Discovering Statistics Using R by Field, Andy -- Read -- Imperial Library of Trantor

Index

Title Page Copyright Page Contents Preface How to use this book Acknowledgements Dedication Symbols used in this book Some maths revision 1 Why is my evil lecturer forcing me to learn statistics?

1.1. What will this chapter tell me? 1.2. What the hell am I doing here? I don’t belong here 1.3. Initial observation: finding something that needs explaining 1.4. Generating theories and testing them 1.5. Data collection 1: what to measure

1.5.1. Variables 1.5.2. Measurement error 1.5.3. Validity and reliability

1.6. Data collection 2: how to measure

1.6.1. Correlational research methods 1.6.2. Experimental research methods 1.6.3. Randomization

1.7. Analysing data

1.7.1. Frequency distributions 1.7.2. The centre of a distribution 1.7.3. The dispersion in a distribution 1.7.4. Using a frequency distribution to go beyond the data 1.7.5. Fitting statistical models to the data

What have I discovered about statistics? Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

2 Everything you ever wanted to know about statistics(well, sort of) (well, sort of)

2.1. What will this chapter tell me? 2.2. Building statistical models 2.3. Populations and samples 2.4. Simple statistical models

2.4.1. The mean: a very simple statistical model 2.4.2. Assessing the fit of the mean: sums of squares, variance and standard deviations 2.4.3. Expressing the mean as a model

2.5. Going beyond the data

2.5.1. The standard error 2.5.2. Confidence intervals

2.6. Using statistical models to test research questions

2.6.1. Test statistics 2.6.2. One- and two-tailed tests 2.6.3. Type I and Type II errors 2.6.4. Effect sizes 2.6.5. Statistical power

What have I discovered about statistics? Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

3 The R environment

3.1. What will this chapter tell me? 3.2. Before you start

3.2.1. The R-chitecture 3.2.2. Pros and cons of R 3.2.3. Downloading and installing R 3.2.4. Versions of R

3.3. Getting started

3.3.1. The main windows in R 3.3.2. Menus in R

3.4. Using R

3.4.1. Commands, objects and functions 3.4.2. Using scripts 3.4.3. The R workspace 3.4.4. Setting a working directory 3.4.5. Installing packages 3.4.6. Getting help

3.5. Getting data into R

3.5.1. Creating variables 3.5.2. Creating dataframes 3.5.3. Calculating new variables from exisiting ones 3.5.4. Organizing your data 3.5.5. Missing values

3.6. Entering data with R Commander

3.6.1. Creating variables and entering data with R Commander 3.6.2. Creating coding variables with R Commander

3.7. Using other software to enter and edit data

3.7.1. Importing data 3.7.2. Importing SPSS data files directly 3.7.3. Importing data with R Commander 3.7.4. Things that can go wrong

3.8. Saving data 3.9. Manipulating data

3.9.1. Selecting parts of a dataframe 3.9.2. Selecting data with the subset() function 3.9.3. Dataframes and matrices 3.9.4. Reshaping data

What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading

4 Exploring data with graphs

4.1. What will this chapter tell me? 4.2. The art of presenting data

4.2.1. Why do we need graphs 4.2.2. What makes a good graph? 4.2.3. Lies, damned lies, and … erm … graphs

4.3. Packages used in this chapter 4.4. Introducing ggplot2

4.4.1. The anatomy of a plot 4.3.2. Geometric objects (geoms) 4.4.3. Aesthetics 4.4.4. The anatomy of the ggplot() function 4.4.5. Stats and geoms 4.4.6. Avoiding overplotting 4.4.7. Saving graphs 4.4.8. Putting it all together: a quick tutorial

4.5. Graphing relationships: the scatterplot

4.5.1. Simple scatterplot 4.5.2. Adding a funky line 4.5.3. Grouped scatterplot

4.6. Histograms: a good way to spot obvious problems 4.7. Boxplots (box–whisker diagrams) 4.8. Density plots 4.9. Graphing means

