DISCOVERING STATISTICS USING R

Transcription

1 DISCOVERING STATISTICS USING R ANDY FIELD I JEREMY MILES I ZOE FIELD Los Angeles London New Delhi Singapore j Washington DC

2 CONTENTS Preface How to use this book Acknowledgements Dedication Symbols used in this book Some maths revision t- xxi xxv xxix xxxi xxxii xxxiv 1 Why is my evil Lecturer forcing me to Learn statistics? What will this chapter tell me? What the hell am I doing here? I don't belong here Initial observation: finding something that needs explaining Generating theories and testing them Data collection 1: what to measure Variables Measurement error Validity and reliability Data collection 2: how to measure Correlational research methods ,2. Experimental research methods Randomization Analysing data Frequency distributions The centre of a distribution The dispersion in a distribution Using a frequency distribution to go beyond the data Fitting statistical models to the data 28 What have I discovered about statistics? 29 Key terms that I've discovered 29 Smart Alex's tasks 30 Further reading 31 Interesting real research 31 2 Everything you ever wanted to know about statistics (well, sort of) What will this chapter tell me? Building statistical models 33

3 vi DISCOVERING STATISTICS USING R 2.3. Populations and samples Simple statistical models * The mean: a very simple statistical model Assessing the fit of the mean: sums of squares, variance and standard deviations Expressing the mean as a model Going beyond the data The standard error Confidence intervals Using statistical models to test research questions Test statistics One- and two-tailed tests Type I and Type II errors Effect sizes ' Statistical power " 58 What have I discovered about statistics? 59 Key terms that I've discovered 60 Smart Alex's tasks 60 Further reading 60 Interesting real research * 61 The R environment What will this chapter tell me? Before you start t The R-chitecture Pros and cons of R Downloading and installing R Versions of R Getting started The main windows in R Menus in R Using R _ Commands, objects and functions Using scripts The R workspace Setting a working directory Installing packages Getting help Getting data into R Creating variables Creating dataframes Calculating new variables from exisiting ones Organizing your data Missing values Entering data with R Commander *1. Creating variables and entering data with R Commander Creating coding variables with R Commander Using other software to enter and edit data Importing data Importing SPSS data files directly 99

4 CONTENTS vii Importing data with R Commander Things that can go wrong Saving data Manipulating data Selecting parts of a dataframe Selecting data with the subset() function Dataframes and matrices Reshaping data 107 What have I discovered about statistics? 113 R packages used in this chapter 113 R functions used in this chapter 113 Key terms that I've discovered 114 Smart Alex's tasks 114 Further reading ' Exploring data with graphs What will this chapter tell me? The art of presenting data Why do we need graphs " What makes a good graph? Lies, damned lies, and... erm... graphs Packages used in this chapter Introducing ggplot The anatomy of a plot Geometric objects (geoms) Aesthetics The anatomy of the ggplot() function Stats and geoms Avoiding overplotting Saving graphs Putting it all together: a quick tutorial Graphing relationships: the scatterplot Simple scatterplot Adding a funky line Grouped scatterplot Histograms: a good way to spot obvious problems Boxplots (box-whisker diagrams) Density plots Graphing means Bar charts and error bars Line graphs Themes and options 161 What have I discovered about statistics? 163 R packages used in this chapter 163 R functions used in this chapter 164 Key terms that I've discovered 164 Smart Alex's tasks 164 Further reading 164 Interesting real research 165

