-groups variance (the variance within
each condition or group) is an estimate of the population error
13. variance.
Now compare the means between the groups. If the independent
variable (rehearsal type) had an effect, we would expect some
of the group means to differ from the grand mean. If the
independent variable had no effect on number of words recalled,
we would still expect the group means to vary from the grand
mean, but only slightly, as a result of error variance attributable
to individual differences. In other words, all subjects in a study
will not score exactly the same. Therefore, even when the
independent variable has no effect, we do not expect that the
group means will exactly equal the grand mean, but they should
be close to the grand mean. If there were no effect of the
independent variable, then the variance between groups would
be due to error.
between-groups variance An estimate of the effect of the
independent variable and error variance.
Between-groups variance may be attributed to several sources.
There could be systematic differences between the groups,
referred to as systematic variance. The systematic variance
between the groups could be due to the effects of the
independent variable (variance due to the experimental
manipulation). However, it could also be due to the influence of
uncontrolled confounding variables (variance due to extraneous
variables). In addition, there will always be some error variance
included in any between-groups variance estimate. In sum,
between-groups variance is an estimate of systematic variance
(the effect of the independent variable and any confounds) and
error variance.
F-ratio The ratio of between-groups variance to within-groups
variance.
By looking at the ratio of between-groups variance to within-
groups variance, known as an F-ratio, we can determine whether
most of the variability is attributable to systematic variance
(due, we hope, to the independent variable and not to
confounds) or to chance and random factors (error variance):
F=Between-groupsvarianceWithin-
14. groupsvariance=Systematicvariance+errorvarianceErrorvariance
F=Between-groups varianceWithin-
groups variance=Systematic variance+ error varianceError varia
nce
Looking at the F-ratio, we can see that if the systematic
variance (which we assume is due to the effect of the
independent variable) is substantially greater than the error
variance, the ratio will be substantially greater than 1. If there
is no systematic variance, then the ratio will be approximately
1.00 (error variance over error variance). There are two points
to remember regarding F-ratios. First, in order for an F-ratio to
be significant (show a statistically meaningful effect of an
independent variable), it must be substantially greater than 1
(we will discuss exactly how much larger than 1 in the next
module). Second, if an F-ratio is approximately 1, then the
between-groups variance equals the within-groups variance and
there is no effect of the independent variable.
Refer back to Table 13.1, and think about the within-groups
versus between-groups variance in this study. Notice that the
amount of variance within the groups is small—the scores
within each group vary from each individual group mean, but
not by very much. The between-groups variance, on the other
hand, is large—two of the means across the three conditions
vary from the grand mean to a greater extent. With these
data, then, it appears that we have a relatively large between-
groups variance and a smaller within-groups variance. Our F-
ratio will therefore be larger than 1.00. To assess how large it
is, we will need to conduct the appropriate calculations
(described in the next module). At this point, however, you
should have a general understanding of how an ANOVA
analyzes variance to determine if there is an effect of the
independent variable.
ONE-WAY BETWEEN-SUBJECTS ANOVA
Concept
Description
15. Null hypothesis (H0)
The independent variable had no effect—the samples all
represent the same population
Alternative hypothesis (Ha)
The independent variable had an effect—at least one of the
samples represents a different population than the others
F-ratio
The ratio formed when the between-groups variance is divided
by the within-groups variance
Between-groups variance
An estimate of the variance of the group means about the grand
mean; includes both systematic variance and error variance
Within-groups variance
An estimate of the variance within each condition in the
experiment; also known as error variance, or variance due to
chance factors
Imagine that the following table represents data from the study
just described (the effects of type of rehearsal on number of
words recalled from the 10 words given). Do you think that the
between-groups and within-groups variances are large,
moderate, or small? Would the corresponding F-ratio be greater
than, equal to, or less than 1.00?
Rote Rehearsal
Imagery
Story
2
4
5
4
2
2
3
17. MODULE EXERCISES
(Answers to odd-numbered questions appear in Appendix B.)
1.What is/are the advantage(s) of conducting a study with three
or more levels of the independent variable?
2.What does the term one-way mean with respect to an
ANOVA?
3.Explain between-groups variance and within-groups variance.
4.If a researcher decides to use multiple comparisons in a study
with three conditions, what is the probability of a Type I error
across these comparisons? Use the Bonferroni adjustment to
determine the suggested alpha level.
5.If H0 is true, what should the F-ratio equal or be close to?
6.If Ha is supported, should the F-ratio be greater than, less
than, or equal to 1?
CRITICAL THINKING CHECK ANSWERS
Critical Thinking Check 13.1
The probability of a Type I error would be 26.5% [1 −(1 −
.05)6] = [1 − (.95)6] = [1 − .735] = 26.5%. Using the Bonferroni
adjustment, the alpha level would be .008 for each comparison.
Critical Thinking Check 13.2
Both the within-groups and between-groups variances are
moderate. This should lead to an F-ratio of approximately 1.
MODULE 14
One-Way Between-Subjects Analysis of Variance (ANOVA)
Learning Objectives
•Identify what a one-way between-subjects ANOVA is and what
it does.
•Use the formulas provided to calculate a one-way between-
subjects ANOVA.
•Interpret the results from a one-way between-subjects ANOVA.
•Calculate η2 for a one-way between-subjects ANOVA.
•Calculate Tukey's post hoc test for a one-way between-subjects
ANOVA.
Calculations for the One-Way Between-Subjects ANOVA
18. To see exactly how ANOVA works, we begin by calculating the
sums of squares (SS). This should sound somewhat familiar to
you because we calculated sums of squares as part of the
calculation for standard deviation in Module 5. The sums of
squares in that formula represented the sum of the squared
deviation of each score from the overall mean. Determining the
sums of squares is the first step in calculating the various types
or sources of variance in an ANOVA.
Several types of sums of squares are used in the calculation of
an ANOVA. In describing them in this module, I
provide definitional formulas for each. The definitional formula
follows the definition for each sum of squares and should give
you the basic idea of how each SS is calculated. When dealing
with very large data sets, however, the definitional formulas can
become somewhat cumbersome. Thus, statisticians have
transformed the definitional formulas into computational
formulas. A computational formula is easier to use in terms of
the number of steps required. However, computational formulas
do not follow the definition of the SS and thus do not
necessarily make sense in terms of the definition for each SS. If
your instructor would prefer that you use the computational
formulas, they are provided in Appendix D.
TABLE 14.1 Calculation of SSTotal using the definitional
formula
ROTE REHEARSAL
IMAGERY
STORY
X
(X−¯¯¯¯XG)2(X−X¯G)2
X
(X−¯¯¯¯XG)2(X−X¯G)2
X
(X−¯¯¯¯XG)2(X−X¯G)2
2
14.69
4
20. 4.70
10
17.36
3
8.03
5
.69
9
10.03
Σ=50.88Σ=50.88
Σ=14.88Σ=14.88
Σ=61.56Σ=61.56
SSTotal = 50.88 + 14.88 + 61.56 = 127.32
Note: All numbers have been rounded to two decimal places.
total sum of squares The sum of the squared deviations of each
score from the grand mean.
The first sum of squares that we need to describe is the total
sum of squares (SSTotal)—the sum of the squared deviations of
each score from the grand mean. In a definitional formula, this
would be represented as Σ=(X−¯¯¯XG)2Σ=(X − X¯G)2,
where X represents each individual score
and ¯¯¯XGX¯G represents the grand mean. In other words, we
determine how much each individual subject varies from the
grand mean, square that deviation score, and sum all of the
squared deviation scores. Using the study from the previous
module on the effects of rehearsal type on memory (see Table
13.1), the total sum of squares (SSTotal) = 127.32. To see
where this number comes from, refer to Table 14.1. (For the
computational formula, see Appendix D.) Once we have
calculated the sum of squares within and between groups (see
below), they should equal the total sum of squares when added
together. In this way, we can check our calculations for
accuracy. If the sum of squares within and between do not equal
21. the sum of squares total, then you know there is an error in at
least one of the calculations.
within-groups sum of squares The sum of the squared deviations
of each score from its group mean.
Because an ANOVA analyzes the variance between groups and
within groups, we need to use different formulas to determine
the amount of variance attributable to these two factors.
