The document describes an experiment examining the effect of different culturing conditions on the growth of Methicillin-resistant Staphylococcus aureus (MRSA) strains. Five MRSA strains were cultured under various time, temperature, and tryptone concentration levels. ANOVA and polynomial regression analyses found that time, temperature, and concentration all significantly affected bacterial counts, with some interaction effects. The optimal conditions estimated were 48 hours for time and 35°C for temperature based on maximizing counts in the regression models.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
The document provides information on the basic principles of experimental design, including replication, randomization, and local control. It then discusses the completely randomized design (CRD) in detail. The CRD allocates treatments randomly across experimental units. It has advantages like maximum use of units and simple analysis, but disadvantages like more experimental error. The document also introduces the randomized block design (RBD) which controls for variation among blocks. The RBD stratifies the experimental area into blocks and allocates treatments randomly within each block.
1. The document describes various experimental designs including completely randomized designs, randomized complete block designs, Latin square designs, and factorial designs. It defines the models and assumptions for each design.
2. Key aspects of each design are discussed such as how to partition sums of squares to conduct analysis of variance, calculate expected mean squares, and test hypotheses about treatment means. Relative efficiencies of designs are also covered.
3. The document explains how to interpret interactions in factorial designs using profile plots and presents tables for analysis of variance for each design. It also discusses estimating treatment differences and multiple comparison procedures.
The document discusses goodness-of-fit tests for categorical data. It introduces notation for categorical variables with multiple categories and hypotheses for goodness-of-fit tests. Expected counts are calculated based on hypothesized proportions. The chi-square statistic is used to calculate test statistics and P-values are found using the chi-square distribution. Examples demonstrate applying goodness-of-fit tests to determine if variable categories occur with equal frequency.
This document provides an overview of two-way analysis of variance (ANOVA). It explains that two-way ANOVA involves two categorical independent variables and one continuous dependent variable. The document outlines the objectives of two-way ANOVA, which are to analyze interactions between the two factors, and evaluate the effects of each factor. It then provides examples of how to set up and perform two-way ANOVA calculations and interpretations.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
The document discusses four nonparametric statistical tests:
1. The Wilcoxon Rank Sum Test (also called the Mann-Whitney U Test) compares the medians of two independent samples and is an alternative to the independent t-test.
2. The Wilcoxon Signed Rank Test compares the medians of two dependent/paired samples and is an alternative to the paired t-test. It calculates the differences between pairs, ranks their absolute values, and sums the ranks of positive differences.
3. The Kruskal-Wallis Test compares more than two independent samples.
4. The Runs Test examines the randomness of a single sample by counting runs, or streaks, of
The document describes a randomized complete block design (RCBD) experimental method. RCBD involves comparing treatments (e.g. fertilizers) applied to experimental units (e.g. corn crops) grouped into blocks (e.g. fields). Treatments are randomly assigned to experimental units within each block. RCBD controls for variability between blocks (e.g. differences in soil between fields) to isolate the effect of treatments. It provides more precise results than a completely randomized design when blocks are homogeneous within and heterogeneous between.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
The document provides information on the basic principles of experimental design, including replication, randomization, and local control. It then discusses the completely randomized design (CRD) in detail. The CRD allocates treatments randomly across experimental units. It has advantages like maximum use of units and simple analysis, but disadvantages like more experimental error. The document also introduces the randomized block design (RBD) which controls for variation among blocks. The RBD stratifies the experimental area into blocks and allocates treatments randomly within each block.
1. The document describes various experimental designs including completely randomized designs, randomized complete block designs, Latin square designs, and factorial designs. It defines the models and assumptions for each design.
2. Key aspects of each design are discussed such as how to partition sums of squares to conduct analysis of variance, calculate expected mean squares, and test hypotheses about treatment means. Relative efficiencies of designs are also covered.
3. The document explains how to interpret interactions in factorial designs using profile plots and presents tables for analysis of variance for each design. It also discusses estimating treatment differences and multiple comparison procedures.
The document discusses goodness-of-fit tests for categorical data. It introduces notation for categorical variables with multiple categories and hypotheses for goodness-of-fit tests. Expected counts are calculated based on hypothesized proportions. The chi-square statistic is used to calculate test statistics and P-values are found using the chi-square distribution. Examples demonstrate applying goodness-of-fit tests to determine if variable categories occur with equal frequency.
This document provides an overview of two-way analysis of variance (ANOVA). It explains that two-way ANOVA involves two categorical independent variables and one continuous dependent variable. The document outlines the objectives of two-way ANOVA, which are to analyze interactions between the two factors, and evaluate the effects of each factor. It then provides examples of how to set up and perform two-way ANOVA calculations and interpretations.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
The document discusses four nonparametric statistical tests:
1. The Wilcoxon Rank Sum Test (also called the Mann-Whitney U Test) compares the medians of two independent samples and is an alternative to the independent t-test.
2. The Wilcoxon Signed Rank Test compares the medians of two dependent/paired samples and is an alternative to the paired t-test. It calculates the differences between pairs, ranks their absolute values, and sums the ranks of positive differences.
3. The Kruskal-Wallis Test compares more than two independent samples.
4. The Runs Test examines the randomness of a single sample by counting runs, or streaks, of
The document describes a randomized complete block design (RCBD) experimental method. RCBD involves comparing treatments (e.g. fertilizers) applied to experimental units (e.g. corn crops) grouped into blocks (e.g. fields). Treatments are randomly assigned to experimental units within each block. RCBD controls for variability between blocks (e.g. differences in soil between fields) to isolate the effect of treatments. It provides more precise results than a completely randomized design when blocks are homogeneous within and heterogeneous between.
