# Post-hoc comparisons & two-way analysis of variance. Two-way ANOVA, II. Post-hoc testing for main effects. Post-hoc testing 9.

Size: px
Start display at page:

Download "Post-hoc comparisons & two-way analysis of variance. Two-way ANOVA, II. Post-hoc testing for main effects. Post-hoc testing 9."

Transcription

1 Two-way ANOVA, II Post-hoc comparisons & two-way analysis of variance 9.7 4/9/4 Post-hoc testing As before, you can perform post-hoc tests whenever there s a significant F But don t bother if it s a main effect and has only two levels you already know the answer We ll just talk about the Tukey s HSD procedure Requires that the n s in all levels of a factor are equal obt Post-hoc testing for main effects This is just like post-hoc testing for the oneway ANOVA

2 Post-hoc testing for main effects is just like what we did for one-way ANOVA MS HSD q wn α = α is the Type I error rate (.5). α n q α MS wn n Is a value from a table of the studentized range statistic based on alpha, df W, and k, the number of levels in the factor you are testing Is the mean square within groups. Is the number of people in each group. I.E. how many numbers did you average to get each mean you are comparing? Our example from last time What effect do a workbook and consumption have on exam performance? Both main effects and the interaction were significant Factor A (the workbook) had only two levels. post-hoc testing required. The workbook helps. Factor B (the ) had three levels. We need to do post-hoc testing. Numbers from our example last time wn = 5.56, 3, MS n = 6 45, 5, 85 q k is a function of df wn and k 3, 4, dfwn = k = 3 So, from the table, qk = 3.77 for α=.5, 45, 55 4, 6, 65 9, 85, 75 Σx = 8 n B = 6 Σx = 345 n B = 6 Σx = 34 n B3 = 6 HSD for this example HSD = qk sqrt(ms wn /n) = 3.77 sqrt(5.56/6) =.7 Differences in means: Level : cups Level : cup Level 3: cups m =3 m = 57.5 m 3 = cups of differ significantly from both and cups of

3 Post-hoc testing for the interaction Involves comparing cell means But we don t compare every possible pair of cell means m = m = 6 m = 3 A) m = 4 Confounded & unconfounded comparisons m = m = 6 m = 3 A) m = 4 Confounded comparison, because the cells differ along more than one factor. If there s a difference, what s the explanation? Is it because of factor A or B? We can t tell, because there s a confound. Confounded & unconfounded comparisons m = m = 6 m = 3 () () A) m = 4 Unconfounded comparisons. The cells differ only in one factor. We can test these with post-hoc tests. Tukey s HSD for interactions. Compute HSD = qk sqrt(ms wn /n) Before, qk was a function of dfwn and k, the number of levels in the factor of interest = # of means being compared For the interaction, we use an adjusted k to account for the actual number of unconfounded comparisons (as opposed to all comparisons of cell means, some of which are confounded). Compare with unconfounded differences in means 3

4 Table from the handout TABLE Values of Adjusted k Design of Number of Cell Adjusted Study Means in Study Value of k x 4 3 x x x x x x 5 Figure by MIT OCW. What s going on here? k is sort of short hand for the number of means you d like to compare In one-way ANOVA or main effects analysis, e.g: 3 means -> + = 3 comparisons 5 means -> = comparisons What s going on here? Two-way interactions x -> 4 comparisons, k=3 is closest te t all stat books bother with this adjusted value of k many just use k = # cell means x3 -> 9 comparisons, k=5 is closest 4

5 Back to our example We had a 3x design, so the adjusted value of k = 5. df wn =. So q k = 4.5 for α=.5 MSwn = 5.56, n = # in each mean = 3, so HSD = 4.5 sqrt(5.56/3) = What unconfounded comparisons lead to differences larger than 37.33? 4 3 A) m = m = 4 5 m = m = 3 A) m = m = 4 m = 6 m = 3 m = m = 6 m = 3 m=36.67 A) m = 4 m=59.44 m=3 m=57.5 m=56.7 m = m = 6 m = 3 m=36.67 A) m = 4 m=59.44 m=3 m=57.5 m=56.7 All significant effects shown (blue = interaction, green = main). What is the interpretation of these results? Interpretation:. If the interaction is not significant, interpretation is easy it s just about what s significant in the main effects. In this case, with no significant interaction, we could say that or cups of are significantly better than cups, and using the workbook is significantly better than not using it. 5