The within-groups sum of squares is the sum of the squared
deviations of each score from its group or condition mean and is
a reflection of the amount of error variance. In the definitional
formula, it would be Σ(X−¯¯¯Xg)2Σ(X−X¯g)2, where X refers
to each individual score and ¯¯¯XgX¯g refers to the mean for
each group or condition. In order to determine this, we find the
difference between each score and its group mean, square these
deviation scores, and then sum all of the squared deviation
scores. The use of this definitional formula to
calculate SSWithin is illustrated in Table 14.2. The
computational formula appears in Appendix D. Thus, rather than
comparing every score in the entire study to the grand mean of
the study (as is done for SSTotal), we compare each score in
each condition to the mean of that condition. Thus, SSWithin is
a reflection of the amount of variability within each condition.
Because the subjects in each condition were treated in a similar
manner, we would expect little variation among the scores
within each group. This means that the within-groups sum of
squares (SSWithin) should be small, indicating a small amount
of error variance in the study. For our memory study, the
within-groups sum of squares (SSWithin) = 62.
TABLE 14.2 Calculation of SSWithin using the definitional
formula
ROTE REHEARSAL
IMAGERY
STORY
X
(X−¯¯¯¯Xg)2(X−X¯g)2
X
23. 2.25
7
1
6
4
8
6.25
10
4
3
1
5
.25
9
1
Σ = 24
Σ = 14
Σ = 24
SSWithin = 24 + 14 + 24 = 62
NOTE: All numbers have been rounded to two decimal places.
between-groups sum of squares The sum of the squared
deviations of each group's mean from the grand mean,
multiplied by the number of subjects in each group.
The between-groups sum of squares is the sum of the squared
deviations of each group's mean from the grand mean,
multiplied by the number of subjects in each group. In the
definitional formula, this would
be Σ[¯¯¯Xg−¯¯¯XG]2n]Σ[X¯g − X¯G]2n],
where ¯¯¯XgX¯g refers to the mean for each
group, ¯¯¯XGX¯G refers to the grand mean, and n refers to the
number of subjects in each group. The use of the definitional
formula to calculate SSBetweenis illustrated in Table 14.3. The
computational formula appears in Appendix D. The between-
24. groups variance is an indication of the systematic variance
across the groups (the variance due to the independent variable
and any confounds) and error. The basic idea behind the
between-groups sum of squares is that if the independent
variable had no effect (if there were no differences between the
groups), then we would expect all the group means to be about
the same. If all the group means were similar, they would also
be approximately equal to the grand mean and there would be
little variance across conditions. If, however, the independent
variable caused changes in the means of some conditions
(caused them to be larger or smaller than other conditions), then
the condition means not only will differ from each other but
will also differ from the grand mean, indicating variance across
conditions. In our memory study, SSBetween = 65.33.
TABLE 14.3 Calculation of SSBetween using the definitional
formula
Rote Rehearsal
(¯¯¯Xg−¯¯¯XG)2n=(4−5.833)28=(−1.833)28=(3.36)8=26.88(X¯
g − X¯G)2n=(4−5.833)28=(−1.833)28=(3.36)8=26.88
Imagery
(¯¯¯Xg−¯¯¯XG)2n=(5.5−5.833)28=(−.333)28=(.11)8=.88(X¯g −
X¯G)2n=(5.5−5.833)28=(−.333)28=(.11)8=.88
Story
(¯¯¯Xg−¯¯¯XG)2n=(8−5.833)28=(2.167)28=(4.696)8=37.57(X¯
g−X¯G)2n=(8−5.833)28=(2.167)28=(4.696)8=37.57
SSBetween = 26.88 + .88 + 37.57 = 65.33
We can check the accuracy of our calculations by adding
the SSWithin to the SSBetween. When added, these numbers
should equal SSTotal. Thus, SSWithin (62)
+ SSBetween (65.33) = 127.33. The SSTotal that we calculated
earlier was 127.32 and is essentially equal
to SSWithin + SSBetween, taking into account rounding errors.
mean square An estimate of either variance between groups or
variance within groups.
Calculating the sums of squares is an important step in the
ANOVA. It is not, however, the end. Now that we have
25. determined SSTotal, SSWithin, and SSBetween, we must
transform these scores into the mean squares. The term mean
square (MS) is an abbreviation of mean squared deviation
scores. The MS scores are estimates of variance between and
within the groups. In order to calculate the MS for each group
(MSWithin and MSBetween), we divide each SS by the
appropriate df (degrees of freedom). The reason for this is that
the MS scores are our variance estimates. You may remember
from Module 5 that when calculating standard deviation and
variance, we divide the sum of squares by N (or N - 1 for the
unbiased estimator) in order to get the average deviation from
the mean. In the same manner, we must divide the SS scores by
their degrees of freedom (the number of scores that contributed
to each SS minus 1).
To do this for the present example, we first need to determine
the degrees of freedom for each type of variance. Let's begin
with the dfTotal, which we will use to check our accuracy when
calculating dfWithin and dfBetween. In other
words, dfWithin and dfBetween should sum to the dfTotal. We
determined SSTotal by calculating the deviations around the
grand mean. We therefore had one restriction on our data—the
grand mean. This leaves us with N − 1 total degrees of freedom
(the total number of participants in the study minus the one
restriction). For our study on the effects of rehearsal type on
memory,
dfTotal = 24 − 1 = 23
Using a similar logic, the degrees of freedom within each group
would then be n − 1 (the number of subjects in each condition
minus 1). However, we have more than one group; we
have k groups, where k refers to the number of groups or
conditions in the study. The degrees of freedom within groups is
therefore k(n − 1), or N − k. For our example,
dfWithin=24−3=21dfWithin=24−3=21
Lastly, the degrees of freedom between groups is the variability
of k means around the grand mean. Therefore, dfBetween equals
the number of groups (k) minus 1, or k − 1. For our study, this
26. would be
dfBetween=3−1=2dfBetween=3−1=2
Notice that the sum of
the dfWithin and df Between equals df total: 21 + 2=23. This
allows you to check your calculations for accuracy. If the
degrees of freedom between and within do not sum to the
degrees of freedom total, you know there is a mistake
somewhere.
Now that we have calculated the sums of squares and their
degrees of freedom, we can use these numbers to calculate our
estimates of the variance between and within groups. As stated
previously, the variance estimates are called mean squares and
are determined by dividing each SS by its corresponding df. In
our example,
MSBetween=SSBetweendfBetween=SSBetweendfBetween=65.3
32=32.67MSWithin=SSWithindfWithin=6221=2.95MSBetween=
SSBetweendfBetween=SSBetweendfBetween=65.332=32.67MS
Within=SSWithindfWithin=6221=2.95
We can now use the estimates of between-groups and within-
groups variance to determine the F-ratio:
F=MSBetweenMSWithin=32.672.95=11.07F=MSBetweenMSWit
hin=32.672.95=11.07
The definitional formulas for the sums of squares, along with
the formulas for the degrees of freedom, mean squares, and the
final F-ratio, are summarized in Table 14.4. The ANOVA
summary table for the F-ratio just calculated is presented
in Table 14.5. This is a common way of summarizing ANOVA
findings. You will sometimes see ANOVA summary tables
presented in journal articles because they provide a concise way
of presenting the data from an analysis of variance.
TABLE 14.4 ANOVA summary table: definitional formulas
source
df
SS
MS
F
27. Between-groups
k − 1
Σ[(¯¯¯Xg−¯¯¯XG)2n]Σ[(X¯g−X¯G)2n]
SSBdfBSSBdfB
MSw MSBMSWMSBMSW
Within-groups
N − k
Σ(X−¯¯¯Xg)2Σ(X −X¯g)2
SSWdfWSSWdfW
Total
N − 1
Σ(X−¯¯¯XG)2Σ(X −X¯G)2
TABLE 14.5 ANOVA summary table for the memory study
SOURCE
df
SS
MS
F
Between-groups
2
65.33
32.67
11.07
Within-groups
21
62
2.95
Total
23
127.33
28. Interpreting the One-Way Between-Subjects ANOVA
Our obtained F-ratio of 11.07 is obviously greater than 1.00.
However, we do not know whether it is large enough to let us
reject the null hypothesis. To make this decision, we need to
compare the obtained F (Fobt) of 11.07 with an Fcv—the
critical value that determines the cutoff for statistical
significance. The underlying F distribution is actually a family
of distributions, each based on the degrees of freedom between
and within each group. Remember that the alternative
hypothesis is that the population means represented by the
sample means are not from the same population. Table
A.4 in Appendix A provides the critical values for the family
of F distributions when α = .05 and when α = .01.