The Chi Square Test is used to determine if observed data fits a hypothesized distribution. It involves calculating the Chi Square statistic by comparing observed and expected values and interpreting the result using a Chi Square table. The document provides an example using Drosophila genetics to test if two traits are independently assorting. The null hypothesis is that the traits are independently assorting. Expected values are calculated based on this. The Chi Square value is found to be not statistically significant, so the null hypothesis that the traits are independently assorting is not rejected.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
InnerSoft STATS - Methods and formulas helpInnerSoft
The document defines several statistical formulas for calculating measures such as mean, variance, skewness, and kurtosis from sample data. It also provides formulas for statistical tests like chi-square, Fisher's exact test, McNemar's test, and odds ratios. Sources are cited for each formula, typically a Wikipedia page on the given statistical measure or test.
The document provides an overview of the chi-square test, including its formula, steps to calculate it, degrees of freedom, and uses. The chi-square test is a statistical test used to compare observed data to expected data. Its formula adds up the squared differences between observed and expected frequencies, divided by the expected frequencies. Degrees of freedom depend on whether the data is in a row/column or contingency table. The chi-square test can test goodness of fit, independence of attributes, and homogeneity.
The document summarizes the key aspects of a Completely Randomized Design (CRD) experiment. It defines a CRD as an experimental design where treatments are randomly assigned to experimental units, giving each unit an equal chance of receiving each treatment. The summary describes some advantages as easy implementation and flexibility, and a disadvantage as not controlling for variation among units. It also outlines the statistical analysis of a CRD using an ANOVA table to partition total variation into treatment and error components.
The chi-square test is used to determine if experimental data fits the results expected from genetic theory. It involves calculating an expected and observed count for each phenotype, then using a chi-square formula to determine how well the observed fits the expected. The result is compared to critical chi-square values from a table based on degrees of freedom and probability to determine if the null hypothesis should be rejected or failed to be rejected.
The document describes the Wilcoxon Rank-Sum Test, a non-parametric statistical hypothesis test used to assess whether one of two independent samples of observations tends to have larger values than the other when normality cannot be assumed. It provides details on running the test, including ranking the combined observations and computing the test statistic to determine if it is less than or equal to the critical value, rejecting the null hypothesis. An example applies the test to compare the nicotine content of two cigarette brands, finding no significant difference between their medians.
This document provides an overview of key concepts in statistics that will be covered in the CHM 235 course, including:
- The normal distribution and how it relates to sampling from populations. Parameters like the mean, standard deviation, and normal curve shape sampling distributions.
- Common statistical tests like confidence intervals, comparing a measured value to a known value, and comparing means of two data sets using t-tests. These tests rely on assumptions of normal distributions and comparing calculated t values to statistical tables.
- Additional concepts like variance, relative standard deviation, average deviation, and F-tests to compare standard deviations before applying t-tests. An example takes the reader through each of these statistical calculations and tests.
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
The document discusses hypothesis testing methods for comparing two population or treatment means. It covers notation, sampling distributions, large sample hypothesis testing, confidence intervals, and paired t-tests. An example compares the mean fill volumes of two beer can filling machines and constructs a 98% confidence interval for the difference in tensile strengths of two thread types.
This document provides guidelines for carrying out statistical analyses in SPSS and R using various datasets. It discusses how to replicate analyses from 2x2 tables using individual level data, and how to perform tests such as the Kappa test, McNemar's test, chi-square tests, tests for independent proportions, Fisher's exact test, Levene's test, Wilcoxon signed-rank tests, Mann-Whitney U tests, t-tests, and Q-Q plots in both SPSS and R. Instructions are provided for reading in SPSS data files into R and accessing variable values.
The document discusses techniques for reducing the number of test cases needed when the number of input variables and their possible values results in too many combinations to test exhaustively. It describes 1-wise and 2-wise testing. 1-wise testing covers each unique value individually, reducing the test cases for three variables with 4, 3, and 2 values from 24 to 9 by reusing test cases. 2-wise testing covers all pairs of variable values, further reducing the number of test cases needed but providing better coverage than 1-wise alone. The optimal approach depends on the specific variables and goals of the testing.
The Chi Square Test is a widely used non-parametric test that does not rely on assumptions about population parameters. It compares observed frequencies to expected frequencies specified by the null hypothesis. The Chi Square value is calculated by summing the squared differences between observed and expected values divided by the expected values. The Chi Square value is then compared to a critical value based on the degrees of freedom. Common applications include tests of goodness of fit, independence of variables, and homogeneity of proportions.
The document discusses parametric and non-parametric statistical tests. It defines parametric tests as those that make assumptions about the population distribution, such as assuming a normal distribution. Non-parametric tests make fewer assumptions. Specific tests covered include the chi-square test, run test, sign test, Kolmogorov-Smirnov test, Cochran Q test, and Friedman F test. Examples are provided for several of the tests.
- Chi-squared (χ2) tests are used to analyze categorical data by comparing observed frequencies to expected frequencies.
- The document provides examples of using the χ2 test to test if frequency distributions match expected distributions (goodness of fit test) and to test for associations between two categorical variables.
- Key assumptions and considerations for χ2 tests are that observations must be independent and expected frequencies should not be too small.
This document discusses analysis of variance (ANOVA) and experimental designs, including complete randomized design (CRD), randomized complete block design (RCBD), and Latin square design (LSD). It provides details on the procedures for ANOVA calculations for one-way and two-way classifications and outlines the advantages and limitations of different experimental designs. The key steps in layout and analysis of a CRD are also demonstrated with an example.
The document discusses hypothesis testing and statistical methods. It defines key terms like the null hypothesis, alternative hypothesis, dependent and independent variables. It then outlines the steps in hypothesis testing which include stating the hypotheses, selecting a test statistic, determining confidence levels, computing and plotting the test statistic to make conclusions. It discusses using p-values and provides an example comparing blood pressure before and after exercise. It also explains how to compare means for one and two independent groups as well as two dependent groups using t-tests.
Test of significance (t-test, proportion test, chi-square test)Ramnath Takiar
The presentation discusses the concept of test of significance including the test of significance examples of t-test, proportion test and chi-square test.