5 viii DISCOVERING STATISTICS USING R Exploring assumptions What will this chapter tell me? What are assumptions? Assumptions of parametric data Packages used in this chapter The assumption of normality Oh no, it's that pesky frequency distribution again; checking normality visually Quantifying normality with numbers Exploring groups of data Testing whether a distribution is normal Doing the Shapiro-Wilk test in R Reporting the Shapiro-Wilk test Testing for homogeneity of variance Levene'stest Reporting Levene's test Hartley's F mg> : the variance ratio Correcting problems in the data < Dealing with outliers Dealing with non-normality and unequal variances Transforming the data using R When it all goes horribly wrong 201 What have I discovered about statistics? 203 R packages used in this chapter 204 R functions used in this chapter 204 Key terms that I've discovered 204 Smart Alex's tasks 204 Further reading Correlation What will this chapter tell me? Looking at relationships How do we measure relationships? A detour into the murky world of covariance Standardization and the correlation coefficient The significance of the correlation coefficient Confidence intervals for/ A word of warning about interpretation: causality Data entry for correlation analysis Bivariate correlation Packages for correlation analysis in R General procedure for correlations using R Commander General procedure for correlations using R Pearson's correlation coefficient ' Spearman's correlation coefficient Kendall's tau (non-parametric) Bootstrapping correlations Biserial and point-biserial correlations 229

6 CONTENTS 6.6. Partial correlation The theory behind part and partial correlation ' Partial correlation using R Semi-partial (or part) correlations Comparing correlations , Comparing independent r$ Comparing dependent r$ Calculating the effect size How to report correlation coefficents 240 What have I discovered about statistics? R packages used in this chapter 243 R functions used in this chapter 243 Key terms that I've discovered 243 Smart Alex's tasks ' 243 Further reading 244 Interesting real research Regression '* What will this chapter tell me? An introduction to regression Some important information about straight lines The method of least squares Assessing the goodness of fit: sums of squares, R and R Assessing individual predictors Packages used in this chapter General procedure for regression in R Doing simple regression using R Commander Regression in R Interpreting a simple regression Overall fit of the object model Model parameters Using the model Multiple regression: the basics An example of a multiple regression model Sums of squares, R and R Parsimony-adjusted measures of fit Methods of regression How accurate is my regression model? Assessing the regression model I: diagnostics Assessing the regression model II: generalization How to do multiple regression using R Commander and R Some things to think about before the analysis Multiple regression: running the basic model Interpreting the basic multiple regression Comparing models Testing the accuracy of your regression model Diagnostic tests using R Commander Outliers and influential cases 288

7 DISCOVERING STATISTICS USING R Assessing the assumption of independence Assessing the assumption of no multicollinearity Checking assumptions about the residuals What if I violate an assumption? _ , Robust regression: bootstrapping , How to report multiple regression , Categorical predictors and multiple regression Dummy coding Regression with dummy variables 305 What have I discovered about statistics?. 308 R packages used in this chapter 309 R functions used in this chapter 309 Key terms that I've discovered 309 Smart Alex's tasks ' 310 Further reading 311 Interesting real research Logistic regression What will this chapter tell me? " Background to logistic regression What are the principles behind logistic regression? Assessing the model: the log-likelihood statistic Assessing the model: the deviance statistic Assessing the model: ftandtf Assessing the model: information criteria Assessing the contribution of predictors: the z-statistic The odds ratio Methods of logistic regression Assumptions and things that can go wrong Assumptions Incomplete information from the predictors Complete separation Packages used in this chapter Binary logistic regression: an example that will make you feel eel Preparing the data The main logistic regression analysis <D Basic logistic regression analysis using R Interpreting a basic logistic regression Model 1: Intervention only Model 2: Intervention and Duration as predictors Casewise diagnostics in logistic regression Calculating the effect size How to report logistic regression Testing assumptions: another example Testing for multicollinearity Testing for linearity of the logit Predicting several categories: multinomial logistic regression Running multinomial logistic regression in R Interpreting the multinomial logistic regression output 350