To use the table, look at the dfWithin running down the left side
of the table and the dfBetween running across the top of the
table. Fcv is found where the row and column of these two
numbers intersect. For our example, dfWithin = 21
and dfBetween = 2. Because there is no 21 in
the dfWithin column, we use the next lower number, 20.
According to Table A.4, the Fcv for the .05 level is 3.49.
Because our Fobt exceeds this, it is statistically significant at
the .05 level. Let's check the .01 level also. The critical value
for the .01 level is 5.85. Our Fobt is larger than this critical
value also. We can therefore conclude that the Fobt is
significant at the .01 level. In APA publication format, this
would be written as F(2, 21) = 11.07, p < .01. This means that
we reject H0 and support Ha. In other words, at least one group
mean differs significantly from the others. The calculation of
this ANOVA using Excel, SPSS, and the TI-84 calculator is
presented in the Statistical Software Resources section at the
end of this chapter.
Let's consider what factors might affect the size of the
final Fobt. Because the Fobt is derived using the between-
groups variance as the numerator and the within-groups
variance as the denominator, anything that increases the
numerator or decreases the denominator will increase the Fobt.
29. What might increase the numerator? Using stronger controls in
the experiment could have this effect because it would make any
differences between the groups more noticeable or larger. This
means that the MSBetween (the numerator in the F-ratio) would
be larger and therefore lead to a larger final F-ratio.
What would decrease the denominator? Once again, using better
control to reduce overall error variance would have this effect;
so would increasing the sample size, which
increases dfWithin and ultimately decreases MSWithin. Why
would each of these affect the F-ratio in this manner? Each
would decrease the size of the MSWithin, which is the
denominator in the F-ratio. Dividing by a smaller number would
lead to a larger final F-ratio and, therefore, a greater chance
that it would be significant.
Graphing the Means and Effect Size
As noted in Module 11, we usually graph the means when a
significant difference is found between them. As in our previous
graphs, the independent variable is placed on the x-axis and the
dependent variable on the y-axis. A graph representing the mean
performance of each group is shown in Figure 14.1. In this
experiment, those in the Rote condition remembered an average
of 4 words, those in the Imagery condition remembered an
average of 5.5 words, and those in the Story condition
remembered an average of 8 words.
eta-squared An inferential statistic for measuring effect size
with an ANOVA.
In addition to graphing the data, we should also assess the
effect size. Based on the Fobt, we know that there was more
variability between groups than within groups. In other words,
the between-groups variance (the numerator in the F-ratio) was
larger than the within-groups variance (the denominator in
the F-ratio). However, it would be useful to know how much of
the variability in the dependent variable can be attributed to the
independent variable. In other words, it would be useful to have
a measure of effect size. For an ANOVA, effect size can be
estimated using eta-squared (η2), which is calculated as
30. follows:
η2=SSBetweenSSTotalη2=SSBetweenSSTotal
Because SSBetween reflects the differences between the means
from the various levels of an independent variable
and SSTotal reflects the total differences between all scores in
the experiment, η2 reflects the proportion of the total
differences in the scores that is associated with differences
between sample means, or how much of the variability in the
dependent variable (memory) is attributable to the manipulation
of the independent variable (rehearsal type). In other words,
η2 indicates how accurately the differences in scores can be
predicted using the levels (conditions) of the independent
variable. Referring to the summary table for our example
in Table 14.5, η2 would be calculated as follows:
FIGURE 14.1Number of words recalled as a function of
rehearsal type
η2=65.33127.33=.51η2=65.33127.33=.51
In other words, approximately 51% of the variance among the
scores can be attributed to the rehearsal condition to which the
participant was assigned. In this example, the independent
variable of rehearsal type is fairly important in determining the
number of words recalled by subjects because the η2 of 51%
represents a considerable effect. According to Cohen (1988), an
η2 of .01 represents a small effect size; .06, a medium effect
size; and .14, a large effect size. Thus, our obtained
η2 represents a large effect size. In addition, as with most
measures of effect size, η2 is useful in determining whether or
not a result of statistical significance is also of practical
significance. In other words, although researchers might find
that an F-ratio is statistically significant, if the corresponding
η2 is negligible, then the result is of little practical significance
(Huck & Cormier, 1996).
Assumptions of the One-Way Between-Subjects ANOVA
As with most statistical tests, certain assumptions must be met
to ensure that the statistic is being used properly. The
assumptions for the between-subjects one-way ANOVA are
31. similar to those for the t test for independent groups:
•The data are on an interval-ratio scale.
•The underlying distribution is normally distributed.
•The variances among the populations being compared are
homogeneous.
Because ANOVA is a robust statistical test, violations of some
of these assumptions do not necessarily affect the results.
Specifically, if the distributions are slightly skewed rather than
normally distributed, it does not affect the results of the
ANOVA. In addition, if the sample sizes are equal, the
assumption of homogeneity of variances can be violated.
However, it is not acceptable to violate the assumption of
interval-ratio data. If the data collected in a study are ordinal or
nominal in scale, other statistical procedures must be used.
These procedures are discussed in later modules.
Tukey's Post Hoc Test
post hoc test When used with an ANOVA, a means of
comparing all possible pairs of groups to determine which ones
differ significantly from each other.
Because the results from our ANOVA indicate that at least one
of the sample means differs significantly from the others
(represents a different population from the others), we must
now compute a post hoc test (a test conducted after the fact—in
this case, after the ANOVA). A post hoc test involves
comparing each of the groups in the study to each of the other
groups to determine which ones differ significantly from each
other. This may sound familiar to you. In fact, you may be
thinking, isn't that what a t test does? In a sense, you are
correct. However, remember that a series of multiple t tests
inflates the probability of a Type I error. A post hoc test is
designed to permit multiple comparisons and still maintain
alpha (the probability of a Type I error) at .05.
Tukey's honestly significant difference (HSD) A post hoc test
used with ANOVAs for making all pairwise comparisons when
conditions have equal n.
The post hoc test presented here is Tukey's honestly significant
32. difference (HSD). This test allows a researcher to make all
pairwise comparisons among the sample means in a study while
maintaining an acceptable alpha (usually .05, but possibly .01)
when the conditions have equal n. If there is not an equal
number of subjects in each condition, then another post hoc test,
such as Fisher's protected t test, would be appropriate. If you
need to use Fisher's protected t test, the formula is provided in
the Computational Supplement in Appendix D.
Tukey's test identifies the smallest difference between any two
means that is significant with α = .05 or α = .01. The formula
for Tukey's HSD is
HSD.05=Q(k,dfWithin)√MSWithinnHSD.05=Q(k,dfWithin)MS
Withinn
Using this formula, we can determine the HSD for the .05 alpha
level. This involves using Table A.5 in Appendix A to look up
the value for Q. To look up Q, we need k (the number of means
being compared—in our study on memory, this is 3)
and dfWithin (found in the ANOVA summary table, Table
14.5). Referring to Table A.5 for k = 3 and dfWithin = 21
(because there is no 20 in Table A.5, we use 20 here as we did
with Table A.4), we find that at the .05 level, Q = 3.58. In
addition, we need MSWithin from Table 14.5 and n (the number
of participants in each group). Using these numbers, we
calculate the HSD as follows:
HSD0.5=(3.58)√2.958=(3.58)√.369=(3.58)(.607)=2.17HSD0.5=(
3.58)2.958=(3.58).369=(3.58)(.607)=2.17
This tells us that a difference of 2.17 or more for any pair of
means is significant at the .05 level. In other words, the
difference between the means is large enough that it is greater
than what would be expected based on chance. Table
14.6 summarizes the differences between the means for each
pairwise comparison. Can you identify which comparisons are
significant using Tukey's HSD?