This document provides an overview of key concepts in statistics for quantitative analysis, including:
- Statistics are mathematical tools used to describe and make judgments about data. The type of statistics discussed assumes data has a normal (bell-shaped) distribution.
- The normal distribution is characterized by a mean (μ) and standard deviation (σ or s). Standard deviation quantifies the spread of data around the mean.
- Common statistical tests covered include confidence intervals, comparing a measured value to a known value using a t-test, and comparing means of two data sets using an F-test and t-test.
- The F-test determines if the standard deviations of two data sets are significantly different before using
The document describes three experimental designs: Complete Randomized Design (CRD), Randomized Block Design (RBD), and Latin Square Design (LSD). CRD randomly assigns treatments to experimental units and is best for homogeneous materials like lab experiments. RBD divides the field into homogeneous blocks to control local variability and is more precise than CRD. LSD arranges treatments in a square layout with equal rows and columns to control variability in two directions and is more efficient than RBD for a limited number of treatments.
The document provides an overview of statistical hypothesis testing and various statistical tests used to analyze quantitative and qualitative data. It discusses types of data, key terms like null hypothesis and p-value. It then outlines the steps in hypothesis testing and describes different tests of significance including standard error of difference between proportions, chi-square test, student's t-test, paired t-test, and ANOVA. Examples are provided to demonstrate how to apply these statistical tests to determine if differences observed in sample data are statistically significant.
This document presents an ordered probability model for predicting outdoor thermal comfort distributions.
1) Data on air temperature, humidity, wind speed, radiation, metabolic rate, clothing values, and thermal sensation votes (TSVs) were collected from 1,549 subjects in a park in Tianjin, China.
2) An ordered probability model is developed using an auxiliary variable z that is a function of the thermal comfort stimuli and coefficients. Different values of z correspond to different TSV categories with probabilities calculated using the normal cumulative distribution function.
3) The coefficients and threshold values linking z to TSV categories were estimated using NLOGIT. Model predictions matched well with surveyed TSV distributions for a range of temperature conditions
The Chi Square Test is used to determine if observed data fits a hypothesized distribution. It involves calculating the Chi Square statistic by comparing observed and expected values and interpreting the result using a Chi Square table. The document provides an example using Drosophila genetics to test if two traits are independently assorting. The null hypothesis is that the traits are independently assorting. Expected values are calculated based on this. The Chi Square value is found to be not statistically significant, so the null hypothesis that the traits are independently assorting is not rejected.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
InnerSoft STATS - Methods and formulas helpInnerSoft
The document defines several statistical formulas for calculating measures such as mean, variance, skewness, and kurtosis from sample data. It also provides formulas for statistical tests like chi-square, Fisher's exact test, McNemar's test, and odds ratios. Sources are cited for each formula, typically a Wikipedia page on the given statistical measure or test.
The document provides an overview of the chi-square test, including its formula, steps to calculate it, degrees of freedom, and uses. The chi-square test is a statistical test used to compare observed data to expected data. Its formula adds up the squared differences between observed and expected frequencies, divided by the expected frequencies. Degrees of freedom depend on whether the data is in a row/column or contingency table. The chi-square test can test goodness of fit, independence of attributes, and homogeneity.
The document summarizes the key aspects of a Completely Randomized Design (CRD) experiment. It defines a CRD as an experimental design where treatments are randomly assigned to experimental units, giving each unit an equal chance of receiving each treatment. The summary describes some advantages as easy implementation and flexibility, and a disadvantage as not controlling for variation among units. It also outlines the statistical analysis of a CRD using an ANOVA table to partition total variation into treatment and error components.
The chi-square test is used to determine if experimental data fits the results expected from genetic theory. It involves calculating an expected and observed count for each phenotype, then using a chi-square formula to determine how well the observed fits the expected. The result is compared to critical chi-square values from a table based on degrees of freedom and probability to determine if the null hypothesis should be rejected or failed to be rejected.
The document describes the Wilcoxon Rank-Sum Test, a non-parametric statistical hypothesis test used to assess whether one of two independent samples of observations tends to have larger values than the other when normality cannot be assumed. It provides details on running the test, including ranking the combined observations and computing the test statistic to determine if it is less than or equal to the critical value, rejecting the null hypothesis. An example applies the test to compare the nicotine content of two cigarette brands, finding no significant difference between their medians.
This document provides an overview of key concepts in statistics that will be covered in the CHM 235 course, including:
- The normal distribution and how it relates to sampling from populations. Parameters like the mean, standard deviation, and normal curve shape sampling distributions.
- Common statistical tests like confidence intervals, comparing a measured value to a known value, and comparing means of two data sets using t-tests. These tests rely on assumptions of normal distributions and comparing calculated t values to statistical tables.
- Additional concepts like variance, relative standard deviation, average deviation, and F-tests to compare standard deviations before applying t-tests. An example takes the reader through each of these statistical calculations and tests.
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
The document discusses hypothesis testing methods for comparing two population or treatment means. It covers notation, sampling distributions, large sample hypothesis testing, confidence intervals, and paired t-tests. An example compares the mean fill volumes of two beer can filling machines and constructs a 98% confidence interval for the difference in tensile strengths of two thread types.
This document provides guidelines for carrying out statistical analyses in SPSS and R using various datasets. It discusses how to replicate analyses from 2x2 tables using individual level data, and how to perform tests such as the Kappa test, McNemar's test, chi-square tests, tests for independent proportions, Fisher's exact test, Levene's test, Wilcoxon signed-rank tests, Mann-Whitney U tests, t-tests, and Q-Q plots in both SPSS and R. Instructions are provided for reading in SPSS data files into R and accessing variable values.