4.9.1. Bar charts and error bars 4.9.2. Line graphs

4.10. Themes and options What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

5 Exploring assumptions

5.1. What will this chapter tell me? 5.2. What are assumptions? 5.3. Assumptions of parametric data 5.4. Packages used in this chapter 5.5. The assumption of normality

5.5.1. Oh no, it’s that pesky frequency distribution again: checking normality visually 5.5.2. Quantifying normality with numbers 5.5.3. Exploring groups of data

5.6. Testing whether a distribution is normal

5.6.1. Doing the Shapiro–Wilk test in R 5.6.2. Reporting the Shapiro–Wilk test

5.7. Testing for homogeneity of variance

5.7.1. Levene’s test 5.7.2. Reporting Levene’s test 5.7.3. Hartley’s Fmax: the variance ratio

5.8. Correcting problems in the data

5.8.1. Dealing with outliers 5.8.2. Dealing with non-normality and unequal variances 5.8.3. Transforming the data using R 5.8.4. When it all goes horribly wrong

What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading

6 Correlation

6.1. What will this chapter tell me? 6.2. Looking at relationships 6.3. How do we measure relationships?

6.3.1. A detour into the murky world of covariance 6.3.2. Standardization and the correlation coefficient 6.3.3. The significance of the correlation coefficient 6.3.4. Confidence intervals for r 6.3.5. A word of warning about interpretation: causality

6.4. Data entry for correlation analysis 6.5. Bivariate correlation

6.5.1. Packages for correlation analysis in R 6.5.2. General procedure for correlations using R Commander 6.5.3. General procedure for correlations using R 6.5.4. Pearson’s correlation coefficient 6.5.5. Spearman’s correlation coefficient 6.5.6. Kendall’s tau (non-parametric) 6.5.7. Bootstrapping correlations 6.5.8. Biserial and point-biserial correlations

6.6. Partial correlation

6.6.1. The theory behind part and partial correlation 6.6.2. Partial correlation using R 6.6.3 Semi-partial (or part) correlations

6.7. Comparing correlations

6.7.1. Comparing independent rs 6.7.2. Comparing dependent rs

6.8. Calculating the effect size 6.9. How to report correlation coefficents What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

7 Regression

7.1. What will this chapter tell me? 7.2. An introduction to regression

7.2.1. Some important information about straight lines 7.2.2. The method of least squares 7.2.3. Assessing the goodness of fit: sums of squares, R and R2 7.2.4. Assessing individual predictors

7.3. Packages used in this chapter 7.4. General procedure for regression in R

7.4.1. Doing simple regression using R Commander 7.4.2. Regression in R

7.5. Interpreting a simple regression

7.5.1. Overall fit of the object model 7.5.2. Model parameters 7.5.3. Using the model

7.6. Multiple regression: the basics

7.6.1. An example of a multiple regression model 7.6.2. Sums of squares, R and R2 7.6.3. Parsimony-adjusted measures of fit 7.6.4. Methods of regression

7.7. How accurate is my regression model?

7.7.1. Assessing the regression model I: diagnostics 7.7.2. Assessing the regression model II: generalization

7.8. How to do multiple regression using R Commander and R

7.8.1. Some things to think about before the analysis 7.8.2. Multiple regression: running the basic model 7.8.3. Interpreting the basic multiple regression 7.8.4. Comparing models

7.9. Testing the accuracy of your regression model

7.9.1. Diagnostic tests using R Commander 7.9.2. Outliers and influential cases 7.9.3. Assessing the assumption of independence 7.9.4. Assessing the assumption of no multicollinearity 7.9.5. Checking assumptions about the residuals 7.9.6. What if I violate an assumption?

7.10. Robust regression: bootstrapping 7.11. How to report multiple regression 7.12. Categorical predictors and multiple regression

7.12.1. Dummy coding 7.12.2. Regression with dummy variables

What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

8 Logistic regression

8.1. What will this chapter tell me? 8.2. Background to logistic regression 8.3. What are the principles behind logistic regression?