TABLE 14.6 Differences between each pair of means in the
memory study
33. ROTE REHEARSAL
IMAGERY
STORY
Rote Rehearsal
—
1.5
4.0
Imagery
—
2.5
Story
—
If you identified the differences between the Story condition
and the Rote Rehearsal condition and between the Story
condition and the Imagery condition as the two honestly
significant differences, you were correct. Because the F-ratio
was significant at α = .01, we should also check the HSD01. To
do this, we use the same formula, but we use Q for the .01 alpha
level (from Table A.5). The calculations are as follows:
HSD.01=(4.64)√2.958=(4.64)(.607)=2.82HSD.01=(4.64)2.958=(
4.64)(.607)=2.82
The only difference significant at this level is between the Rote
Rehearsal condition and the Story condition. Thus, based on
these data, those in the Story condition recalled significantly
more words than those in the Imagery condition (p < .05) and
those in the Rote Rehearsal condition (p < .01).
ONE-WAY BETWEEN-SUBJECTS ANOVA
Concept
Description
F-ratio
The ratio formed when the between-groups variance is divided
by the within-groups variance
34. Between-groups variance
An estimate of the variance of the group means about the grand
mean; includes both systematic variance and error variance
Within-groups variance
An estimate of the variance within each condition in the
experiment; also known as error variance, or variance due to
chance factors
Eta-squared
A measure of effect size—the variability in the dependent
variable attributable to the independent variable
Tukey's post hoc test
A test conducted to determine which conditions in a study with
more than two groups differ significantly from each other
1.Of the following four F-ratios, which appears to indicate that
the independent variable had an effect on the dependent
variable?
a.1.25/1.11
b.0.91/1.25
c.1.95/0.26
d.0.52/1.01
2.The following ANOVA summary table represents the results
from a study on the effects of exercise on stress. There were
three conditions in the study: a control group, a moderate
exercise group, and a high exercise group. Each group had 10
subjects and the mean stress levels for each group were Control
= 75.0, Moderate Exercise = 44.7, and High Exercise = 63.7.
Stress was measured using a 100-item stress scale, with 100
representing the highest level of stress. Complete the ANOVA
summary table, and determine whether the F-ratio is significant.
In addition, calculate eta-squared and Tukey's HSD if
necessary.
ANOVA Summary Table
Source
Between
Within
35. Total
df
SS
4,689.27
82,604.20
MS
F
REVIEW OF KEY TERMS
between-groups sum of squares (p. 229)
eta-squared (p. 233)
mean square (p. 230)
post hoc test (p. 235)
total sum of squares (p. 228)
Tukey's honestly significant difference (HSD) (p. 235)
within-groups sum of squares (p. 228)
MODULE EXERCISES
(Answers to odd-numbered questions appear in Appendix B.)
1.A researcher conducts a study on the effects of amount of
sleep on creativity. The creativity scores for four levels of sleep
(2 hours, 4 hours, 6 hours, and 8 hours) for n = 5 subjects (in
each group) are presented next.
Amount of Sleep (in hours)
2
4
6
8
3
4
10
10
5
7
11
13
6
8
37. d.What conclusions can be drawn from the F-ratio and the post
hoc comparisons?
e.What is the effect size, and what does this mean?
f.Graph the means.
2.In a study on the effects of stress on illness, a researcher
tallied the number of colds people contracted during a 6-month
period as a function of the amount of stress they reported during
the same time period. There were three stress levels: minimal,
moderate, and high stress. The sums of squares appear in the
following ANOVA summary table. The mean for each condition
and the number of participants per condition are also noted.
Source
df
SS
MS
F
Between-groups
22.167
Within-groups
14.750
Total
36.917
Stress Level
Mean
n
Minimal
38. 3
4
Moderate
4
4
High
6
4
a.Complete the ANOVA summary table.
b.Is Fobt significant at α = .05? at α = .01?
c.Perform post hoc comparisons if necessary.
d.What conclusions can be drawn from the F-ratio and the post
hoc comparisons?
e.What is the effect size, and what does this mean?
f.Graph the means.
3.A researcher interested in the effects of exercise on stress had
subjects exercise for 30, 60, or 90 minutes per day. The mean
stress level on a 100-point stress scale (with 100 indicating high
stress) for each condition appears next, along with the ANOVA
summary table with the sums of squares indicated.
Source
df
SS
MS
F
Between-groups
4,689.27
Within-groups
82,604.20
Total
39. 87,293.47
Exercise Level
Mean
n
30 Minutes
75.0
10
60 Minutes
44.7
10
90 Minutes
63.7
10
a.Complete the ANOVA summary table.
b.Is Fobt significant at α = .05? at α = .01?
c.Perform post hoc comparisons if necessary.
d.What conclusions can be drawn from the F-ratio and the post
hoc comparisons?
e.What is the effect size, and what does this mean?
f.Graph the means.
4.A researcher conducted an experiment on the effects of a new
“drug” on depression. The control group received nothing and
the placebo group received a placebo pill. An experimental
group received the “drug.” A depression inventory that provided
a measure of depression on a 50-point scale was used (50
indicates that an individual is very high on the depression
variable). The ANOVA summary table appears next, along with
the mean depression score for each condition.
Source
df
SS
MS
41. 5.When should post hoc tests be performed?
6.What information does eta-squared (η2) provide?
CRITICAL THINKING CHECK ANSWERS
Critical Thinking Check 14.1
1.The F-ratio 1.95/0.26 = 7.5 suggests that the independent
variable had an effect on the dependent variable.
2.
ANOVA Summary Table
Source
df
SS
MS
F
Between
2
4,689.27
2,344.64
.766
Within
27
82,604.20
3,059.41
Total
29
87,293.47
The resulting F-ratio is less than 1 and thus not significant.
Although stress levels differed across some of the groups, the
difference was not large enough to be significant.
CHAPTER SEVEN SUMMARY AND REVIEW
Introduction to Analysis of Variance (ANOVA)
CHAPTER SUMMARY
42. In this chapter we discussed designs using more than two levels
of an independent variable. Advantages to such designs include
being able to compare more than two kinds of treatment, using
fewer subjects, comparing all treatments to a control group, and
using placebo groups. When interval-ratio data are collected
using such a design, the parametric statistical analysis most
appropriate for use is the ANOVA (analysis of variance). A
one-way between-subjects ANOVA would be used for between-
subjects designs. Appropriate post hoc tests (Tukey's HSD) and
measures of effect size (eta-squared) were also discussed.
CHAPTER 7 REVIEW EXERCISES
(Answers to exercises appear in Appendix B.)
Fill-in Self-Test
Answer the following questions. If you have trouble answering
any of the questions, restudy the relevant material before going
on to the multiple-choice self-test.
1.The______________provides a means of setting a more
stringent alpha level for multiple tests in order to minimize
Type I errors.
2.A(n) ______________ is an inferential statistical test for
comparing the means of three or more groups.
3.The mean performance across all subjects is represented by
the ______________.
4.The ______________ variance is an estimate of the effect of
the independent variable, confounds, and error variance.
5.The sum of squared deviations of each score from the grand
mean is the ______________
6.When we divide an SS score by its degrees of freedom, we
have calculated a ______________
7.______________ is an inferential statistic for measuring
effect size with an ANOVA.
8.For an ANOVA, we use ______________ to compare all
possible pairs of groups to determine which ones differ
significantly from each other.
Multiple-Choice Self-Test
Select the single best answer for each of the following
43. questions. If you have trouble answering any of the questions,
restudy the relevant material.
1.The F-ratio is determined by dividing ______________ by
______________.
a.error variance; systematic variance
b.between-groups variance; within-groups variance
c.within-groups variance; between-groups variance
d.systematic variance; error variance
2.If between-groups variance is large, then we have observed
a.experimenter effects.
b.large systematic variance.
c.large differences due to confounds.
d.possibly both large systematic variance and large differences
due to confounds.
3.The larger the F-ratio, the greater the chance that
a.a mistake has been made in the computation.
b.there are large systematic effects present.
c.the experimental manipulation probably did not have the
predicted effects.
d.the between-groups variation is no larger than would be
expected by chance and no larger than the within-groups
variance.
4.One reason to use an ANOVA over a t test is to reduce the
risk of
a.a Type II error.
b.a Type I error.
c.confounds.
d.error variance.
5.If the null hypothesis for an ANOVA is false, then the F-ratio
should be
a.greater than 1.00.
b.a negative number.
c.0.00.
d.1.00.
6.If in a between-subjects ANOVA, there are four groups with
15 participants in each group, then the df for the F-ratio is equal
44. to
a.60.
b.59.
c.3, 56.
d.3, 57.
7.For an F-ratio with df = (3, 20), the Fcv for α = .05 would be
a.3.10.
b.4.94.
c.8.66.
d.5.53.