The document discusses techniques for reducing the number of test cases needed when the number of input variables and their possible values results in too many combinations to test exhaustively. It describes 1-wise and 2-wise testing. 1-wise testing covers each unique value individually, reducing the test cases for three variables with 4, 3, and 2 values from 24 to 9 by reusing test cases. 2-wise testing covers all pairs of variable values, further reducing the number of test cases needed but providing better coverage than 1-wise alone. The optimal approach depends on the specific variables and goals of the testing.
The Chi Square Test is a widely used non-parametric test that does not rely on assumptions about population parameters. It compares observed frequencies to expected frequencies specified by the null hypothesis. The Chi Square value is calculated by summing the squared differences between observed and expected values divided by the expected values. The Chi Square value is then compared to a critical value based on the degrees of freedom. Common applications include tests of goodness of fit, independence of variables, and homogeneity of proportions.
The document discusses parametric and non-parametric statistical tests. It defines parametric tests as those that make assumptions about the population distribution, such as assuming a normal distribution. Non-parametric tests make fewer assumptions. Specific tests covered include the chi-square test, run test, sign test, Kolmogorov-Smirnov test, Cochran Q test, and Friedman F test. Examples are provided for several of the tests.
- Chi-squared (χ2) tests are used to analyze categorical data by comparing observed frequencies to expected frequencies.
- The document provides examples of using the χ2 test to test if frequency distributions match expected distributions (goodness of fit test) and to test for associations between two categorical variables.
- Key assumptions and considerations for χ2 tests are that observations must be independent and expected frequencies should not be too small.
This document discusses analysis of variance (ANOVA) and experimental designs, including complete randomized design (CRD), randomized complete block design (RCBD), and Latin square design (LSD). It provides details on the procedures for ANOVA calculations for one-way and two-way classifications and outlines the advantages and limitations of different experimental designs. The key steps in layout and analysis of a CRD are also demonstrated with an example.
The document discusses hypothesis testing and statistical methods. It defines key terms like the null hypothesis, alternative hypothesis, dependent and independent variables. It then outlines the steps in hypothesis testing which include stating the hypotheses, selecting a test statistic, determining confidence levels, computing and plotting the test statistic to make conclusions. It discusses using p-values and provides an example comparing blood pressure before and after exercise. It also explains how to compare means for one and two independent groups as well as two dependent groups using t-tests.
Test of significance (t-test, proportion test, chi-square test)Ramnath Takiar
The presentation discusses the concept of test of significance including the test of significance examples of t-test, proportion test and chi-square test.
This document provides an overview of key concepts in statistics for quantitative analysis, including:
- Statistics are mathematical tools used to describe and make judgments about data. The type of statistics discussed assumes data has a normal (bell-shaped) distribution.
- The normal distribution is characterized by a mean (μ) and standard deviation (σ or s). Standard deviation quantifies the spread of data around the mean.
- Common statistical tests covered include confidence intervals, comparing a measured value to a known value using a t-test, and comparing means of two data sets using an F-test and t-test.
- The F-test determines if the standard deviations of two data sets are significantly different before using
The document describes three experimental designs: Complete Randomized Design (CRD), Randomized Block Design (RBD), and Latin Square Design (LSD). CRD randomly assigns treatments to experimental units and is best for homogeneous materials like lab experiments. RBD divides the field into homogeneous blocks to control local variability and is more precise than CRD. LSD arranges treatments in a square layout with equal rows and columns to control variability in two directions and is more efficient than RBD for a limited number of treatments.
The document provides an overview of statistical hypothesis testing and various statistical tests used to analyze quantitative and qualitative data. It discusses types of data, key terms like null hypothesis and p-value. It then outlines the steps in hypothesis testing and describes different tests of significance including standard error of difference between proportions, chi-square test, student's t-test, paired t-test, and ANOVA. Examples are provided to demonstrate how to apply these statistical tests to determine if differences observed in sample data are statistically significant.
This document presents an ordered probability model for predicting outdoor thermal comfort distributions.
1) Data on air temperature, humidity, wind speed, radiation, metabolic rate, clothing values, and thermal sensation votes (TSVs) were collected from 1,549 subjects in a park in Tianjin, China.
2) An ordered probability model is developed using an auxiliary variable z that is a function of the thermal comfort stimuli and coefficients. Different values of z correspond to different TSV categories with probabilities calculated using the normal cumulative distribution function.
3) The coefficients and threshold values linking z to TSV categories were estimated using NLOGIT. Model predictions matched well with surveyed TSV distributions for a range of temperature conditions
I am Luke M. I love exploring new topics. Academic writing seemed an interesting option for me. After working for many years with statisticsassignmentexperts.com. I have assisted many students with their assignments. I can proudly say, each student I have served is happy with the quality of the solution that I have provided. I have acquired my Master’s Degree in Statistics, from Arizona University, United States.
This document outlines key concepts for designing and conducting effective biology experiments, including formulating a testable hypothesis, identifying independent and dependent variables, controlling other factors, collecting precise measurements, analyzing sources of error, interpreting results, and drawing valid conclusions. Key aspects are designing experiments to test hypotheses, minimizing random and identifying systematic errors to improve reliability and accuracy, and repeating experiments to verify findings.
This document provides information about statistical tests that can be used to make inferences when comparing two samples or populations. Specifically, it discusses:
- Tests for comparing two proportions, means, variances or standard deviations from independent and dependent samples using z-tests, t-tests and F-tests.
- The assumptions and procedures for each test, including how to determine critical values and calculate test statistics.
- Examples of how to perform hypothesis tests and construct confidence intervals for various statistical comparisons between two samples or populations using a TI calculator.
DESIGN OF EXPERIMENTS (DOE)
DOE is invented by Sir Ronald Fisher in 1920’s and 1930’s.
The following designs of experiments will be usually followed:
Completely randomised design(CRD)
Randomised complete block design(RCBD)
Latin square design(LSD)
Factorial design or experiment
Confounding
Split and strip plot design
FACTORIAL DESIGN
When a several factors are investigated simultaneously in a single experiment such experiments are known as factorial experiments. Though it is not an experimental design, indeed any of the designs may be used for factorial experiments.