8 CONTENTS xi Reporting the results 355 What have I discovered about statistics? ' 355 R packages used in this chapter 356 R functions used in this chapter 356 Key terms that I've discovered 356 Smart Alex's tasks 357 Further reading 358 Interesting real research Comparing two means What will this chapter tell me? Packages used in this chapter Looking at differences A problem with error bar graphs of repeated-measures designs Step 1: calculate the mean for each participant Step 2: calculate the grand mean Step 3: calculate the adjustment factor Step 4: create adjusted values for each variable <* TheMest.* Rationale for the Mest The Mest as a general linear model Assumptions of the f-test The independent Mest The independent f-test equation explained Doing the independent Mest The dependent f-test Sampling distributions and the standard error The dependent Mest equation explained Dependent Mests using R Between groups or repeated measures? 394 What have I discovered about statistics? 395 R packages used in this chapter 396 R functions used in this chapter 396 Key terms that I've discovered 396 Smart Alex's tasks 396 Further reading 397 Interesting real research Comparing several means: ANOVA (GLM 1) What will this chapter tell me? , The theory behind ANOVA Inflated error rates Interpreting F ANOVA as regression Logic of the F-ratio Total sum of squares (SS T ) Model sum of squares (SSJ Residual sum of squares (SS R ) Mean squares 411

9 xii DISCOVERING STATISTICS USING R TheF-ratio Assumptions of ANOVA > Homogeneity of variance Is ANOVA robust? Planned contrasts 414~ Choosing which contrasts to do Defining contrasts using weights Non-orthogonal comparisons Standard contrasts Polynomial contrasts: trend analysis Post hoc procedures Post hoc procedures and Type I (a) and Type II error rates Post hoc procedures and violations of test assumptions Summary of post hoc procedures One-way ANOVA using R Packages for one-way ANOVA in R General procedure for one-way ANOVA Entering data One-way ANOVA using R Commander * Exploring the data The main analysis Planned contrasts using R Post hoc tests using R Calculating the effect size Reporting results from one-way independent ANOVA 457 What have I discovered about statistics? 458 R packages used in this chapter 459 R functions used in this chapter 459 Key terms that I've discovered 459 Smart Alex's tasks 459 Further reading 461 Interesting real research Analysis of covariance, ANC0VA (GLM 2) What will this chapter tell me? WhatisANCOVA? 463.1,3. Assumptions and issues in ANCOVA Independence of the covariate and treatment effect Homogeneity of regression slopes ANCOVA using R Packages for ANCOVA in R General procedure for ANCOVA Entering data ANCOVA using R Commander Exploring the data Are the predictor variable and covariate independent? Fitting an ANCOVA model Interpreting the main ANCOVA model 477

10 CONTENTS xiii Planned contrasts in ANCOVA Interpreting the covariate " Post hoc tests in ANCOVA Plots in ANCOVA Some final remarks Testing for homogeneity of regression slopes Robust ANCOVA Calculating the effect size Reporting results 494 What have! discovered about statistics? 495 R packages used in this chapter 495 R functions used in this chapter 496 Key terms that I've discovered 496 Smart Alex's tasks 496 Further reading 497 Interesting real research Factorial ANOVA (GLM 3) ^ What will this chapter tell me?.* Theory of factorial ANOVA (independent design) ,1, Factorial designs Factorial ANOVA as regression An example with two independent variables Extending the regression model Two-way ANOVA: behind the scenes Total sums of squares (SS T ), The model sum of squares (SSJ ' The residua! sum of squares (SS R ) The F-ratios Factorial ANOVA using R Packages for factorial ANOVA in R General procedure for factorial ANOVA Factorial ANOVA using R Commander Entering the data Exploring the data Choosing contrasts Fitting a factorial ANOVA model Interpreting factorial ANOVA Interpreting contrasts Simple effects analysis Posf hoc analysis Overall conclusions Plots in factorial ANOVA Interpreting interaction graphs Robust factorial ANOVA Calculating effect sizes Reporting the results of two-way ANOVA 544 What have I discovered about statistics? 546