8.If a researcher reported an F-ratio with df = (2, 21) for a
between-subjects one-way ANOVA, then there were
______________ conditions in the experiment and
______________ total subjects.
a.2; 21
b.3; 23
c.2; 24
d.3; 24
9.Systematic variance and error variance comprise the
______________ variance.
a.within-groups
b.total
c.between-groups
d.subject
10.If a between-subjects one-way ANOVA
produced MSBetween = 25 and MSWithin = 5, then the F-ratio
would be
a.25/5 = 5.
b.5/25 = .20.
c.25/30 = .83.
d.30/5 = 6.
Self-Test Problems
Calculate Tukey's HSD and eta-squared for the following
ANOVA.
Anova Summary Table
Source
45. df
SS
MS
F
Between
2
150
Error
18
100
Total
20
CHAPTER SEVEN
Statistical Software Resources
If you need help getting started with Excel or SPSS, please
see Appendix C: Getting Started with Excel and SPSS.
MODULES 13 AND 14 One-Way Between-Subjects ANOVA
The problem we'll be using to illustrate how to calculate the
one-way between-subjects ANOVA appears in Modules
13 and 14.
Let's use the example from Modules 13 and 14 in which a
researcher wants to study the effects on memory performance of
rehearsal type. Three types of rehearsal are used (Rote,
Imagery, and Story) by three different groups of subjects. The
dependent variable is the subjects' scores on a 10-item test of
the material. These scores are listed in Table 13.1in Module 13.
46. Using Excel
We'll use the data from Table 13.1 (in Module 13) to illustrate
the use of Excel to compute a one-way between-subjects
ANOVA. In this study, we had participants use one of three
different types of rehearsal (rote, imagery, or story) and then
had them perform a recall task. Thus we manipulated rehearsal
and measured memory for the 10 words subjects studied.
Because there were different participants in each condition, we
use a between-subjects ANOVA. We begin by entering the data
into Excel, with the data from each condition appearing in a
different column. This can be seen next.
Next, with the Data ribbon highlighted, click on the Data
Analysis tab in the top right corner. You should receive the
following dialog box:
Select Anova: Single Factor, as in the preceding box, and
click OK. The following dialog box will appear:
With the cursor in the Input Range box, highlight the three
columns of data so that they are entered into the Input Range
box as they are in the preceding box (highlight only the data,
not the column headings). Then click OK. The output from the
ANOVA will appear on a new Worksheet, as seen next.
You can see from the ANOVA Summary Table provided by
Excel that F(2, 21) = 11.06, p = .000523. In addition to the full
ANOVA Summary Table, Excel also provides the mean and
variance for each condition.
Using SPSS
We'll again use the data from Table 13.1 (in Module 13) to
illustrate the use of SPSS to compute a one-way between-
subjects ANOVA. In this study, we had subjects use one of
three different types of rehearsal (rote, imagery, or story) and
then had them perform a recall task. Thus we manipulated
rehearsal and measured memory for the 10 words subjects
47. studied. Because there were different participants in each
condition, we use a between-subjects ANOVA. We begin by
entering the data into SPSS. The first column is labeled
Rehearsaltype and indicates which type of rehearsal the subjects
used (1 for rote, 2 for imagery, and 3 for story). The recall data
for each of the three conditions appear in the second column,
labeled Recall.
Next, click on Analyze, followed by Compare Means, and
then One-Way ANOVA, as illustrated next.
You should receive the following dialog box:
Enter Rehearsaltype into the Factor box by highlighting it and
using the appropriate arrow. Do the same for Recall by entering
it into the Dependent List box. After doing this, the dialog box
should appear as follows:
Next click on the Options button and
select Descriptive and Continue. Then click on
the Post H oc button and select T ukey and then Continue. Then
click on OK. The output from the ANOVA will appear in a new
Output window as seen next.
You can see that the descriptive statistics for each condition are
provided, followed by the ANOVA Summary Table in which
F(2, 21) = 11.065, p = .001. In addition to the full ANOVA
Summary Table, SPSS also calculated Tukey's HSD and
provides all pairwise comparisons between the three conditions
along with whether or not the comparison was significant.
Using the TI-84
Let's use the data from Table 13.1 (in Module 13) to conduct the
analysis.
1.With the calculator on, press the STAT key.
2.EDIT will be highlighted. Press the ENTER key.
3.Under L1 enter the data from Table 13.1 for the rote group.
48. 4.Under L2 enter the data from Table 13.1 for the imagery
group.
5.Under L3 enter the data from Table 13.1 for the story group.
6.Press the STAT key once again and highlight TESTS.
7.Scroll down to ANOVA. Press the ENTER key.
8.Next to “ANOVA” enter (L1,L2,L3) using the 2nd function
key with the appropriate number keys. Make sure that you use
commas. The finished line should read “ANOVA(L1,L2,L3).”
9.Press ENTER.
The F score of 11.065 should be displayed followed by the
significance level of .0005.
Discussion: Please discuss, elaborate and give example on the
topic below. Please use the Module/reference I provided.
Professor will not allow outside sources.
Author: Jackson, S. L. (2017). Statistics plain and simple, (4th
ed.). Boston, MA: Cengage Learning
Topic:
Using the sample provided, address the following:
· How would you interpret the results of the two-way ANOVA?
· What does the p value tell you?
· The results mention df. What does that term represent? How is
it calculated? Write a plainly stated sentence that explains what
these results tell you about the groups.
Sample
Sum of Squares df Mean Square F Sig.
SCORES Between Groups 351.520 4 87.880 9.085 .000
Within Groups 435.300 45 9.673
Total 7 86.820 49
Module/reference
49. Module 13: Comparing More Than Two Groups
Using Designs with Three or More Levels of an Independent
Variable
Comparing More than Two Kinds of Treatment in One Study
Comparing Two or More Kinds of Treatment with a Control
Group
Comparing a Placebo Group to the Control and Experimental
Groups
Analyzing the Multiple-Group Design
One-Way Between-Subjects ANOVA: What It Is and What It
Does
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 14: One-Way Between-Subjects Analysis of Variance
(ANOVA)
Calculations for the One-Way Between-Subjects ANOVA
Interpreting the One-Way Between-Subjects ANOVA
Graphing the Means and Effect Size
Assumptions of the One-Way Between-Subjects ANOVA
Tukey's Post Hoc Test
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 7 Summary and Review
Chapter 7 Statistical Software Resources
In this chapter, we discuss the common types of statistical
analyses used with designs involving more than two groups. The
inferential statistics discussed in this chapter differ from those
presented in the previous two chapters. In Chapter 5, single
samples were being compared to populations (z test and t test),
and in Chapter 6, two independent or correlated samples were
being compared. In this chapter, the statistics are designed to
test differences between more than two equivalent groups of
subjects.
50. Several factors influence which statistic should be used to
analyze the data collected. For example, the type of data
collected and the number of groups being compared must be
considered. Moreover, the statistic used to analyze the data will
vary depending on whether the study involves a between-
subjects design (designs in which different subjects are used in
each group) or a correlated-groups design. (Remember,
correlated-groups designs are of two types: within-subjects
designs, in which the same subjects are used repeatedly in each
group, and matched-subjects designs, in which different
subjects are matched between conditions on variables that the
researcher believes are relevant to the study.)
We will look at the typical inferential statistics used to analyze
interval-ratio data for between-subjects designs. In Module
13 we discuss the advantages and rationale for studying more
than two groups; in Module 14 we discuss the statistics
appropriate for use with between-subjects designs involving
more than two groups.
MODULE 13
Comparing More Than Two Groups
Learning Objectives
•Explain what additional information can be gained by using
designs with more than two levels of an independent variable.
•Explain and be able to use the Bonferroni adjustment.
•Identify what a one-way between-subjects ANOVA is and what
it does.
•Describe what between-groups variance is.
•Describe what within-groups variance is.
•Understand conceptually how an F-ratio is derived.
The experiments described so far have involved manipulating
one independent variable with only two levels—either a control
group and an experimental group or two experimental groups. In
this module, we discuss experimental designs involving one
independent variable with more than two levels. Examining
51. more levels of an independent variable allows us to address
more complicated and interesting questions. Often, experiments
begin as two-group designs and then develop into more complex
designs as the questions asked become more elaborate and
sophisticated.