8.3.1. Assessing the model: the log-likelihood statistic 8.3.2. Assessing the model: the deviance statistic 8.3.3. Assessing the model: R and R2 8.3.4. Assessing the model: information criteria 8.3.5. Assessing the contribution of predictors: the z-statistic 8.3.6. The odds ratio 8.3.7. Methods of logistic regression

8.4. Assumptions and things that can go wrong

8.4.1. Assumptions 8.4.2. Incomplete information from the predictors 8.4.3. Complete separation

8.5. Packages used in this chapter 8.6. Binary logistic regression: an example that will make you feel eel

8.6.1. Preparing the data 8.6.2. The main logistic regression analysis 8.6.3. Basic logistic regression analysis using R 8.6.4. Interpreting a basic logistic regression 8.6.5. Model 1: Intervention only 8.6.6. Model 2: Intervention and Duration as predictors 8.6.7. Casewise diagnostics in logistic regression 8.6.8. Calculating the effect size

8.7. How to report logistic regression 8.8. Testing assumptions: another example

8.8.1. Testing for multicollinearity 8.8.2. Testing for linearity of the logit

8.9. Predicting several categories: multinomial logistic regression

8.9.1. Running multinomial logistic regression in R 8.9.2. Interpreting the multinomial logistic regression output 8.9.3. Reporting the results

What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

9 Comparing two means

9.1. What will this chapter tell me? 9.2. Packages used in this chapter 9.3. Looking at differences

9.3.1. A problem with error bar graphs of repeated-measures designs 9.3.2. Step 1: calculate the mean for each participant 9.3.3. Step 2: calculate the grand mean 9.3.4. Step 3: calculate the adjustment factor 9.3.5. Step 4: create adjusted values for each variable

9.4. The t-test

9.4.1. Rationale for the t-test 9.4.2. The t-test as a general linear model 9.4.3. Assumptions of the t-test

9.5. The independent t-test

9.5.1. The independent t-test equation explained 9.5.2. Doing the independent t-test

9.6. The dependent t-test

9.6.1. Sampling distributions and the standard error 9.6.2. The dependent t-test equation explained 9.6.3. Dependent t-tests using R

9.7. Between groups or repeated measures? What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

10 Comparing several means: ANOVA (GLM 1)

10.1. What will this chapter tell me? 10.2. The theory behind ANOVA

10.2.1 Inflated error rates 10.2.2. Interpreting F 10.2.3. ANOVA as regression 10.2.4. Logic of the F-ratio 10.2.5. Total sum of squares (SST) 10.2.6. Model sum of squares (SSM) 10.2.7. Residual sum of squares (SSR) 10.2.8. Mean squares 10.2.9. The F-ratio

10.3. Assumptions of ANOVA

10.3.1. Homogeneity of variance 10.3.2. Is ANOVA robust?

10.4. Planned contrasts

10.4.1. Choosing which contrasts to do 10.4.2. Defining contrasts using weights 10.4.3. Non-orthogonal comparisons 10.4.4. Standard contrasts 10.4.5. Polynomial contrasts: trend analysis

10.5. Post hoc procedures

10.5.1. Post hoc procedures and Type I (α) and Type II error rates 10.5.2. Post hoc procedures and violations of test assumptions 10.5.3. Summary of post hoc procedures

10.6. One-way ANOVA using R

10.6.1. Packages for one-way ANOVA in R 10.6.2. General procedure for one-way ANOVA 10.6.3. Entering data 10.6.4. One-way ANOVA using R Commander 10.6.5. Exploring the data 10.6.6. The main analysis 10.6.7. Planned contrasts using R 10.6.8. Post hoc tests using R

10.7. Calculating the effect size 10.8. Reporting results from one-way independent ANOVA What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

