Multiple Linear Regression II and ANOVA IJames Neill
Explains advanced use of multiple linear regression, including residuals, interactions and analysis of change, then introduces the principles of ANOVA starting with explanation of t-tests.
Multiple Linear Regression II and ANOVA IJames Neill
Explains advanced use of multiple linear regression, including residuals, interactions and analysis of change, then introduces the principles of ANOVA starting with explanation of t-tests.
a full lecture presentation on ANOVA .
areas covered include;
a. definition and purpose of anova
b. one-way anova
c. factorial anova
d. mutiple anova
e MANOVA
f. POST-HOC TESTS - types
f. easy step by step process of calculating post hoc test.
This presentation explains the concept of ANOVA, ANCOVA, MANOVA and MANCOVA. This presentation also deals about the procedure to do the ANOVA, ANCOVA and MANOVA with the use of SPSS.
Statistical inference: Statistical Power, ANOVA, and Post Hoc testsEugene Yan Ziyou
This deck was used in the IDA facilitation of the John Hopkins' Data Science Specialization course for Statistical Inference. It covers the topics in week 4 (statistical power, ANOVA, and post hoc tests).
The data and R script for the lab session can be found here: https://github.com/eugeneyan/Statistical-Inference
a full lecture presentation on ANOVA .
areas covered include;
a. definition and purpose of anova
b. one-way anova
c. factorial anova
d. mutiple anova
e MANOVA
f. POST-HOC TESTS - types
f. easy step by step process of calculating post hoc test.
This presentation explains the concept of ANOVA, ANCOVA, MANOVA and MANCOVA. This presentation also deals about the procedure to do the ANOVA, ANCOVA and MANOVA with the use of SPSS.
Statistical inference: Statistical Power, ANOVA, and Post Hoc testsEugene Yan Ziyou
This deck was used in the IDA facilitation of the John Hopkins' Data Science Specialization course for Statistical Inference. It covers the topics in week 4 (statistical power, ANOVA, and post hoc tests).
The data and R script for the lab session can be found here: https://github.com/eugeneyan/Statistical-Inference
(Individuals With Disabilities Act Transformation Over the Years)DSilvaGraf83
(Individuals With Disabilities Act Transformation Over the Years)
Discussion Forum Instructions:
1. You must post at least three times each week.
2. Your initial post is due Tuesday of each week and the following two post are due before Sunday.
3. All post must be on separate days of the week.
4. Post must be at least 150 words and cite all of your references even it its the book.
Discussion Topic:
Describe how the lives of students with disabilities from culturally and/or linguistically diverse backgrounds have changed since the advent of IDEA. What do you feel are some things that can or should be implemented to better assist with students that have disabilities? Tell me about these ideas and how would you integrate them?
ANOVA
ANOVA
• Analysis of Variance
• Statistical method to analyzes variances to determine if the means from more than
two populations are the same
• compare the between-sample-variation to the within-sample-variation
• If the between-sample-variation is sufficiently large compared to the within-sample-
variation it is likely that the population means are statistically different
• Compares means (group differences) among levels of factors. No
assumptions are made regarding how the factors are related
• Residual related assumptions are the same as with simple regression
• Explanatory variables can be qualitative or quantitative but are categorized
for group investigations. These variables are often referred to as factors
with levels (category levels)
ANOVA Assumptions
• Assume populations , from which the response values for the groups
are drawn, are normally distributed
• Assumes populations have equal variances
• Can compare the ratio of smallest and largest sample standard deviations.