Using Designs with three or More Levels of an Independent
Variable
Researchers may decide to use a design with more than two
levels of an independent variable for several reasons. First, it
allows them to compare multiple treatments. Second, it allows
them to compare multiple treatments to no treatment (the
control group). Third, more complex designs allow researchers
to compare a placebo group to control and experimental groups
(Mitchell & Jolley, 2001).
Comparing More than Two Kinds of Treatment in One Study
To illustrate this advantage of more complex experimental
designs, imagine that we want to compare the effects of various
types of rehearsal on memory. We have participants study a list
of 10 words using either rote rehearsal (repetition) or some
form of elaborative rehearsal. In addition, we specify the type
of elaborative rehearsal to be used in the different experimental
groups. Group 1 (the control group) uses rote rehearsal, Group 2
uses an imagery mnemonic technique, and Group 3 uses a story
mnemonic device. You may be wondering why we do not simply
conduct three studies or comparisons. Why don't we compare
Group 1 to Group 2, Group 2 to Group 3, and Group 1 to Group
3 in three different experiments? There are several reasons why
this is not recommended.
You may remember from Module 11 that a t test is used to
compare performance between two groups. If we do three
experiments, we need to use three t tests to determine any
differences. The problem is that using multiple tests inflates the
Type I error rate. Remember, a Type I error means that we
reject the null hypothesis when we should have failed to reject
it; that is, we claim that the independent variable has an effect
when it does not. For most statistical tests, we use the .05 alpha
52. level, meaning that we are willing to accept a 5% risk of making
a Type I error. Although the chance of making a Type I error on
one t test is .05, the overall chance of making a Type I error
increases as more tests are conducted.
Imagine that we conducted three t tests or comparisons among
the three groups in the memory experiment. The probability of a
Type I error on any single comparison is .05. The probability of
a Type I error on at least one of the three tests, however, is
considerably greater. To determine the chance of a Type I error
when making multiple comparisons, we use the formula 1 − (1 −
α)c, where c equals the number of comparisons performed.
Using this formula for the present example, we get
1 − (1 − .05)3 = 1 − (.95 )3 = 1 − .86 = .14
Thus, the probability of a Type I error on at least one of the
three tests is .14, or 14%.
Bonferroni adjustment A means of setting a more stringent
alpha level in order to minimize Type | errors.
One way of counteracting the increased chance of a Type I error
is to use a more stringent alpha level. The Bonferroni
adjustment, in which the desired alpha level is divided by the
number of tests or comparisons, is typically used to accomplish
this. For example, if we were using the t test to make the three
comparisons just described, we would divide .05 by 3 and get
.017. By not accepting the result as significant unless the alpha
level is .017 or less, we minimize the chance of a Type I error
when making multiple comparisons. We know from discussions
in previous modules, however, that although using a more
stringent alpha level decreases the chance of a Type I error, it
increases the chance of a Type II error (failing to reject the null
hypothesis when it should have been rejected—missing an effect
of an independent variable). Thus, the Bonferroni adjustment is
not the best method of handling the problem. A better method is
to use a single statistical test that compares all groups rather
than using multiple comparisons and statistical tests. Luckily
for us, there is a statistical technique that will do this—the
analysis of variance (ANOVA), which will be discussed shortly.
53. Another advantage of comparing more than two kinds of
treatment in one experiment is that it reduces both the number
of experiments conducted and the number of subjects needed.
Once again, refer back to the three-group memory experiment.
If we do one comparison with three groups, we can conduct only
one experiment, and we need subjects for only three groups. If,
however, we conduct three comparisons, each with two groups,
we need to perform three experiments, and we need participants
for six groups or conditions.
Comparing Two or More Kinds of Treatment with a Control
Group
Using more than two groups in an experiment also allows
researchers to determine whether each treatment is more or less
effective than no treatment (the control group). To illustrate
this, imagine that we are interested in the effects of aerobic
exercise on anxiety. We hypothesize that the more aerobic
activity engaged in, the more anxiety will be reduced. We use a
control group that does not engage in any aerobic activity and a
high aerobic activity group that engages in 50 minutes per day
of aerobic activity—a simple two-group design. Assume,
however, that when using this design, we find that both those in
the control group and those in the experimental group have high
levels of anxiety at the end of the study—not what we expected
to find. How could a design with more than two groups provide
more information? Suppose we add another group to this
study—a moderate aerobic activity group (25 minutes per
day)—and get the following results:
Control Group
High Anxiety
Moderate Aerobic Activity
Low Anxiety
High Aerobic Activity
High Anxiety
Based on these data, we have a V-shaped function. Up to a
certain point, aerobic activity reduces anxiety. However, when
the aerobic activity exceeds a certain level, anxiety increases
54. again. If we had conducted only the original study with two
groups, we would have missed this relationship and erroneously
concluded that there was no relationship between aerobic
activity and anxiety. Using a design with multiple groups allows
us to see more of the relationship between the variables.
Figure 13.1 illustrates the difference between the results
obtained with the three-group versus the two-group design in
this hypothetical study. It also shows the other two-group
comparisons—control compared to moderate aerobic activity,
and moderate aerobic activity compared to high aerobic activity.
This set of graphs allows you to see how two-group designs
limit our ability to see the full relationship between variables.
FIGURE 13.1FIGURE>Determining relationships with three-
group versus two-group designs: (a) three-group design; (b)
two-group comparison of control to high aerobic activity; (c)
two-group comparison of control to moderate aerobic activity;
(d) two-group comparison of moderate aerobic activity to high
aerobic activity
Figure 13.1a shows clearly how the three-group design allows
us to assess more fully the relationship between the variables. If
we had only conducted a two-group study, such as those
illustrated in Figure 13.1b, c, or d, we would have drawn a
much different conclusion from that drawn from the three-group
design. Comparing only the control to the high aerobic activity
group (Figure 13.1b) would have led us to conclude that aerobic
activity does not affect anxiety. Comparing only the control to
the moderate aerobic activity group (Figure 13.1c) would have
led to the conclusion that increasing aerobic activity reduces
anxiety. Comparing only the moderate aerobic activity group to
the high aerobic activity group (Figure 13.1d) would have led to
the conclusion that increasing aerobic activity increases
anxiety.
Being able to assess the relationship between the variables
means that we can determine the type of relationship that exists.
In the previous example, the variables produced a V-shaped
function. Other variables may be related in a straight linear
55. manner or in an alternative curvilinear manner (for example, a
J-shaped or S-shaped function). In summary, adding levels to
the independent variable allows us to determine more accurately
the type of relationship that exists between the variables.
Comparing a Placebo Group to the Control and Experimental
Groups
A final advantage of designs with more than two groups is that
they allow for the use of a placebo group—a group of subjects
who believe they are receiving treatment but in reality are not.
A placebo is an inert substance that participants believe is a
treatment. How can adding a placebo group improve an
experiment? Consider an often-cited study by Paul (1966, 1967)
involving children who suffered from maladaptive anxiety in
public-speaking situations. Paul used a control group, which
received no treatment; a placebo group, which received a
placebo that they were told was a potent tranquilizer; and an
experimental group, which received desensitization therapy. Of
the participants in the experimental group, 85% showed
improvement, compared with only 22% in the control condition.
If the placebo group had not been included, the difference
between the therapy and control groups (85% − 22% = 63%)
would overestimate the effectiveness of the desensitization
program. The placebo group showed 50% improvement,
meaning that the therapy's true effectiveness is much less (85%
− 50% = 35%). Thus, a placebo group allows for a more
accurate assessment of a therapy's effectiveness because, in
addition to spontaneous remission, it controls for participant
expectation effects.
DESIGNS WITH MORE THAN TWO LEVELS OF AN
INDEPENDENT VARIABLE
Advantages
Considerations
Allows comparisons of more than two types of treatment
Type of statistical analysis (e.g., multiple t tests or ANOVA)
Fewer subjects are needed
56. Multiple t tests increase chance of Type I error; Bonferroni
adjustment increases chance of Type II error
Allows comparisons of all treatments to control condition
Allows for use of a placebo group with control and experimental
groups
Imagine that a researcher wants to compare four different types
of treatment. The researcher decides to conduct six individual
studies to make these comparisons. What is the probability of a
Type I error, with α = .05, across these six comparisons? Use
the Bonferroni adjustment to determine the suggested alpha
level for these six tests.