For example, the yield of a product depends on the particular type of synthetic substance used and also on the type of chemical used.
ADVANTAGES OF FACTORIAL DESIGN.
Factorial experiments are advantageous to study the combined effect of two or more factors simultaneously and analyze their interrelationships. Such factorial experiments are economic in nature and provide a lot of relevant information about the phenomenon under study. It also increases the efficiency of the experiment.
It is an advantageous because a wide range of factor combination are used. This will give us an idea to predict about what will happen when two or more factors are used in combination.
DISADVANTAGES
It is disadvantageous because the execution of the experiment and the statistical analysis becomes more complex when several treatments combinations or factors are involved simultaneously.
It is also disadvantageous in cases where may not be interested in certain treatment combinations but we are forced to include them in the experiment. This will lead to wastage of time and also the experimental material.
2(square) FACTORIAL EXPERIMENT
A special set of factorial experiment consist of experiments in which all factors have 2 levels such experiments are referred to generally as 2n factorials.
If there are four factors each at two levels the experiment is known as 2x2x2x2 or 24 factorial experiment. On the other hand if there are 2 factors each with 3 levels the experiment is known as 3x3 or 32 factorial experiment. In general if there are n factors each with p levels then it is known as pn factorial experiment.
The calculation of the sum of squares is as follows:
Correction factor (CF) = (𝐺𝑇)2/𝑛
GT = grand total
n = total no of observations
Total sum of squares = ∑▒〖𝑥2−𝐶𝐹〗
Replication sum of squares (RSS) = ((𝑅1)2+(𝑅2)2+…+(𝑅𝑛)2)/𝑛 - CF
Or
1/𝑛 ∑▒𝑅2−𝐶𝐹
2(Cube) FACTORIAL DESIGN
In this type of design, one independent variable has 2 levels, and the other independent variable has 3 levels.
Estimating the effect:
In a factorial design the main effect of an independent variable is its overall effect averaged across all other independent variable.
Effect of a factor A is the average of the runs where A is at the high level minus the average of the runs
This document provides information about parametric statistics tests. It defines parametric tests as those applied to normally distributed interval or ratio data. The main parametric tests discussed are t-tests, z-tests, ANOVA, correlation, and regression. It explains that parametric tests are used when the data are normally distributed and measured on an interval or ratio scale. Parametric tests are more powerful than nonparametric alternatives. The document provides details on how to determine if data meet the assumptions for parametric tests and how to perform t-tests, including the steps and formulas for independent and correlated samples t-tests. It also defines interval and ratio levels of measurement.
- Measurements were taken of the length, width, size, and mass of 60 eggs to develop a model predicting mass from the other variables.
- Initial models using just length and width or adding size did not meet assumptions of constant variance.
- The final adequate model included length, width, and a calculated volume variable, meeting assumptions and performing well on cross-validation.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.4: Two Variances or Standard Deviations
This document introduces parametric tests and provides information about the t-test. It defines parametric tests as those applied to normally distributed data measured on interval or ratio scales. Parametric tests make inferences about the parameters of the probability distribution from which the sample data were drawn. Examples of common parametric tests are provided, including the t-test. The t-test is used to compare two means from independent samples or correlated samples. Steps for conducting a t-test are outlined, including calculating the t-statistic and making decisions based on critical t-values. Two examples of using a t-test on experimental data are shown.
The document discusses statistical tests such as the t-test and F-test. The t-test is used to compare means of two samples, such as comparing sample means before and after treatment. There are different types of t-tests, including paired samples and independent samples t-tests. The F-test, also called the F-ratio, compares variances between samples and is used in analysis of variance (ANOVA) to test differences between two or more groups. Examples are provided to demonstrate how to perform t-tests and F-tests to analyze data and test hypotheses.
1) The document presents information on different types of t-tests including the single sample t-test, independent sample t-test, and dependent/paired sample t-test. Equations and examples are provided for each.
2) The single sample t-test compares the mean of a sample to a hypothesized population mean. The independent t-test compares the means of two independent samples. The dependent t-test compares the means of two related samples, such as pre-and post-test scores.
3) A z-test is also discussed and compared to t-tests. The z-test is used when the population standard deviation is known and sample sizes are large, while t-tests are used
1. The document discusses Chi-square tests for categorical variables, including tests for goodness of fit, comparison of proportions between independent samples, and contingency tables.
2. Examples are provided to illustrate Chi-square tests for independence between binary variables, comparison of proportions between paired samples using McNemar's test, and tests on R×C contingency tables.
3. Pre-requisites for Chi-square tests are discussed, including minimum expected cell frequencies and using Fisher's exact test for small sample sizes.
The document discusses categorical data analysis and various statistical tests that can be used to analyze relationships between categorical variables. It defines categorical variables as those that can take on nominal or ordinal levels. Contingency tables are used to represent combinations of levels between an explanatory and response categorical variable. Several statistical tests are introduced, including chi-square tests to assess associations between categorical variables, relative risk/odds ratios to compare outcomes between groups, and McNemar's test for paired categorical data. Examples are provided to demonstrate calculation and interpretation of these tests.
1. The document discusses hypothesis testing, including defining the null and alternative hypotheses, types of errors, test statistics, and testing differences between population means and differences between two samples.
2. Examples are provided to demonstrate hypothesis testing for one and two sample means. This includes stating the hypotheses, significance level, test statistic, critical region, and conclusion.
3. Assignments are given applying hypothesis testing to compare lung destruction between smokers and non-smokers, serum complement activity between disease and normal subjects, and podiatric problems between elderly diabetic and non-diabetic patients.
1. The document discusses Chi-square tests for categorical variables, including tests for goodness of fit, comparing proportions between independent samples, and contingency tables.