11 DISCOVERING STATISTICS USING R R packages used in this chapter 546 R functions used in this chapter 546 Key terms that I've discovered 547 Smart Alex's tasks 547 Further reading ' 548 Interesting reai research Repeated-measures designs (GLM 4) What will this chapter tell me? Introduction to repeated-measures designs The assumption of sphericity How is sphericity measured? Assessing the severity of departures from sphericity What is the effect of violating the assumption of sphericity What do you do if you violate sphericity? Theory of one-way repeated-measures ANOVA The total sum of squares (SS T ) The within-participant sum of squares (SS W ) « The model sum of squares (SSJ The residual sum of squares (SS R ) The mean squares TheF-ratio The between-participant sum of squares One-way repeated-measures designs using R Packages for repeated measures designs in R ' General procedure for repeated-measures designs Repeated-measures ANOVA using R Commander Entering the data Exploring the data Choosing contrasts Analysing repeated measures: two ways to skin a.dat Robust one-way repeated-measures ANOVA Effect sizes for repeated-measures designs Reporting one-way repeated-measures designs Factorial repeated-measures designs Entering the data Exploring the data Setting contrasts Factorial repeated-measures ANOVA Factorial repeated-measures designs as a GLM Robust factorial repeated-measures ANOVA Effect sizes for factorial repeated-measures designs Reporting the results from factorial repeated-measures designs 600 What have I discovered about statistics? 601 R packages used in this chapter 602 R functions used in this chapter 602 Key terms that I've discovered 602 Smart Alex's tasks 602

12 CONTENTS xv Further reading 603 Interesting real research Mixed designs (GLM 5) What will this chapter tell me? Mixed designs What do men and women look for in a partner? Entering and exploring your data Packages for mixed designs in R ,4.2. General procedure for mixed designs Entering the data Exploring the data Mixed ANOVA, Mixed designs as a GLM Setting contrasts Building the model ,6.3. The main effect of gender The main effect of looks * The main effect of personality,* The interaction between gender and looks The interaction between gender and personality The interaction between looks and personality The interaction between looks, personality and gender Conclusions Calculating effect sizes Reporting the results of mixed ANOVA Robust analysis for mixed designs 643 What have I discovered about statistics? 650 R packages used in this chapter 650 R functions used in this chapter 651 Key terms that I've discovered 651 Smart Alex's tasks 651 Further reading 652 Interesting real research Non-parametric tests What will this chapter tell me? When to use non-parametric tests Packages used in this chapter Comparing two independent conditions: the Wilcoxon rank-sum test Theory of the Wilcoxon rank-sum test Inputting data and provisional analysis Running the analysis using R Commander Running the analysis using R Output from the Wilcoxon rank-sum test Calculating an effect size Writing the results 666

13 xvi DISCOVERING STATISTICS USING R Comparing two related conditions: the Wilcoxon signed-rank test Theory of the Wilcoxon signed-rank test Running the analysis with R Commander Running the analysis using R Wilcoxon signed-rank test output Calculating an effect size Writing the results Differences between several independent groups: the Kruskal-Wallis test Theory of the Kruskal-Wallis test Inputting data and provisional analysis Doing the Kruskal-Wallis test using R Commander Doing the Kruskal-Wallis test using R Output from the Kruskal-Wallis test Posf hoc tests for the Kruskal-Wallis test Testing for trends: the Jonckheere-Terpstra test Calculating an effect size Writing and interpreting the results Differences between several related groups: Friedman's ANOVA Theory of Friedman's ANOVA '" Inputting data and provisional analysis Doing Friedman's ANOVA in R Commander Friedman's ANOVA using R Output from Friedman's ANOVA Posf hoc tests for Friedman's ANOVA Calculating an effect size Writing and interpreting the results 692 What have I discovered about statistics? 693 R packages used in this chapter 693 R functions used in this chapter 693 Key terms that I've discovered 694 Smart Alex's tasks 694 Further reading 695 Interesting real research MuLtivariate analysis of variance (MANOVA) What will this chapter tell me? When to use MANOVA Introduction: similarities to and differences from ANOVA Words of warning The example for this chapter Theory of MANOVA Introduction to matrices Some important matrices and their functions Calculating MANOVA by hand: a worked example ' Principle of the MANOVA test statistic Practical issues when conducting MANOVA Assumptions and how to check them 717