6 m = m = 6 m = 3 m=36.67 A) m = 4 m=59.44 m=3 m=57.5 m=56.7 m = m = 6 m = 3 m=36.67 A) m = 4 m=59.44 m=3 m=57.5 m=56.7 Interpretation: Interpretation:. However, if there is a significant interaction, then the main interpretation of the experiment has to do with the interaction. Would we still say that or cups of are better than? That using the workbook is better than not using it? NO. It depends on the level of the other factor. Increasing consumption improves exam scores, where without the workbook there s an improvement going from to cups, and with the workbook there s an improvement in going from to cups. The workbook leads to significant improvement in exam scores, but only for students drinking cups of. Within-subjects experimental design Within-subjects (one-way) ANOVA Also known as repeated-measures Instead of having a bunch of people each try out one tennis racket, so you can compare two kinds of racket (between-subjects), you instead have a bunch of people each try out both rackets (within-subjects) 6

7 Why within-subjects designs can be useful Subjects may differ in ways that influence the dependent variable, e.g. some may be better tennis players than others In a between-subjects design, these differences add to the noise in the experiment, i.e. they increase the variability we cannot account for by the independent variable. As a result, it can be more difficult to see a significant effect. In a within-subjects design, we can discount the variability due to subject differences, and thus perhaps improve the power of the significance test How to do a within-subjects ANOVA (and why we didn t cover it until now) A one-way within-subjects ANOVA looks an awful lot like the two-way ANOVA we did in (my) last lecture We just use a different measure for MS error, the denominator of our Fobt, and a corresponding different df error An example The data Factor A: Type of dress How does your style of dress affect your comfort level when you are acting as a greeter in a social situation? 3 styles of dress: casual, semiformal, and formal. 5 subjects. Each subject wears each style of dress, one on each of 3 days. Order is randomized. Comfort level is measured by a questionnaire Subj Subj Subj3 Subj4 Subj5 Casual Σx=5 Σx =33 Semiformal Σx=45 Σx =4 Formal Σx=6 Σx =66 Σx=7 Σx=4 Σx=6 Σx=7 Σx= Total: Σx=86 Σx =6 7

8 Two-way between-subjects vs. oneway within-subjects The table on the previous slide looks a lot like we re doing a two-way ANOVA, with subject as one of the factors However, cell (i, ) is not necessarily independent of cell (i, ) and cell (i, 3) Also, there is only, in this case, one data point per cell we can t calculate MSerror = MS wn the way we did with two-way ANOVA Sum of squared differences between the scores in each cell and the mean for that cell We have to estimate the error variance in some other way Error variance is the variation we can t explain by one of the other factors So it s clearly not variance in the data for the different levels of factor A, and it s not the variance in the data due to the different subjects We use as our estimate of the error variance the MS for the interaction between subject and factor A The difference between the cell means not accounted for by the main effects Steps. Compute SS A, as before (see other lectures for the equation) = 5 / /5 + 6 /5 86 /5 = Similarly, compute SSsubj = 7 /3 + 4 /3 + 6 /3 + 7 /3 + /3 86 /5 = Compute SStot as usual, = 6 86 /5 = 6.93 Steps 4. SS tot = SS A + SS subj + SS Axsubj -> SS Axsubj = SS tot SS A + SS subj = = Compute degrees of freedom: df A = k A = df Axsubj = (k A )(k subj ) = ()(4) = 8 8