Between .05 and 2 are typically not considered evidence of a violation
assumption
• Assumes the response data are independent
• For large sample sizes, or for factor level sample sizes that are equal,
the ANOVA test is robust to assumption violations of normality and
unequal variances
ANOVA and Variance
Fixed or Random Factors
• A factor is fixed if its levels are chosen before the ANOVA investigation
begins
• Difference in groups are only investigated for the specific pre-selected factors
and levels
• A factor is random if its levels are choosen randomly from the
population before the ANOVA investigation begins
Randomization
• Assigning subjects to treatment groups or treatments to subjects
randomly reduces the chance of bias selecting results
ANOVA hypotheses statements
One-way ANOVA
One-Way ANOVA
Hypotheses statements
Test statistic
=
𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
𝑊𝑖𝑡ℎ𝑖𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Under the null hypothesis both the between and within group variances estimate the
variance of the random error so the ratio is assumed to be close to 1.
Null Hypothesis
Alternate Hypothesis
One-Way ANOVA
One-Way ANOVA
One-Way ANOVA Excel Output
Treatme
(Individuals With Disabilities Act Transformation Over the Years)DMoseStaton39
(Individuals With Disabilities Act Transformation Over the Years)
Discussion Forum Instructions:
1. You must post at least three times each week.
2. Your initial post is due Tuesday of each week and the following two post are due before Sunday.
3. All post must be on separate days of the week.
4. Post must be at least 150 words and cite all of your references even it its the book.
Discussion Topic:
Describe how the lives of students with disabilities from culturally and/or linguistically diverse backgrounds have changed since the advent of IDEA. What do you feel are some things that can or should be implemented to better assist with students that have disabilities? Tell me about these ideas and how would you integrate them?
ANOVA
ANOVA
• Analysis of Variance
• Statistical method to analyzes variances to determine if the means from more than
two populations are the same
• compare the between-sample-variation to the within-sample-variation
• If the between-sample-variation is sufficiently large compared to the within-sample-
variation it is likely that the population means are statistically different
• Compares means (group differences) among levels of factors. No
assumptions are made regarding how the factors are related
• Residual related assumptions are the same as with simple regression
• Explanatory variables can be qualitative or quantitative but are categorized
for group investigations. These variables are often referred to as factors
with levels (category levels)
ANOVA Assumptions
• Assume populations , from which the response values for the groups
are drawn, are normally distributed
• Assumes populations have equal variances
• Can compare the ratio of smallest and largest sample standard deviations.
Between .05 and 2 are typically not considered evidence of a violation
assumption
• Assumes the response data are independent
• For large sample sizes, or for factor level sample sizes that are equal,
the ANOVA test is robust to assumption violations of normality and
unequal variances
ANOVA and Variance
Fixed or Random Factors
• A factor is fixed if its levels are chosen before the ANOVA investigation
begins
• Difference in groups are only investigated for the specific pre-selected factors
and levels
• A factor is random if its levels are choosen randomly from the
population before the ANOVA investigation begins
Randomization
• Assigning subjects to treatment groups or treatments to subjects
randomly reduces the chance of bias selecting results
ANOVA hypotheses statements
One-way ANOVA
One-Way ANOVA
Hypotheses statements
Test statistic
=
𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
𝑊𝑖𝑡ℎ𝑖𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Under the null hypothesis both the between and within group variances estimate the
variance of the random error so the ratio is assumed to be close to 1.
Null Hypothesis
Alternate Hypothesis
One-Way ANOVA
One-Way ANOVA
One-Way ANOVA Excel Output
Treatme
univariate and bivariate analysis in spss Subodh Khanal
this slide will help to perform various tests in spss targeting univariate and bivariate analysis along with the way of entering and analyzing multiple responses.
Inferential statistics are techniques that allow us to use these samples to make generalizations about the populations from which the samples were drawn. ... The methods of inferential statistics are (1) the estimation of parameter(s) and (2) testing of statistical hypotheses.