11 Analysis of covariance, ANCOVA (GLM 2)

11.1. What will this chapter tell me? 11.2. What is ANCOVA? 11.3. Assumptions and issues in ANCOVA

11.3.1. Independence of the covariate and treatment effect 11.3.2. Homogeneity of regression slopes

11.4. ANCOVA using R

11.4.1. Packages for ANCOVA in R 11.4.2. General procedure for ANCOVA 11.4.3. Entering data 11.4.4. ANCOVA using R Commander 11.4.5. Exploring the data 11.4.6. Are the predictor variable and covariate independent? 11.4.7. Fitting an ANCOVA model 11.4.8. Interpreting the main ANCOVA model 11.4.9. Planned contrasts in ANCOVA 11.4.10. Interpreting the covariate 11.4.11. Post hoc tests in ANCOVA 11.4.12. Plots in ANCOVA 11.4.13. Some final remarks 11.4.14. Testing for homogeneity of regression slopes

11.5. Robust ANCOVA 11.6. Calculating the effect size 11.7. Reporting results What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

12 Factorial ANOVA (GLM 3)

12.1. What will this chapter tell me? 12.2. Theory of factorial ANOVA (independent design)

12.2.1. Factorial designs

12.3. Factorial ANOVA as regression

12.3.1. An example with two independent variables 12.3.2. Extending the regression model

12.4. Two-way ANOVA: behind the scenes

12.4.1. Total sums of squares (SST) 12.4.2. The model sum of squares (SSM) 12.4.3. The residual sum of squares (SSR) 12.4.4. The F-ratios

12.5. Factorial ANOVA using R

12.5.1. Packages for factorial ANOVA in R 12.5.2. General procedure for factorial ANOVA 12.5.3. Factorial ANOVA using R Commander 12.5.4. Entering the data 12.5.5. Exploring the data 12.5.6. Choosing contrasts 12.5.7. Fitting a factorial ANOVA model 12.5.8. Interpreting factorial ANOVA 12.5.9. Interpreting contrasts 12.5.10. Simple effects analysis 12.5.11. Post hoc analysis 12.5.12. Overall conclusions 12.5.13. Plots in factorial ANOVA

12.6. Interpreting interaction graphs 12.7. Robust factorial ANOVA 12.8. Calculating effect sizes 12.9. Reporting the results of two-way ANOVA What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

13 Repeated-measures designs (GLM 4)

13.1. What will this chapter tell me? 13.2. Introduction to repeated-measures designs

13.2.1. The assumption of sphericity 13.2.2. How is sphericity measured? 13.2.3. Assessing the severity of departures from sphericity 13.2.4. What is the effect of violating the assumption of sphericity? 13.2.5. What do you do if you violate sphericity?

13.3. Theory of one-way repeated-measures ANOVA

13.3.1. The total sum of squares (SST) 13.3.2. The within-participant sum of squares (SSW) 13.3.3. The model sum of squares (SSM) 13.3.4. The residual sum of squares (SSR) 13.3.5. The mean squares 13.3.6. The F-ratio 13.3.7. The between-participant sum of squares

13.4. One-way repeated-measures designs using R

13.4.1. Packages for repeated measures designs in R 13.4.2. General procedure for repeated-measures designs 13.4.3. Repeated-measures ANOVA using R Commander 13.4.4. Entering the data 13.4.5. Exploring the data 13.4.6. Choosing contrasts 13.4.7. Analysing repeated measures: two ways to skin a .dat 13.4.8. Robust one-way repeated-measures ANOVA

13.5. Effect sizes for repeated-measures designs 13.6. Reporting one-way repeated-measures designs 13.7. Factorial repeated-measures designs

13.7.1. Entering the data 13.7.2. Exploring the data 13.7.3. Setting contrasts 13.7.4. Factorial repeated-measures ANOVA 13.7.5. Factorial repeated-measures designs as a GLM 13.7.6. Robust factorial repeated-measures ANOVA

13.8. Effect sizes for factorial repeated-measures designs 13.9. Reporting the results from factorial repeated-measures designs What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

14 Mixed designs (GLM 5)

14.1. What will this chapter tell me? 14.2. Mixed designs 14.3. What do men and women look for in a partner? 14.4. Entering and exploring your data

14.4.1. Packages for mixed designs in R 14.4.2. General procedure for mixed designs 14.4.3. Entering the data 14.4.4. Exploring the data