14 CONTENTS xvii Choosing a test statistic Follow-up analysis * MANOVA using R Packages for factorial ANOVA in R General procedure for MANOVA MANOVA using R Commander Entering the data Exploring the data Setting contrasts The MANOVA model Follow-up analysis: univariate test statistics Contrasts Robust MANOVA Reporting results from MANOVA Following up MANOVA with discriminant analysis Reporting results from discriminant analysis Some final remarks The final interpretation Univariate ANOVA or discriminant analysis? n ' 745 What have I discovered about statistics? " 745 R packages used in this chapter 746 R functions used in this chapter 746 Key terms that I've discovered 747 Smart Alex's tasks 747 Further reading 748 Interesting real research Exploratory factor analysis What will this chapter tell me? When to use factor analysis Factors ,3,1. Graphical representation of factors Mathematical representation of factors Factor scores Choosing a method Communality Factor analysis vs. principal components analysis Theory behind principal components analysis Factor extraction: eigenvalues and the scree plot Improving interpretation: factor rotation Research example Samplesize Correlations between variables The distribution of data Running the analysis with R Commander Running the analysis with R Packages used in this chapter Initial preparation and analysis 772

15 xvih DISCOVERING STATISTICS USING R Factor extraction using R Rotation ' Factor scores Summary How to report factor analysis ,8. Reliability analysis Measures of reliability Interpreting Cronbach's a (some cautionary tales...) Reliability analysis with R Commander Reliability analysis using R Interpreting the output Reporting reliability analysis 806 What have I discovered about statistics? 807 R packages used in this chapter 807 R functions used in this chapter Key terms that I've discovered Smart Alex's tasks Further reading 810 Interesting real research ** Categorical data What will this chapter tell me? Packages used in this chapter Analysing categorical data Theory of analysing categorical data Pearson's chi-square test Fisher's exact test The likelihood ratio Yates's correction Assumptions of the chi-square test Doing the chi-square test using R Entering data: raw scores Entering data: the contingency table Running the analysis with R Commander Running the analysis using R 821 * Output from the CrossTableQ function Breaking down a significant chi-square test with standardized residuals Calculating an effect size Reporting the results of chi-square Several categorical variables: loglinear analysis Chi-square as regression Loglinear analysis Assumptions in loglinear analysis Loglinear analysis using R Initial considerations Loglinear analysis as a chi-square test Output from loglinear analysis as a chi-square test 843

16 CONTENTS xix Loglinear analysis Following up loglinear analysis ' Effect sizes in loglinear analysis Reporting the results of loglinear analysis 851 What have I discovered about statistics? 852 R packages used in this chapter 853 R functions used in this chapter 853 Key terms that I've discovered 853 Smart Alex's tasks 853 Further reading. 854 Interesting real research Multilevel linear models What will this chapter tell me? Hierarchical data The intraclass correlation Benefits of multilevel models Theory of multilevel linear models <D '" An example * ,3.2. Fixed and random coefficients The multilevel model Assessing the fit and comparing multilevel models Types of covariance structures Some practical issues Assumptions Sample size and power Centring variables Multilevel modelling in R Packages for multilevel modelling in R Entering the data Picturing the data Ignoring the data structure; ANOVA Ignoring the data structure: ANCOVA Assessing the need for a multilevel model Adding infixed effects Introducing random slopes Adding an interaction term to the model Growth models Growth curves (polynomials) An example: the honeymoon period Restructuring the data Setting up the basic model Adding in time as a fixed effect Introducing random slopes ,7.7. Modelling the covariance structure Comparing models Adding higher-order polynomials Further analysis 905

17 XX DISCOVERING STATISTICS USING R How to report a multilevel model 906 What have I discovered about statistics? * 907 R packages used in this chapter 908 R functions used in this chapter 908 Key terms that I've discovered 908 Smart Alex's tasks 908 Further reading 909 Interesting real research 909 Epilogue: life after discovering statistics 910 Troubleshooting R 912 Glossary 913 Appendix. 929 A.1. Table of the standard normal distribution. 929 A.2. Critical values of the f-distribution 935 A.3. Critical values of the F-distribution 936 A.4. Critical values of the chi-square distribution 940 References,> 941 Index 948 Functions in R 956 Packages in R 957