2. Examples are provided to illustrate Chi-square tests for comparing the results of two medical testing methods and evaluating whether patient outcomes are independent of a treatment.
3. Requirements for using Chi-square tests are outlined, along with alternative tests like Fisher's exact test that are used when Chi-square assumptions are not met.
1. The document discusses categorical data analysis and goodness-of-fit tests. It introduces concepts such as univariate categorical data, expected counts, the chi-square test statistic, and assumptions of the chi-square test.
2. An example analyzes faculty status data from a university using a goodness-of-fit test to determine if the proportions are equal across categories. The test fails to reject the null hypothesis that the proportions are equal.
3. Tests for homogeneity and independence in two-way tables are described. Examples calculate expected counts and perform chi-square tests to compare populations' category proportions.
- The document outlines the concept of robust design and tools like Taguchi method to achieve it. Robust design aims to reduce variability and optimize performance even under extreme conditions.
- Taguchi method is described as a tool for robust design involving orthogonal arrays to perform experiments efficiently and analyze results using signal-to-noise ratios to find optimal settings.
- An example applies Taguchi method to optimize a CVD process for silicon wafer production by selecting factor levels for temperature, pressure, time and cleaning method that minimize surface defects based on experiment results.
BIOSTATISTICS MEAN MEDIAN MODE SEMESTER 8 AND M PHARMACY BIOSTATISTICS.pptxPayaamvohra1
1. The document provides information about biostatistics including measures of central tendency, dispersion, correlation, and regression. It defines terms like mean, median, mode, range, and standard deviation.
2. Examples of calculating mean, median, and mode from individual data sets, grouped frequency distributions, and continuous series are shown step-by-step.
3. Parametric tests like t-test, ANOVA, and tests of significance are also introduced. Overall, the document covers fundamental concepts in biostatistics through examples.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation
Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Travis Hills of MN is Making Clean Water Accessible to All Through High Flux ...Travis Hills MN
By harnessing the power of High Flux Vacuum Membrane Distillation, Travis Hills from MN envisions a future where clean and safe drinking water is accessible to all, regardless of geographical location or economic status.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
2. Introduction
Staphylococcus aureus is a bacterium, commonly found on skin and in the
respiratory tract, that can cause ailments such as skin infections and respiratory
diseases.
Like other bacteria, Staphylococcus aureus can be grown in medical laboratories
to aid in identifying and treating skin conditions.
Poor growth rates of Methicillin resistant Staphylococcus aureus (MRSA) in one
laboratory prompted the investigators to experiment with different culturing
conditions.
Five strains of MRSA were examined in this experiment. Due to their complex
names, they are referred to as 1, 2, 3, 4, and 5 in the data.
3. Data Description
The tryptone dataset contains bacteria counts after the culturing of five strains
of Staphylococcus aureus.
The data was collected by Gavin Cooper at the Auckland University of
Technology, New Zealand. The full dataset:
http://www.amstat.org/publications/jse/datasets/Tryptone.dat.txt
No missing values.
Tests on (a) factorial models with interactions to identify significant factors,
(b) optimal conditions estimated by partial differentiation.
4. Data Description
Treatments:
Time - In hours: 24 and 48
Temperature - Temperature of incubation in degrees Celcius: 27, 35, 43
Concentration - The concentration of the nutrient tryptone as a
percentage: 0.6, 0.8, 1.0, 1.2, 1.4
Block:
Count column - Five count columns: 1, 2, 3, 4, 5
Redundant variable:
Row - this is the case number
Response (dependent) variable:
Strain counts - Bacteria counts: 3 to 284
5. Data Management
Data transformation
The original dataset shows aspects of both multivariate data, where the count
column variable is arranged in columns, and univariate data, where the levels of
the time, temperature and concentration variables respectively are listed in three
columns.
Row Count1 Count2 Count3 Count4 Count5 Time Temp Conc
1 9 3 10 14 33 24 27 0.6
2 16 12 26 20 31 24 27 0.8
Strain counts, which are analyzed in a univariate procedure, are recorded in
different count columns: they must be placed in a single column. The count
column variable should be in its own single column as well.
Data was transformed by SAS code:
Input row count1 count2 count3 count4 count5 time temp conc;
column = 1; count = count1; output strain;
column = 2; count = count2; output strain;
The new dataset:
The new dataset strain and the complete SAS code are in the output files.
Obs time temp conc column count
1 24 27 0.6 1 9
2 24 27 0.6 2 3
3 24 27 0.6 3 10
6. Data Management
Balance check:
When fixing the treatment “time”, the tables below demonstrate that all 12
combinations of the other two treatments exist, and that the frequency of
replicates in each combination is the same.
Similarly, when fixing variable concentration or temperature, the frequency
tables show that the experiment is balanced. (These results are shown in the
output files.)
α = 0.05 is used for the entire analysis.
Table 1 of temp by conc
Controlling for time=24
temp conc
Frequency 0.6 0.8 1 1.2 1.4 Total
27 5 5 5 5 5 25
35 5 5 5 5 5 25
43 5 5 5 5 5 25
Total 15 15 15 15 15 75
Table 2 of temp by conc
Controlling for time=48
temp conc
Frequency 0.6 0.8 1 1.2 1.4 Total
27 5 5 5 5 5 25
35 5 5 5 5 5 25
43 5 5 5 5 5 25
Total 15 15 15 15 15 75
7. Data Summary
Differences in means? Symmetric data? Homogeneous variances?
Figures below (left to right): distribution of count by time, temperature and
concentration.
First impressions from the box plots:
In each treatment, means at different levels are quite different.
In temperature treatments, the data is less symmetric, so possibly not normal.
The other two treatments looks more symmetric.
In each treatment, the variances may not be equal to each other.
8. Method Description
Step 1: Test on factorial models with interactions to identify significant factors.
ANOVA test on factorial RBD, full model:
The variances are separated.