14.5. Mixed ANOVA 14.6. Mixed designs as a GLM

14.6.1. Setting contrasts 14.6.2. Building the model 14.6.3. The main effect of gender 14.6.4. The main effect of looks 14.6.5. The main effect of personality 14.6.6. The interaction between gender and looks 14.6.7. The interaction between gender and personality 14.6.8. The interaction between looks and personality 14.6.9. The interaction between looks, personality and gender 14.6.10. Conclusions

14.7. Calculating effect sizes 14.8. Reporting the results of mixed ANOVA 14.9. Robust analysis for mixed designs What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

15 Non-parametric tests

15.1. What will this chapter tell me? 15.2. When to use non-parametric tests 15.3. Packages used in this chapter 15.4. Comparing two independent conditions: the Wilcoxon rank-sum test

15.4.1. Theory of the Wilcoxon rank-sum test 15.4.2. Inputting data and provisional analysis 15.4.3. Running the analysis using R Commander 15.4.4. Running the analysis using R 15.4.5. Output from the Wilcoxon rank-sum test 15.4.6. Calculating an effect size 15.4.7. Writing the results

15.5. Comparing two related conditions: the Wilcoxon signed-rank test

15.5.1. Theory of the Wilcoxon signed-rank test 15.5.2. Running the analysis with R Commander 15.5.3. Running the analysis using R 15.5.4. Wilcoxon signed-rank test output 15.5.5. Calculating an effect size 15.5.6. Writing the results

15.6. Differences between several independent groups: the Kruskal–Wallis test

15.6.1. Theory of the Kruskal–Wallis test 15.6.2. Inputting data and provisional analysis 15.6.3. Doing the Kruskal–Wallis test using R Commander 15.6.4. Doing the Kruskal–Wallis test using R 15.6.5. Output from the Kruskal–Wallis test 15.6.6. Post hoc tests for the Kruskal–Wallis test 15.6.7. Testing for trends: the Jonckheere–Terpstra test 15.6.8. Calculating an effect size 15.6.9. Writing and interpreting the results

15.7. Differences between several related groups: Friedman’s ANOVA

15.7.1. Theory of Friedman’s ANOVA 15.7.2. Inputting data and provisional analysis 15.7.3. Doing Friedman’s ANOVA in R Commander 15.7.4. Friedman’s ANOVA using R 15.7.5. Output from Friedman’s ANOVA 15.7.6. Post hoc tests for Friedman’s ANOVA 15.7.7. Calculating an effect size 15.7.8. Writing and interpreting the results

What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

16 Multivariate analysis of variance (MANOVA)

16.1. What will this chapter tell me? 16.2. When to use MANOVA 16.3. Introduction: similarities to and differences from ANOVA

16.3.1. Words of warning 16.3.2. The example for this chapter

16.4. Theory of MANOVA

16.4.1. Introduction to matrices 16.4.2. Some important matrices and their functions 16.4.3. Calculating MANOVA by hand: a worked example 16.4.4. Principle of the MANOVA test statistic

16.5. Practical issues when conducting MANOVA

16.5.1. Assumptions and how to check them 16.5.2. Choosing a test statistic 16.5.3. Follow-up analysis

16.6. MANOVA using R

16.6.1. Packages for factorial ANOVA in R 16.6.2. General procedure for MANOVA 16.6.3. MANOVA using R Commander 16.6.4. Entering the data 16.6.5. Exploring the data 16.6.6. Setting contrasts 16.6.7. The MANOVA model 16.6.8. Follow-up analysis: univariate test statistics 16.6.9. Contrasts

16.7. Robust MANOVA 16.8. Reporting results from MANOVA 16.9. Following up MANOVA with discriminant analysis 16.10. Reporting results from discriminant analysis 16.11. Some final remarks