9 Steps We are doing this to check whether there s a significant effect of factor A, so: 6. MS A = SS A /df A = 88.3/ = MSerror = MS Axsubj = SS Axsubj /df Axsubj = 3.87/8 = Compute F obt = MS A /MS error = Compare with Fcrit for df = (df A, df error ) = (, 8). In this case, we wont bother, because it s clearly significant. What if we had done this the between-subjects way? SS tot = 6.9, SS bn = SS A = 88.3 SS wn = SS tot SSbn = 8.79 df bn =, df wn = 5 3 = MSbn = 88.3/ = 44.7 MS wn = 8.79/ =.4 F obt = 8.37 Still, no doubt significant, but not as huge of an F value as before. The extent to which the within-subjects design will have more statistical power is a function of how dependent the samples are for the different conditions, for each subject Other two-sample parametric tests (Some of) what this course did not cover (This will not be on the exam; I just think it can be helpful to know what other sorts of tests are out there.) We talked about z- and t-tests for whether or not two means differ, assuming that the underlying distributions were approximately normal Recall that only two parameters are necessary to describe a normal distribution: mean and variance F-tests (which we used in ANOVA) can test whether the variances of two distributions differ significantly 9

10 Multiple regression and correlation We ve talked about regression and correlation, in which we looked at linear prediction of Y given X, and how much of the variance in Y is accounted for by X (or vice versa) Sometimes several X variables help us more accurately predict Y E.G. height and practise both affect a person s ability to shoot baskets in basketball This is like fitting a best fit plane instead of a best fit line Multiple regression and correlation Put another way, sometimes we want to know the strength of relationship between 3 or more variables If we want to simultaneously study how several X variables affect the Y variable, we use multiple regression & multiple correlation n-linear regression And, as mentioned, you can fit curves other than lines and planes to the data Correlation for ranked data To what extent do two rankings agree? Spearman s rank correlation coefficient, or Kendall s Tau

11 Other variants on ANOVA We talked about one-way and two-way between subjects ANOVA, and one-way within-subjects ANOVA You can, of course, also do two-way withinsubjects ANOVA, and n-way ANOVA (though this gets complicated to interpret after n>3) Designs can also be mixed-design, meaning some factors are within-subjects factors, and others are between-subjects factors And there are all sorts of other complications as well What if our data don t meet the requirements for ANOVA? Recall for t-tests we talked about what to do when the data violate the assumption that the two groups have equal variance we adjusted the degrees of freedom to account for this For ANOVA, there is a similar adjustment if the equivalent sphericity assumption is violated n-parametric procedures like t- tests The chi-square test was, in a sense, a nonparametric version of a t-test A t-test tested whether a mean differed from what was expected, or whether two means were significantly different A chi-square test tests whether cell values differ significantly from predicted, or whether two distributions were significantly different Other non-parametric procedures like t-tests The equivalent of a t-test for ranked data is either the Mann-Whitney U test, or the rank sums test The Wilcoxon t-test is a test for related samples of ranked data E.G. rank subjects reaction times on each of two tasks. Each subject participates in both tasks.

12 n-parametric version of ANOVA In addition, in some special cases there are parametric techniques like t-tests that assume some distribution other than a normal distribution Kruskal-Wallis H test Like a one-way, between-subjects ANOVA for ranked data Friedman χ test Like a one-way, within-subjects ANOVA for ranked data There are also more advanced techniques ANCOVA: ANalysis of COVAriance Provides a type of after-the-fact control for one or more variables that may have affected the dependent variable in an experiment The aim of this technique is to find out what the analysis of variance results might have been like if these variables had been held constant

13 More on ANCOVA Suppose factor A affects the response Y, but Y is also affected by a nuisance variable, X Ideally, you d have run your experiment so that groups for different levels of factor A all had the same value of X But sometimes this isn t feasible, for whatever reason, & under certain conditions you can use ANCOVA to adjust things after the fact, as if X had been held constant E.G. After the fact, adjust for the effects of intelligence on a training program n-parametric bootstrapping techniques Pulling yourself up by your bootstraps Use information gained in the experiment (e.g. about the distribution of the data) to create a nonparametric test that s basically designed for the distribution of your data These are basically Montecarlo techniques they involve estimating the distribution of the data, and then generating multiple samples of new data with that distribution ( Montecarlo techniques are what you did in MATLAB in the beginning of class) Why bootstrapping? Because now we can This is a recent technique (97 s), made feasible by computers It s conceptually and computationally simple The distribution assumptions depend upon the observed distribution of actual data, instead of upon large sample approximations like most of our parametric tests Why not: distribution estimates will change from one run of the experiment to another, and if the data does closely follow, say, a normal distribution, this technique will not do as well 3