Statistics for Anaesthesiologists covers basic to intermediate level statistics for researchers especially commonly used study designs or tests in Anaesthesiology research.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
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1. Chapter 12
Introduction to Analysis of Variance
PowerPoint Lecture Slides
Essentials of Statistics for the
Behavioral Sciences
Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
2. Chapter 12 Learning Outcomes
• Explain purpose and logic of Analysis of
Variance1
• Perform Analysis of Variance on data from
single-factor study2
• Know when and why to use post hoc tests
(posttests)3
• Compute Tukey’s HSD and Scheffé test post
hoc tests4
• Compute η2 to measure effect size5
3. Tools You Will Need
• Variability (Chapter 4)
– Sum of squares
– Sample variance
– Degrees of freedom
• Introduction to hypothesis testing (Chapter 8)
– The logic of hypothesis testing
• Independent-measures t statistic (Chapter 10)
4. 12.1 Introduction to Analysis
of Variance
• Analysis of variance
– Used to evaluate mean differences between two
or more treatments
– Uses sample data as basis for drawing general
conclusions about populations
• Clear advantage over a t test: it can be used
to compare more than two treatments at the
same time
6. Terminology
• Factor
– The independent (or quasi-independent) variable
that designates the groups being compared
• Levels
– Individual conditions or values that make up
a factor
• Factorial design
– A study that combines two or more factors
8. Statistical Hypotheses for ANOVA
• Null hypothesis: the level or value on the
factor does not affect the dependent variable
– In the population, this is equivalent to saying that
the means of the groups do not differ from each
other
•
3210 : H
9. Alternate Hypothesis for ANOVA
• H1: There is at least one mean difference
among the populations (Acceptable
shorthand is “Not H0”)
• Issue: how many ways can H0 be wrong?
– All means are different from every other mean
– Some means are not different from some others,
but other means do differ from some means
10. Test statistic for ANOVA
• F-ratio is based on variance instead of sample
mean differences
effecttreatmentnowithexpectedes)(differencvariance
meanssamplebetweenes)(differencvariance
F
11. Test statistic for ANOVA
• Not possible to compute a sample mean
difference between more than two samples
• F-ratio based on variance instead of sample
mean difference
– Variance used to define and measure the size of
differences among sample means (numerator)
– Variance in the denominator measures the mean
differences that would be expected if there is no
treatment effect
12. Type I Errors and
Multiple-Hypothesis tests
• Why ANOVA (if t can compare two means)?
– Experiments often require multiple hypothesis
tests—each with Type I error (testwise alpha)
– Type I error for a set of tests accumulates testwise
alpha experimentwise alpha > testwise alpha
• ANOVA evaluates all mean differences
simultaneously with one test—regardless of
the number of means—and thereby avoids
the problem of inflated experimentwise alpha
13. 12.2 Analysis of Variance Logic
• Between-treatments variance
– Variability results from general differences
between the treatment conditions
– Variance between treatments measures
differences among sample means
• Within-treatments variance
– Variability within each sample
– Individual scores are not the same within each
sample
14. Sources of Variability
Between Treatments
• Systematic differences caused
by treatments
• Random, unsystematic differences
– Individual differences
– Experimental (measurement) error
15. Sources of Variability
Within Treatments
• No systematic differences related to treatment
groups occur within each group
• Random, unsystematic differences
– Individual differences
– Experimental (measurement) error
effectstreatmentnowithsdifference
effectstreatmentanyincludingsdifference
F
17. F-ratio
• If H0 is true:
– Size of treatment effect is near zero
– F is near 1.00
• If H1 is true:
– Size of treatment effect is more than 0.