Analyzing the Multiple-Group Design
ANOVA (analysis of variance) An inferential parametric
statistical test for comparing the means of three or more groups.
As noted previously, t tests are not recommended for comparing
performance across groups in a multiple-group design because
of the increased probability of a Type I error. For multiple-
group designs in which interval-ratio data are collected, the
recommended parametric statistical analysis is the ANOVA
(analysis of variance). As its name indicates, this procedure
allows us to analyze the variance in a study. You should be
familiar with variance from Chapter 3 on descriptive statistics.
Nonparametric analyses are also available for designs in which
ordinal data are collected (the Kruskal-Wallis analysis of
variance) and for designs in which nominal data are collected
(chi-square test). We will discuss some of these tests in later
modules.
We will begin our coverage of statistics appropriate for
multiple-group designs by discussing those used with data
collected from a between-subjects design. Recall that a
between-subjects design is one in which different participants
serve in each condition. Imagine that we conducted the study
mentioned at the beginning of the module in which subjects are
57. asked to study a list of 10 words using rote rehearsal or one of
two forms of elaborative rehearsal. A total of 24 participants are
randomly assigned, 8 to each condition. Table 13.1 lists the
number of words correctly recalled by each participant.
one-way between-subjects ANOVA An inferential statistical test
for comparing the means of three or more groups using a
between-subjects design.
Because these data represent an interval-ratio scale of
measurement and because there are more than two groups, an
ANOVA is the appropriate statistical test to analyze the data. In
addition, because this is a between-subjects design, we use
a one-way between-subjects ANOVA. As discussed earlier, the
term between-subjects indicates that participants have been
randomly assigned to conditions in a between-subjects design.
The term one-way indicates that the design uses only one
independent variable—in this case, type of rehearsal. We will
discuss statistical tests appropriate for correlated-groups
designs and tests appropriate for designs with more than one
independent variable in Chapter 8. Please note that although the
study used to illustrate the ANOVA procedure in this section
has an equal number of subjects in each condition, this is not
necessary to the procedure.
TABLE 13.1 Number of words recalled correctly in rote,
imagery, and story conditions
ROTE REHEARSAL
IMAGERY
STORY
2
4
6
4
5
5
58. 3
7
9
5
6
10
2
5
8
7
4
7
6
8
10
3
5
9
¯¯¯X=4X¯=4
¯¯¯X=5.5X¯=5.5
¯¯¯X=8X¯=8
Grand Mean = 5.833
One-Way Between-Subjects ANOVA: What It Is and What It
Does
The analysis of variance (ANOVA) is an inferential statistical
test for comparing the means of three or more groups. In
addition to helping maintain an acceptable Type I error rate, the
ANOVA has the added advantage over using multiple t tests of
being more powerful and thus less susceptible to a Type II
error. In this section, we will discuss the simplest use of
59. ANOVA—a design with one independent variable with three
levels.
Let's continue to use the experiment and data presented in Table
13.1. Remember that we are interested in the effects of
rehearsal type on memory. The null hypothesis (H0) for an
ANOVA is that the sample means represent the same population
(H0: μ1 = μ2 = μ3). The alternative hypothesis (Ha) is that they
represent different populations (Ha: at least one μ ≠ another μ).
When a researcher rejects H0 using an ANOVA, it means that
the independent variable affected the dependent variable to the
extent that at least one group mean differs from the others by
more than would be expected based on chance. Failing to
reject H0 indicates that the means do not differ from each other
more than would be expected based on chance. In other words,
there is not enough evidence to suggest that the sample means
represent at least two different populations.
grand mean The mean performance across all participants in a
study.
error variance The amount of variability among the scores
caused by chance or uncontrolled variables.
In our example, the mean number of words recalled in the rote
rehearsal condition is 4, for the imagery condition it is 5.5, and
in the story condition it is 8. If you look at the data from each
condition, you will notice that most subjects in each condition
did not score exactly at the mean for that condition. In other
words, there is variability within each condition. The overall
mean, or grand mean, across all participants in all conditions is
5.83. Because none of the participants in any condition recalled
exactly 5.83 words, there is also variability between conditions.
What we are interested in is whether this variability is due
primarily to the independent variable (differences in rehearsal
type) or to error variance—the amount of variability among the
scores caused by chance or uncontrolled variables (such as
individual differences between subjects).
within-groups variance The variance within each condition; an
estimate of the population error variance.
60. The amount of error variance can be estimated by looking at the
amount of variability within each condition. How will this give
us an estimate of error variance? Each subject in each condition
was treated similarly—they were each instructed to rehearse the
words in the same manner. Because the subjects in each
condition were treated in the same manner, any differences
observed in the number of words recalled are attributable only
to error variance. In other words, some participants may have
been more motivated, or more distracted, or better at memory
tasks—all factors that would contribute to error variance in this
case. Therefore, the within-groups variance (the variance within
each condition or group) is an estimate of the population error
variance.
Now compare the means between the groups. If the independent
variable (rehearsal type) had an effect, we would expect some
of the group means to differ from the grand mean. If the
independent variable had no effect on number of words recalled,
we would still expect the group means to vary from the grand
mean, but only slightly, as a result of error variance attributable
to individual differences. In other words, all subjects in a study
will not score exactly the same. Therefore, even when the
independent variable has no effect, we do not expect that the
group means will exactly equal the grand mean, but they should
be close to the grand mean. If there were no effect of the
independent variable, then the variance between groups would
be due to error.
between-groups variance An estimate of the effect of the
independent variable and error variance.
Between-groups variance may be attributed to several sources.
There could be systematic differences between the groups,
referred to as systematic variance. The systematic variance
between the groups could be due to the effects of the
independent variable (variance due to the experimental
manipulation). However, it could also be due to the influence of
uncontrolled confounding variables (variance due to extraneous
variables). In addition, there will always be some error variance
61. included in any between-groups variance estimate. In sum,
between-groups variance is an estimate of systematic variance
(the effect of the independent variable and any confounds) and
error variance.
F-ratio The ratio of between-groups variance to within-groups
variance.
By looking at the ratio of between-groups variance to within-
groups variance, known as an F-ratio, we can determine whether
most of the variability is attributable to systematic variance
(due, we hope, to the independent variable and not to
confounds) or to chance and random factors (error variance):
F=Between-groupsvarianceWithin-
groupsvariance=Systematicvariance+errorvarianceErrorvariance
F=Between-groups varianceWithin-
groups variance=Systematic variance+ error varianceError varia
nce
Looking at the F-ratio, we can see that if the systematic
variance (which we assume is due to the effect of the
independent variable) is substantially greater than the error
variance, the ratio will be substantially greater than 1. If there
is no systematic variance, then the ratio will be approximately
1.00 (error variance over error variance). There are two points
to remember regarding F-ratios. First, in order for an F-ratio to
be significant (show a statistically meaningful effect of an
independent variable), it must be substantially greater than 1
(we will discuss exactly how much larger than 1 in the next
module). Second, if an F-ratio is approximately 1, then the
between-groups variance equals the within-groups variance and
there is no effect of the independent variable.
Refer back to Table 13.1, and think about the within-groups
versus between-groups variance in this study. Notice that the
amount of variance within the groups is small—the scores
within each group vary from each individual group mean, but
not by very much. The between-groups variance, on the other
hand, is large—two of the means across the three conditions
vary from the grand mean to a greater extent. With these
62. data, then, it appears that we have a relatively large between-
groups variance and a smaller within-groups variance. Our F-
ratio will therefore be larger than 1.00. To assess how large it
is, we will need to conduct the appropriate calculations
(described in the next module). At this point, however, you
should have a general understanding of how an ANOVA
analyzes variance to determine if there is an effect of the
independent variable.
ONE-WAY BETWEEN-SUBJECTS ANOVA
Concept
Description
Null hypothesis (H0)
The independent variable had no effect—the samples all
represent the same population
Alternative hypothesis (Ha)
The independent variable had an effect—at least one of the
samples represents a different population than the others
F-ratio
The ratio formed when the between-groups variance is divided
by the within-groups variance
Between-groups variance
An estimate of the variance of the group means about the grand
mean; includes both systematic variance and error variance
Within-groups variance
An estimate of the variance within each condition in the
experiment; also known as error variance, or variance due to
chance factors
Imagine that the following table represents data from the study
just described (the effects of type of rehearsal on number of
words recalled from the 10 words given). Do you think that the
between-groups and within-groups variances are large,
moderate, or small? Would the corresponding F-ratio be greater
than, equal to, or less than 1.00?