### Research Methods & Experimental Design

Research Methods & Experimental Design 16.422 Human Supervisory Control April 2004 Research Methods Qualitative vs. quantitative Understanding the relationship between objectives (research question) and

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### SPSS Tests for Versions 9 to 13

SPSS Tests for Versions 9 to 13 Chapter 2 Descriptive Statistic (including median) Choose Analyze Descriptive statistics Frequencies... Click on variable(s) then press to move to into Variable(s): list

### Two-sample hypothesis testing, II 9.07 3/16/2004

Two-sample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For two-sample tests of the difference in mean, things get a little confusing, here,

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### Rank-Based Non-Parametric Tests

Rank-Based Non-Parametric Tests Reminder: Student Instructional Rating Surveys You have until May 8 th to fill out the student instructional rating surveys at https://sakai.rutgers.edu/portal/site/sirs

### INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

### Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

### Overview of Non-Parametric Statistics PRESENTER: ELAINE EISENBEISZ OWNER AND PRINCIPAL, OMEGA STATISTICS

Overview of Non-Parametric Statistics PRESENTER: ELAINE EISENBEISZ OWNER AND PRINCIPAL, OMEGA STATISTICS About Omega Statistics Private practice consultancy based in Southern California, Medical and Clinical

### Analysis of Variance ANOVA

Analysis of Variance ANOVA Overview We ve used the t -test to compare the means from two independent groups. Now we ve come to the final topic of the course: how to compare means from more than two populations.

### SPSS Explore procedure

SPSS Explore procedure One useful function in SPSS is the Explore procedure, which will produce histograms, boxplots, stem-and-leaf plots and extensive descriptive statistics. To run the Explore procedure,

### Projects Involving Statistics (& SPSS)

Projects Involving Statistics (& SPSS) Academic Skills Advice Starting a project which involves using statistics can feel confusing as there seems to be many different things you can do (charts, graphs,

### Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

### CHAPTER 14 NONPARAMETRIC TESTS

CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences

### Research Methodology: Tools

MSc Business Administration Research Methodology: Tools Applied Data Analysis (with SPSS) Lecture 11: Nonparametric Methods May 2014 Prof. Dr. Jürg Schwarz Lic. phil. Heidi Bruderer Enzler Contents Slide

### Analysis of Data. Organizing Data Files in SPSS. Descriptive Statistics

Analysis of Data Claudia J. Stanny PSY 67 Research Design Organizing Data Files in SPSS All data for one subject entered on the same line Identification data Between-subjects manipulations: variable to

### EPS 625 INTERMEDIATE STATISTICS FRIEDMAN TEST

EPS 625 INTERMEDIATE STATISTICS The Friedman test is an extension of the Wilcoxon test. The Wilcoxon test can be applied to repeated-measures data if participants are assessed on two occasions or conditions

### Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk

Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk Structure As a starting point it is useful to consider a basic questionnaire as containing three main sections:

### Difference tests (2): nonparametric

NST 1B Experimental Psychology Statistics practical 3 Difference tests (): nonparametric Rudolf Cardinal & Mike Aitken 10 / 11 February 005; Department of Experimental Psychology University of Cambridge

### T-test & factor analysis

Parametric tests T-test & factor analysis Better than non parametric tests Stringent assumptions More strings attached Assumes population distribution of sample is normal Major problem Alternatives Continue

### UNIVERSITY OF NAIROBI

UNIVERSITY OF NAIROBI MASTERS IN PROJECT PLANNING AND MANAGEMENT NAME: SARU CAROLYNN ELIZABETH REGISTRATION NO: L50/61646/2013 COURSE CODE: LDP 603 COURSE TITLE: RESEARCH METHODS LECTURER: GAKUU CHRISTOPHER

### UNDERSTANDING THE TWO-WAY ANOVA

UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### Introduction to Minitab and basic commands. Manipulating data in Minitab Describing data; calculating statistics; transformation.