– F is noticeably larger than 1.00
• Denominator of the F-ratio is called the
error term
18. Learning Check
• Decide if each of the following statements
is True or False
• ANOVA allows researchers to compare
several treatment conditions without
conducting several hypothesis tests
T/F
• If the null hypothesis is true, the
F-ratio for ANOVA is expected (on
average) to have a value of 0
T/F
19. Learning Check - Answers
• Several conditions can be
compared in one testTrue
• If the null hypothesis is true, the
F-ratio will have a value near 1.00
False
20. 12.3 ANOVA Notation
and Formulas
• Number of treatment conditions: k
• Number of scores in each treatment: n1, n2…
• Total number of scores: N
– When all samples are same size, N = kn
• Sum of scores (ΣX) for each treatment: T
• Grand total of all scores in study: G = ΣT
• No universally accepted notation for ANOVA;
Other sources may use other symbols
24. Degrees of Freedom Analysis
• Total degrees of freedom
dftotal= N – 1
• Within-treatments degrees of freedom
dfwithin= N – k
• Between-treatments degrees of freedom
dfbetween= k – 1
26. Mean Squares and F-ratio
within
within
withinwithin
df
SS
sMS 2
between
between
betweenbetween
df
SS
sMS 2
within
between
within
between
MS
MS
s
s
F 2
2
27. ANOVA Summary Table
Source SS df MS F
Between Treatments 40 2 20 10
Within Treatments 20 10 2
Total 60 12
•Concise method for presenting ANOVA results
•Helps organize and direct the analysis process
•Convenient for checking computations
•“Standard” statistical analysis program output
28. Learning Check
• An analysis of variance produces SStotal = 80
and SSwithin = 30. For this analysis, what is
SSbetween?
• 50A
• 110B
• 2400C
• More information is neededD
29. Learning Check - Answer
• An analysis of variance produces SStotal = 80
and SSwithin = 30. For this analysis, what is
SSbetween?
• 50A
• 110B
• 2400C
• More information is neededD
30. 12.4 Distribution of F-ratios
• If the null hypothesis is true, the value of F will
be around 1.00
• Because F-ratios are computed from two
variances, they are always positive numbers
• Table of F values is organized by two df
– df numerator (between) shown in table columns
– df denominator (within) shown in table rows
32. 12.5 Examples of Hypothesis
Testing and Effect Size
• Hypothesis tests use the same four steps that
have been used in earlier hypothesis tests.
• Computation of the test statistic F is done
in stages
– Compute SStotal, SSbetween, SSwithin
– Compute MStotal, MSbetween, MSwithin
– Compute F
34. Measuring Effect size for
ANOVA
• Compute percentage of variance accounted
for by the treatment conditions
• In published reports of ANOVA, effect size is
usually called η2 (“eta squared”)
– r2 concept (proportion of variance explained)
total
treatmentsbetween
SS
SS
2
35. In the Literature
• Treatment means and standard deviations are
presented in text, table or graph
• Results of ANOVA are summarized, including
– F and df
– p-value
– η2
• E.g., F(3,20) = 6.45, p<.01, η2 = 0.492
37. MSwithin and Pooled Variance
• In the t-statistic and in the F-ratio, the
variances from the separate samples are
pooled together to create one average value
for the sample variance
• Numerator of F-ratio measures how much
difference exists between treatment means.
• Denominator measures the variance of the
scores inside each treatment
38. 12.6 post hoc Tests
• ANOVA compares all individual mean
differences simultaneously, in one test
• A significant F-ratio indicates that at least one
difference in means is statistically significant
– Does not indicate which means differ significantly
from each other!
• post hoc tests are follow up tests done to
determine exactly which mean differences are
significant, and which are not
39. Experimentwise Alpha
• post hoc tests compare two individual means
at a time (pairwise comparison)
– Each comparison includes risk of a Type I error
– Risk of Type I error accumulates and is called the
experimentwise alpha level.
• Increasing the number of hypothesis tests
increases the total probability of a Type I error
• post hoc (“posttests”) use special methods to
try to control experimentwise Type I error rate
40. Tukey’s Honestly Significant
Difference
• A single value that determines the minimum
difference between treatment means that is
necessary to claim statistical significance–a
difference large enough that p < αexperimentwise
– Honestly Significant Difference (HSD)
n
MS
qHSD within
41. The Scheffé Test
• The Scheffé test is one of the safest of all
possible post hoc tests
– Uses an F-ratio to evaluate significance of the
difference between two treatment conditions
groupstwoofSSwithcalculatedBA versus
within
between
MS
MS
F
42. Learning Check
• Which combination of factors is most likely to
produce a large value for the F-ratio?