Rote Rehearsal
64. ¯¯¯X=3.88X¯=3.88
¯¯¯X=4X¯=4
Grand Mean = 3.96
REVIEW OF KEY TERMS
ANOVA (analysis of variance) (p. 222)
between-groups variance (p. 224)
Bonferroni adjustment (p. 218)
error variance (p. 223)
F-ratio (p. 224)
grand mean (p. 223)
one-way between-subjects ANOVA (p. 222)
within-groups variance (p. 224)
MODULE EXERCISES
(Answers to odd-numbered questions appear in Appendix B.)
1.What is/are the advantage(s) of conducting a study with three
or more levels of the independent variable?
2.What does the term one-way mean with respect to an
ANOVA?
3.Explain between-groups variance and within-groups variance.
4.If a researcher decides to use multiple comparisons in a study
with three conditions, what is the probability of a Type I error
across these comparisons? Use the Bonferroni adjustment to
determine the suggested alpha level.
5.If H0 is true, what should the F-ratio equal or be close to?
6.If Ha is supported, should the F-ratio be greater than, less
than, or equal to 1?
CRITICAL THINKING CHECK ANSWERS
Critical Thinking Check 13.1
The probability of a Type I error would be 26.5% [1 −(1 −
.05)6] = [1 − (.95)6] = [1 − .735] = 26.5%. Using the Bonferroni
adjustment, the alpha level would be .008 for each comparison.
Critical Thinking Check 13.2
Both the within-groups and between-groups variances are
moderate. This should lead to an F-ratio of approximately 1.
65. MODULE 14
One-Way Between-Subjects Analysis of Variance (ANOVA)
Learning Objectives
•Identify what a one-way between-subjects ANOVA is and what
it does.
•Use the formulas provided to calculate a one-way between-
subjects ANOVA.
•Interpret the results from a one-way between-subjects ANOVA.
•Calculate η2 for a one-way between-subjects ANOVA.
•Calculate Tukey's post hoc test for a one-way between-subjects
ANOVA.
Calculations for the One-Way Between-Subjects ANOVA
To see exactly how ANOVA works, we begin by calculating the
sums of squares (SS). This should sound somewhat familiar to
you because we calculated sums of squares as part of the
calculation for standard deviation in Module 5. The sums of
squares in that formula represented the sum of the squared
deviation of each score from the overall mean. Determining the
sums of squares is the first step in calculating the various types
or sources of variance in an ANOVA.
Several types of sums of squares are used in the calculation of
an ANOVA. In describing them in this module, I
provide definitional formulas for each. The definitional formula
follows the definition for each sum of squares and should give
you the basic idea of how each SS is calculated. When dealing
with very large data sets, however, the definitional formulas can
become somewhat cumbersome. Thus, statisticians have
transformed the definitional formulas into computational
formulas. A computational formula is easier to use in terms of
the number of steps required. However, computational formulas
do not follow the definition of the SS and thus do not
necessarily make sense in terms of the definition for each SS. If
your instructor would prefer that you use the computational
formulas, they are provided in Appendix D.
TABLE 14.1 Calculation of SSTotal using the definitional
formula
67. .69
8
4.70
7
1.36
4
3.36
7
1.36
6
.03
8
4.70
10
17.36
3
8.03
5
.69
9
10.03
Σ=50.88Σ=50.88
Σ=14.88Σ=14.88
Σ=61.56Σ=61.56
SSTotal = 50.88 + 14.88 + 61.56 = 127.32
Note: All numbers have been rounded to two decimal places.
total sum of squares The sum of the squared deviations of each
score from the grand mean.
The first sum of squares that we need to describe is the total
sum of squares (SSTotal)—the sum of the squared deviations of
each score from the grand mean. In a definitional formula, this
would be represented as Σ=(X−¯¯¯XG)2Σ=(X − X¯G)2,
where X represents each individual score
68. and ¯¯¯XGX¯G represents the grand mean. In other words, we
determine how much each individual subject varies from the
grand mean, square that deviation score, and sum all of the
squared deviation scores. Using the study from the previous
module on the effects of rehearsal type on memory (see Table
13.1), the total sum of squares (SSTotal) = 127.32. To see
where this number comes from, refer to Table 14.1. (For the
computational formula, see Appendix D.) Once we have
calculated the sum of squares within and between groups (see
below), they should equal the total sum of squares when added
together. In this way, we can check our calculations for
accuracy. If the sum of squares within and between do not equal
the sum of squares total, then you know there is an error in at
least one of the calculations.
within-groups sum of squares The sum of the squared deviations
of each score from its group mean.
Because an ANOVA analyzes the variance between groups and
within groups, we need to use different formulas to determine
the amount of variance attributable to these two factors.
The within-groups sum of squares is the sum of the squared
deviations of each score from its group or condition mean and is
a reflection of the amount of error variance. In the definitional
formula, it would be Σ(X−¯¯¯Xg)2Σ(X−X¯g)2, where X refers
to each individual score and ¯¯¯XgX¯g refers to the mean for
each group or condition. In order to determine this, we find the
difference between each score and its group mean, square these
deviation scores, and then sum all of the squared deviation
scores. The use of this definitional formula to
calculate SSWithin is illustrated in Table 14.2. The
computational formula appears in Appendix D. Thus, rather than
comparing every score in the entire study to the grand mean of
the study (as is done for SSTotal), we compare each score in
each condition to the mean of that condition. Thus, SSWithin is
a reflection of the amount of variability within each condition.
Because the subjects in each condition were treated in a similar
manner, we would expect little variation among the scores
69. within each group. This means that the within-groups sum of
squares (SSWithin) should be small, indicating a small amount
of error variance in the study. For our memory study, the
within-groups sum of squares (SSWithin) = 62.
TABLE 14.2 Calculation of SSWithin using the definitional
formula
ROTE REHEARSAL
IMAGERY
STORY
X
(X−¯¯¯¯Xg)2(X−X¯g)2
X
(X−¯¯¯¯Xg)2(X−X¯g)2
X
(X−¯¯¯¯Xg)2(X−X¯g)2
2
4
4
2.25
6
4
4
0
5
.25
5
9
3
1
7
2.25
9
1
5
1
6
71. deviations of each group's mean from the grand mean,
multiplied by the number of subjects in each group.
The between-groups sum of squares is the sum of the squared
deviations of each group's mean from the grand mean,
multiplied by the number of subjects in each group. In the
definitional formula, this would
be Σ[¯¯¯Xg−¯¯¯XG]2n]Σ[X¯g − X¯G]2n],
where ¯¯¯XgX¯g refers to the mean for each
group, ¯¯¯XGX¯G refers to the grand mean, and n refers to the
number of subjects in each group. The use of the definitional
formula to calculate SSBetweenis illustrated in Table 14.3. The
computational formula appears in Appendix D. The between-
groups variance is an indication of the systematic variance
across the groups (the variance due to the independent variable
and any confounds) and error. The basic idea behind the
between-groups sum of squares is that if the independent
variable had no effect (if there were no differences between the
groups), then we would expect all the group means to be about
the same. If all the group means were similar, they would also
be approximately equal to the grand mean and there would be
little variance across conditions. If, however, the independent
variable caused changes in the means of some conditions
(caused them to be larger or smaller than other conditions), then
the condition means not only will differ from each other but
will also differ from the grand mean, indicating variance across
conditions. In our memory study, SSBetween = 65.33.
TABLE 14.3 Calculation of SSBetween using the definitional
formula
Rote Rehearsal
(¯¯¯Xg−¯¯¯XG)2n=(4−5.833)28=(−1.833)28=(3.36)8=26.88(X¯
g − X¯G)2n=(4−5.833)28=(−1.833)28=(3.36)8=26.88
Imagery
(¯¯¯Xg−¯¯¯XG)2n=(5.5−5.833)28=(−.333)28=(.11)8=.88(X¯g −
X¯G)2n=(5.5−5.833)28=(−.333)28=(.11)8=.88
Story
(¯¯¯Xg−¯¯¯XG)2n=(8−5.833)28=(2.167)28=(4.696)8=37.57(X¯