Computer Workshop 1 Part I Introduction to Minitab and basic commands. Manipulating data in Minitab Describing data; calculating statistics; transformation. Outlier testing Problem: 1. Five months of nickel

### statistics Chi-square tests and nonparametric Summary sheet from last time: Hypothesis testing Summary sheet from last time: Confidence intervals

Summary sheet from last time: Confidence intervals Confidence intervals take on the usual form: parameter = statistic ± t crit SE(statistic) parameter SE a s e sqrt(1/n + m x 2 /ss xx ) b s e /sqrt(ss

### QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.

### Reporting Statistics in Psychology

This document contains general guidelines for the reporting of statistics in psychology research. The details of statistical reporting vary slightly among different areas of science and also among different

### SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

### One-Way ANOVA using SPSS 11.0. SPSS ANOVA procedures found in the Compare Means analyses. Specifically, we demonstrate

1 One-Way ANOVA using SPSS 11.0 This section covers steps for testing the difference between three or more group means using the SPSS ANOVA procedures found in the Compare Means analyses. Specifically,

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Come scegliere un test statistico

Come scegliere un test statistico Estratto dal Capitolo 37 of Intuitive Biostatistics (ISBN 0-19-508607-4) by Harvey Motulsky. Copyright 1995 by Oxfd University Press Inc. (disponibile in Iinternet) Table

### ANOVA ANOVA. Two-Way ANOVA. One-Way ANOVA. When to use ANOVA ANOVA. Analysis of Variance. Chapter 16. A procedure for comparing more than two groups

ANOVA ANOVA Analysis of Variance Chapter 6 A procedure for comparing more than two groups independent variable: smoking status non-smoking one pack a day > two packs a day dependent variable: number of

### SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011

SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011 Statistical techniques to be covered Explore relationships among variables Correlation Regression/Multiple regression Logistic regression Factor analysis

### MASTER COURSE SYLLABUS-PROTOTYPE PSYCHOLOGY 2317 STATISTICAL METHODS FOR THE BEHAVIORAL SCIENCES

MASTER COURSE SYLLABUS-PROTOTYPE THE PSYCHOLOGY DEPARTMENT VALUES ACADEMIC FREEDOM AND THUS OFFERS THIS MASTER SYLLABUS-PROTOTYPE ONLY AS A GUIDE. THE INSTRUCTORS ARE FREE TO ADAPT THEIR COURSE SYLLABI

### Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Comparing Means in Two Populations

Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### STATISTICAL ANALYSIS WITH EXCEL COURSE OUTLINE

STATISTICAL ANALYSIS WITH EXCEL COURSE OUTLINE Perhaps Microsoft has taken pains to hide some of the most powerful tools in Excel. These add-ins tools work on top of Excel, extending its power and abilities

### Correlation key concepts:

CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)

### COMP6053 lecture: Relationship between two variables: correlation, covariance and r-squared. jn2@ecs.soton.ac.uk

COMP6053 lecture: Relationship between two variables: correlation, covariance and r-squared jn2@ecs.soton.ac.uk Relationships between variables So far we have looked at ways of characterizing the distribution

### Statistical tests for SPSS

Statistical tests for SPSS Paolo Coletti A.Y. 2010/11 Free University of Bolzano Bozen Premise This book is a very quick, rough and fast description of statistical tests and their usage. It is explicitly

### The Dummy s Guide to Data Analysis Using SPSS

The Dummy s Guide to Data Analysis Using SPSS Mathematics 57 Scripps College Amy Gamble April, 2001 Amy Gamble 4/30/01 All Rights Rerserved TABLE OF CONTENTS PAGE Helpful Hints for All Tests...1 Tests

Chapter 12 Nonparametric Tests Chapter Table of Contents OVERVIEW...171 Testing for Normality...... 171 Comparing Distributions....171 ONE-SAMPLE TESTS...172 TWO-SAMPLE TESTS...172 ComparingTwoIndependentSamples...172