ANOVA test on factorial RBD, reduced model:
Homogeneous variance is assumed and the variance is pooled.
Step 2: Test for optimal conditions estimated by partial differentiation.
Multiple polynomial regression
The current protocols for culturing this bacteria have the time at 24 hours, the
temperature at 35 degrees Celsius and the tryptone concentration at 1.0%.
9. Step 1: Test on factorial models with
interactions to identify significant factors
Full model vs. reduced model: which one is better?
Fit Statistics
-2 Res Log Likelihood 1107.3
AIC (Smaller is Better) 1169.3
AICC (Smaller is Better) 1191.9
BIC (Smaller is Better) 1157.2
Full model: Reduced
model:
Fit Statistics
-2 Res Log Likelihood 1148.5
AIC (Smaller is Better) 1152.5
AICC (Smaller is Better) 1152.6
BIC (Smaller is Better) 1151.7
The reduced model has the smaller AIC value, which indicates that it is the better
model.
The sources of variation and degrees of freedom:
Assumptions: Independence, normal distribution of residuals, homogeneity of
variances
Source degrees of freedom d.f.
Tmt1 (Time) t1-1 1
Tmt2 (Temperature) t2-1 2
Tmt3 (Concentration) t3-1 4
Block (Count column) b-1 4
Interaction1 (Tmt1 * Tmt2) (t1-1)(t2-1) 2
Interaction2 (Tmt1 * Tmt3) (t1-1)(t3-1) 4
Interaction3 (Tmt1 * Tmt2) (t2-1)(t3-1) 8
Interaction4 (Tmt1 * Tmt2 * Tmt3) (t1-1)(t2-1)(t3-1) 8
Experimental Error (b-1)[(t1-1) + (t2-1) + (t3-1) (t1-1)(t2-1) + (t1-1)(t2-1)
+ (t1-1)(t2-1) + (t1-1)(t2-1)(t3-1)]
116
Total bt1t2t3-1 149
Block interactions are pooled into a single error term because of the assumption of no block interaction in RBD
10. ANOVA Test on factorial RBD, reduced
model
Yes, as p-values of all three treatments are <0.05, we reject H0: μ1 = μ2=…= μt in
each treatment.
According to the factorial RBD (reduced) model, do different levels in each
treatment have significantly different effects on strain counts?
Type 3 Tests of Fixed Effects
Effect
Num
DF
Den
DF F Value Pr > F
time 1 116 444.27 <.0001
temp 2 116 80.12 <.0001
conc 4 116 64.86 <.0001
Is there interaction between treatments?
Type 3 Tests of Fixed Effects
Effect
Num
DF
Den
DF F Value Pr > F
time*temp 2 116 38.07 <.0001
time*conc 4 116 3.99 0.0046
temp*conc 8 116 0.85 0.5613
time*temp*conc 8 116 2.17 0.0343
The hypothesis of no significant interaction effect between time & temp was rejected.
The hypothesis of no significant interaction effect between time & conc was rejected.
The hypothesis of no significant interaction effect between temp & conc was NOT
rejected.
The hypothesis of no significant interaction effect between three treatments was
11. ANOVA Test on factorial RBD, reduced
model
Saxton’s Macro was applied to do a range test with the LSMeans output. e.g.:
Least Squares Means table gives the least squares estimate, the standard error of
the estimate, etc.:
Which pairs of means in the one treatment are different, at a certain condition of
other treatment levels?
Pairwise comparisons with TUKEY adjustments are shown in the “Differences
of Least Squares Means” table.
Least Squares Means
Effect time temp conc Estimate
Standard
Error DF t Value Pr > |t| Alpha Lower Upper
time 24 82.2800 3.4399 116 23.92 <.0001 0.05 75.4668 89.0932
time 48 162.75 3.4399 116 47.31 <.0001 0.05 155.93 169.56
temp 27 91.1200 3.9340 116 23.16 <.0001 0.05 83.3281 98.9119
Obs time temp conc Estimate
Standard
Error Alpha Lower Upper
Letter
Group
1 48 _ _ 162.75 3.4399 0.05 155.93 169.56 A
2 24 _ _ 82.2800 3.4399 0.05 75.4668 89.0932 B
Effect=time Method=Tukey-Kramer(P<0.05) Set=1
The complete tables mentioned above are available in the output file.
12. ANOVA Test on factorial RBD, reduced
model
Last part of the ANOVA is testing the hypothesis of normality:
P-value >0.05, so we fail to reject the hypothesis of normality in the residual
distribution.
Contrasts to test linear/curved trend
Temperature and concentration treatments are quantitative and equally spaced,
having 3 levels and 5 levels respectively. (Time has only 2 levels)
The results of the contrasts indicate that both linear and curved models can fit the
data.
Contrasts
Label
Num
DF
Den
DF F Value Pr > F
linear 1 116 57.32 <.0001
quadratic 1 116 102.93 <.0001
linear 1 116 189.36 <.0001
quadratic 1 116 19.69 <.0001
cubic 1 116 32.80 <.0001
quartic 1 116 17.59 <.0001
First two rows are test results for the treatment Temp.
Last four rows are test results for the treatment Conc.
Tests for Normality
Test Statistic p Value
Shapiro-Wilk W 0.988251 Pr < W 0.2392
Kolmogorov-Smirnov D 0.050081 Pr > D >0.1500
Cramer-von Mises W-Sq 0.040777 Pr > W-Sq >0.2500
Anderson-Darling A-Sq 0.333665 Pr > A-Sq >0.2500
13. Step 2: Test for optimal conditions estimated
by partial differentiation
Multiple polynomial regression
Three simple polynomial regressions are done separately, each treatment with
one polynomial regression.
Sequentially adjusted Type I SS were used to determine whether the
polynomial model is as good as the one with a higher order term.
Regression model:
Y = β0 + β1 Xi + β2 X2i +…+ βk Xki + ei
Based on the regression model, partial differentiation is used to determine the
optimal conditions. (Not displayed in this presentation.)