• large mean differences
and large sample variancesA
• large mean differences
and small sample variancesB
• small mean differences
and large sample variancesC
• small mean differences
and small sample variancesD
43. Learning Check - Answer
• Which combination of factors is most likely to
produce a large value for the F-ratio?
• large mean differences
and large sample variancesA
• large mean differences
and small sample variancesB
• small mean differences
and large sample variancesC
• small mean differences
and small sample variancesD
44. Learning Check
• Decide if each of the following statements
is True or False
• Post tests are needed if the decision from
an analysis of variance is “fail to reject the
null hypothesis”
T/F
• A report shows ANOVA results: F(2, 27) =
5.36, p < .05. You can conclude that the
study used a total of 30 participants
T/F
45. Learning Check - Answers
• post hoc tests are needed only if
you reject H0 (indicating at least
one mean difference is significant)
False
• Because dftotal = N-1 and
• Because dftotal = dfbetween + dfwithin
True
46. 12.7 Relationship between
ANOVA and t tests
• For two independent samples, either t or F
can be used
– Always result in same decision
– F = t2
• For any value of α, (tcritical)2 = Fcritical
48. Independent Measures
ANOVA Assumptions
• The observations within each sample must be
independent
• The population from which the samples are
selected must be normal
• The populations from which the samples are
selected must have equal variances
(homogeneity of variance)
• Violating the assumption of homogeneity of
variance risks invalid test results
FIGURE 12.1 A typical situation in which ANOVA would be used. Three separate samples are obtained to evaluate the mean differences among three populations (or treatments) with unknown means.
FIGURE 12.2 A research design with two factors. The research study uses two factors: One factor uses two levels of therapy technique (I versus II), and the second factor uses three levels of time (before, after and 6 months after). Also notice that the therapy factor uses two separate groups (independent measures) and the time factor uses the same group for all three levels (repeated measures).
Some instructors may want to spend some time listing some of the ways that the null hypothesis cold be wrong that do NOT include μ1 ≠ μ2 ≠ μ3 because many students make the logic error of assuming that if null is wrong then the ONLY way it can be wrong is if all the groups are significantly different from each other.
Testwise Type I error rate and Experimentwise Type I error rate are sufficiently complex concepts that some instructors may want to expand the lecture materials on this topic.
FIGURE 12.3 The independent-measures ANOVA partitions, or analyzes, the total variability into two components: variance between treatments and variance within treatments.
FIGURE 12.4 The structure and sequence of calculations for the ANOVA.
FIGURE 12.5 Partitioning the sum of squares (SS) for the independent-measures ANOVA.
FIGURE 12.6 Partitioning degrees of freedom (df) for the independent-measures ANOVA.
Some instructors may want to begin the discussion of the computational procedures with this slide. Students sometimes get lost in all the formulas, but if they can see that the summary table is the “goal” of the process, they can break it down into the small steps required to fill in each part of the table.
FIGURE 12.7 The distribution of F-ratios with df = 2, 12. Of all the values in the distribution, only 5% are larger than F = 3.88, and only 1% are larger than F = 6.93.
FIGURE 12.8 The distribution of F-ratios with df = 3, 20. The critical values for α = .01 is 4.94.
A visual representation of the between-treatments variability and the within0-treatments variability that form the numerator and denominator, respectively, of the F-ratio. In (a), the difference between treatments is relatively large and easy to see. In (b), the same 4-point difference between treatments is relatively small and is overwhelmed by the within-treatments variability.
FIGURE 12.10 The distribution of t statistics with df = 18 and the corresponding distribution of F-ratios with df = 1, 18. Notice that the critical values for α = .05 are t = ±2.101 and F = 2.1012 = 4.41.
FIGURE 12.10 Formulas for ANOVA.
FIGURE 12.12 SPSS Output of the ANOVA for the studying strategy experiment in Example 12.1.