### UNDERSTANDING ANALYSIS OF COVARIANCE (ANCOVA)

UNDERSTANDING ANALYSIS OF COVARIANCE () In general, research is conducted for the purpose of explaining the effects of the independent variable on the dependent variable, and the purpose of research design

### THE KRUSKAL WALLLIS TEST

THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08 THE KRUSKAL-WALLIS TEST: The non-parametric alternative to ANOVA: testing for difference between several independent groups 2 NON

### POLYNOMIAL AND MULTIPLE REGRESSION. Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model.

Polynomial Regression POLYNOMIAL AND MULTIPLE REGRESSION Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model. It is a form of linear regression

### 1 Nonparametric Statistics

1 Nonparametric Statistics When finding confidence intervals or conducting tests so far, we always described the population with a model, which includes a set of parameters. Then we could make decisions

### Mixed 2 x 3 ANOVA. Notes

Mixed 2 x 3 ANOVA This section explains how to perform an ANOVA when one of the variables takes the form of repeated measures and the other variable is between-subjects that is, independent groups of participants

### One-Way Analysis of Variance

One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We

### HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

### Using Excel for inferential statistics

FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

### STA-201-TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance

Principles of Statistics STA-201-TE This TECEP is an introduction to descriptive and inferential statistics. Topics include: measures of central tendency, variability, correlation, regression, hypothesis

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

### Deciding which statistical test to use:

Deciding which statistical test to use: (b) Parametric tests: z-scores (one score compared against the distribution of scores to which it belongs) Relationship between two IV s - Pearson s r (correlation

### Two-Way ANOVA tests. I. Definition and Applications...2. II. Two-Way ANOVA prerequisites...2. III. How to use the Two-Way ANOVA tool?...

Two-Way ANOVA tests Contents at a glance I. Definition and Applications...2 II. Two-Way ANOVA prerequisites...2 III. How to use the Two-Way ANOVA tool?...3 A. Parametric test, assume variances equal....4

### 13: Additional ANOVA Topics. Post hoc Comparisons

13: Additional ANOVA Topics Post hoc Comparisons ANOVA Assumptions Assessing Group Variances When Distributional Assumptions are Severely Violated Kruskal-Wallis Test Post hoc Comparisons In the prior

### Nonparametric Statistics

Nonparametric Statistics J. Lozano University of Goettingen Department of Genetic Epidemiology Interdisciplinary PhD Program in Applied Statistics & Empirical Methods Graduate Seminar in Applied Statistics

Lecture 16: Generalized Additive Models Regression III: Advanced Methods Bill Jacoby Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Goals of the Lecture Introduce Additive Models

### Statistics. One-two sided test, Parametric and non-parametric test statistics: one group, two groups, and more than two groups samples

Statistics One-two sided test, Parametric and non-parametric test statistics: one group, two groups, and more than two groups samples February 3, 00 Jobayer Hossain, Ph.D. & Tim Bunnell, Ph.D. Nemours

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### CORRELATIONAL ANALYSIS: PEARSON S r Purpose of correlational analysis The purpose of performing a correlational analysis: To discover whether there

CORRELATIONAL ANALYSIS: PEARSON S r Purpose of correlational analysis The purpose of performing a correlational analysis: To discover whether there is a relationship between variables, To find out the

### Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

### DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,

### INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

### Introduction to Analysis of Variance (ANOVA) Limitations of the t-test

Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One- Way ANOVA Limitations of the t-test Although the t-test is commonly used, it has limitations Can only

### MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance

### Financial Risk Management Exam Sample Questions/Answers

Financial Risk Management Exam Sample Questions/Answers Prepared by Daniel HERLEMONT 1 2 3 4 5 6 Chapter 3 Fundamentals of Statistics FRM-99, Question 4 Random walk assumes that returns from one time period

### (and sex and drugs and rock 'n' roll) ANDY FIELD

DISCOVERING USING SPSS STATISTICS THIRD EDITION (and sex and drugs and rock 'n' roll) ANDY FIELD CONTENTS Preface How to use this book Acknowledgements Dedication Symbols used in this book Some maths revision