Also, the fit plots are useful in finding the maxima.
Assumptions: Independence, normal distribution of residuals, homogeneity of
variances
14. Polynomial regression with “Time”
Is the linear effect significant?
Fit plot (count vs time)
Time has only 2 levels, fit with a linear model.
Source DF Type I SS Mean Square F Value Pr > F
time 1 242808.1667 242808.1667 99.48 <.0001
Yes: p-value for linear <0.05, reject H0: β1 = 0.
15. Polynomial regression with “Time”
Polynomial regression model
Normality test: p-value <0.05, reject the hypothesis of normality
Tests for Normality
Test Statistic p Value
Shapiro-Wilk W 0.97416 Pr < W 0.0063
Kolmogorov-Smirnov D 0.064387 Pr > D 0.1302
Cramer-von Mises W-Sq 0.15658 Pr > W-Sq 0.0204
Anderson-Darling A-Sq 1.023263 Pr > A-Sq 0.0106
Parameter Estimate
Standard
Error t Value Pr > |t|
Intercept 1.813333333 12.75597911 0.14 0.8872
time 3.352777778 0.33614956 9.97 <.0001
Count = 1.813 + 3.352*Time
According to the regression model, the strain count increases with the time increase:
48 hours might get a higher strain count than 24 hours. The current protocol for
culturing this bacteria has the time at 24 hours, so the statistical results do NOT
support this protocol.
16. Polynomial regression with
“Temperature”
Is the quadratic effect significant?
Fit plot (count vs. temperature)
Temperature has 3 levels, so it is fit with a quadratic model.
Yes: p-value for quadratic <0.05, reject H0: β2 = 0.
Source DF Type I SS Mean Square F Value Pr > F
temp 1 31329.00000 31329.00000 8.92 0.0033
temp*temp 1 56252.21333 56252.21333 16.01 <.0001
17. Polynomial regression with
“Temperature”
Polynomial regression model
Normality test: p-value <0.05, so we reject the hypothesis of normality
Count = -713.834 + 47.144*Temp – 0.642*Temp2
According to the regression model, the strain count has a maximum at Temp = 35
degrees. The current protocol for culturing this bacteria has the temperature at 35
degrees, so the results support this protocol.
Tests for Normality
Test Statistic p Value
Shapiro-Wilk W 0.966924 Pr < W 0.0011
Kolmogorov-Smirnov D 0.067926 Pr > D 0.0888
Cramer-von Mises W-Sq 0.182315 Pr > W-Sq 0.0089
Anderson-Darling A-Sq 1.229754 Pr > A-Sq <0.0050
Parameter Estimate
Standard
Error t Value Pr > |t|
Intercept -713.8343750 191.4866910 -3.73 0.0003
temp 47.1437500 11.2532848 4.19 <.0001
temp*temp -0.6418750 0.1604124 -4.00 <.0001
18. Polynomial regression with
“Concentration”
Is the quartic effect significant?
Temperature has 5 levels, so we fit it with a quartic model.
No: p-value for quartic >0.05, do not reject H0: β4 = 0.
Source DF Type I SS Mean Square F Value Pr > F
conc 1 103490.6133 103490.6133 32.46 <.0001
conc*conc 1 10761.6095 10761.6095 3.38 0.0682
conc*conc*conc 1 17925.8700 17925.8700 5.62 0.0190
conc*conc*conc*conc 1 9612.8805 9612.8805 3.02 0.0846
19. Polynomial regression with
“Concentration”
Is the cubic effect significant?
Fit plot (count vs. concentration)
Now fit it with a cubic model.
Yes: p-value for quartic <0.05, reject H0: β3 = 0.
Source DF Type I SS Mean Square F Value Pr > F
conc 1 103490.6133 103490.6133 32.02 <.0001
conc*conc 1 10761.6095 10761.6095 3.33 0.0701
conc*conc*conc 1 17925.8700 17925.8700 5.55 0.0198
20. Polynomial regression with
“Concentration”
Polynomial regression model
Normality test: p-value <0.05, reject the hypothesis of normality
Count = 608.923 – 1960.155*Conc + 2289.077*Conc2 – 805.208*Conc3
According to the regression model, the strain count has a maximum at Conc = 1.2%.
The current protocol for culturing this bacteria has the concentration at 1.0%, so the
results do NOT support this protocol.
Tests for Normality
Test Statistic p Value
Shapiro-Wilk W 0.978016 Pr < W 0.0166
Kolmogorov-Smirnov D 0.069177 Pr > D 0.0787
Cramer-von Mises W-Sq 0.161641 Pr > W-Sq 0.0179
Anderson-Darling A-Sq 1.095717 Pr > A-Sq 0.0073
Parameter Estimate
Standard
Error t Value Pr > |t|
Intercept 608.922857 302.692620 2.01 0.0461
conc -1960.154762 989.107532 -1.98 0.0494
conc*conc 2289.077381 1028.037019 2.23 0.0275
conc*conc*conc -805.208333 341.898415 -2.36 0.0198
21. Conclusion
Polynomial regression models support the temperature in the current protocol for
culturing Staphylococcus aureus. However, the models do not support the time and
concentration in the protocol.
An ANOVA test on the factorial RBD was done, and the reduced model is better.
Different levels in each treatment have significantly different effects on strain counts.
There is a significant interaction effect between temperature & concentration. Other pair-
wise comparisons can be found in the output.
The polynomial regression models did not meet the assumption of normality
according to the Shapiro-Wilk criteria (although they do according to the
Kolmogorov-Smirnov criteria). This might make the data analysis less reliable.
22. Reference
“Using EDA, ANOVA and Regression to Optimize some Microbiology Data.”
Journal of Statistics Education, Volume 12, Number 2 (July 2004)
http://www.amstat.org/publications/jse/v12n2/datasets.binnie.html