### Statistics for Sports Medicine

Statistics for Sports Medicine Suzanne Hecht, MD University of Minnesota (suzanne.hecht@gmail.com) Fellow s Research Conference July 2012: Philadelphia GOALS Try not to bore you to death!! Try to teach

### Chapter 14: Repeated Measures Analysis of Variance (ANOVA)

Chapter 14: Repeated Measures Analysis of Variance (ANOVA) First of all, you need to recognize the difference between a repeated measures (or dependent groups) design and the between groups (or independent

### Types of Data, Descriptive Statistics, and Statistical Tests for Nominal Data. Patrick F. Smith, Pharm.D. University at Buffalo Buffalo, New York

Types of Data, Descriptive Statistics, and Statistical Tests for Nominal Data Patrick F. Smith, Pharm.D. University at Buffalo Buffalo, New York . NONPARAMETRIC STATISTICS I. DEFINITIONS A. Parametric

### Chapter 10. Key Ideas Correlation, Correlation Coefficient (r),

Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

### Non-Parametric Tests (I)

Lecture 5: Non-Parametric Tests (I) KimHuat LIM lim@stats.ox.ac.uk http://www.stats.ox.ac.uk/~lim/teaching.html Slide 1 5.1 Outline (i) Overview of Distribution-Free Tests (ii) Median Test for Two Independent

### Univariate Regression

Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

### Module 5: Multiple Regression Analysis

Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

### Elementary Statistics Sample Exam #3

Elementary Statistics Sample Exam #3 Instructions. No books or telephones. Only the supplied calculators are allowed. The exam is worth 100 points. 1. A chi square goodness of fit test is considered to

### Chapter Eight: Quantitative Methods

Chapter Eight: Quantitative Methods RESEARCH DESIGN Qualitative, Quantitative, and Mixed Methods Approaches Third Edition John W. Creswell Chapter Outline Defining Surveys and Experiments Components of

### Outline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares

Topic 4 - Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test - Fall 2013 R 2 and the coefficient of correlation

### 1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

### Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular

### Multivariate Analysis of Variance. The general purpose of multivariate analysis of variance (MANOVA) is to determine

2 - Manova 4.3.05 25 Multivariate Analysis of Variance What Multivariate Analysis of Variance is The general purpose of multivariate analysis of variance (MANOVA) is to determine whether multiple levels

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test

The t-test Outline Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test - Dependent (related) groups t-test - Independent (unrelated) groups t-test Comparing means Correlation

### First-year Statistics for Psychology Students Through Worked Examples. 3. Analysis of Variance

First-year Statistics for Psychology Students Through Worked Examples 3. Analysis of Variance by Charles McCreery, D.Phil Formerly Lecturer in Experimental Psychology Magdalen College Oxford Copyright

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### One-Way Analysis of Variance (ANOVA) Example Problem

One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means

### Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

### Statistiek II. John Nerbonne. October 1, 2010. Dept of Information Science j.nerbonne@rug.nl

Dept of Information Science j.nerbonne@rug.nl October 1, 2010 Course outline 1 One-way ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic

### X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)

CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

### Statistics Review PSY379

Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

### Parametric and non-parametric statistical methods for the life sciences - Session I

Why nonparametric methods What test to use? Rank Tests Parametric and non-parametric statistical methods for the life sciences - Session I Liesbeth Bruckers Geert Molenberghs Interuniversity Institute

### Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:

T O P I C 1 2 Techniques and tools for data analysis Preview Introduction In chapter 3 of Statistics In A Day different combinations of numbers and types of variables are presented. We go through these

### business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar

business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel

### SPSS Guide How-to, Tips, Tricks & Statistical Techniques

SPSS Guide How-to, Tips, Tricks & Statistical Techniques Support for the course Research Methodology for IB Also useful for your BSc or MSc thesis March 2014 Dr. Marijke Leliveld Jacob Wiebenga, MSc CONTENT

### Sample Size and Power in Clinical Trials

Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance

### individualdifferences

1 Simple ANalysis Of Variance (ANOVA) Oftentimes we have more than two groups that we want to compare. The purpose of ANOVA is to allow us to compare group means from several independent samples. In general,