16.11.1. The final interpretation 16.11.2. Univariate ANOVA or discriminant analysis?

What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

17 Exploratory factor analysis

17.1. What will this chapter tell me? 17.2. When to use factor analysis 17.3. Factors

17.3.1. Graphical representation of factors 17.3.2. Mathematical representation of factors 17.3.3. Factor scores 17.3.4. Choosing a method 17.3.5. Communality 17.3.6. Factor analysis vs. principal components analysis 17.3.7. Theory behind principal components analysis 17.3.8. Factor extraction: eigenvalues and the scree plot 17.3.9. Improving interpretation: factor rotation

17.4. Research example

17.4.1. Sample size 17.4.2. Correlations between variables 17.4.3. The distribution of data

17.5. Running the analysis with R Commander 17.6. Running the analysis with R

17.6.1. Packages used in this chapter 17.6.2. Initial preparation and analysis 17.6.3. Factor extraction using R 17.6.4. Rotation 17.6.5. Factor scores 17.6.6. Summary

17.7. How to report factor analysis 17.8. Reliability analysis

17.8.1. Measures of reliability 17.8.2. Interpreting Cronbach’s α (some cautionary tales …) 17.8.3. Reliability analysis with R Commander 17.8.4. Reliability analysis using R 17.8.5. Interpreting the output

17.9. Reporting reliability analysis What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

18 Categorical data

18.1. What will this chapter tell me? 18.2. Packages used in this chapter 18.3. Analysing categorical data 18.4. Theory of analysing categorical data

18.4.1. Pearson’s chi-square test 18.4.2. Fisher’s exact test 18.4.3. The likelihood ratio 18.4.4. Yates’s correction

18.5. Assumptions of the chi-square test 18.6. Doing the chi-square test using R

18.6.1. Entering data: raw scores 18.6.2. Entering data: the contingency table 18.6.3. Running the analysis with R Commander 18.6.4. Running the analysis using R 18.6.5. Output from the CrossTable() function 18.6.6. Breaking down a significant chi-square test with standardized residuals 18.6.7. Calculating an effect size 18.6.8. Reporting the results of chi-square

18.7. Several categorical variables: loglinear analysis

18.7.1. Chi-square as regression 18.7.2. Loglinear analysis

18.8. Assumptions in loglinear analysis 18.9. Loglinear analysis using R

18.9.1. Initial considerations 18.9.2. Loglinear analysis as a chi-square test 18.9.3. Output from loglinear analysis as a chi-square test 18.9.4. Loglinear analysis

18.10. Following up loglinear analysis 18.11. Effect sizes in loglinear analysis 18.12. Reporting the results of loglinear analysis What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

19 Multilevel linear models

19.1. What will this chapter tell me? 19.2. Hierarchical data

19.2.1. The intraclass correlation 19.2.2. Benefits of multilevel models

19.3. Theory of multilevel linear models

19.3.1. An example 19.3.2. Fixed and random coefficients

19.4. The multilevel model

19.4.1. Assessing the fit and comparing multilevel models 19.4.2. Types of covariance structures

19.5. Some practical issues

19.5.1. Assumptions 19.5.2. Sample size and power 19.5.3. Centring variables

19.6. Multilevel modelling in R

19.6.1. Packages for multilevel modelling in R 19.6.2. Entering the data 19.6.3. Picturing the data 19.6.4. Ignoring the data structure: ANOVA 19.6.5. Ignoring the data structure: ANCOVA 19.6.6. Assessing the need for a multilevel model 19.6.7. Adding in fixed effects 19.6.8. Introducing random slopes 19.6.9. Adding an interaction term to the model

19.7. Growth models

19.7.1. Growth curves (polynomials) 19.7.2. An example: the honeymoon period 19.7.3. Restructuring the data 19.7.4. Setting up the basic model 19.7.5. Adding in time as a fixed effect 19.7.6. Introducing random slopes 19.7.7. Modelling the covariance structure 19.7.8. Comparing models 19.7.9. Adding higher-order polynomials 19.7.10. Further analysis

19.8. How to report a multilevel model What have I discovered about statistics? R packages used in this chapter R functions used in this chapter Key terms that I’ve discovered Smart Alex’s tasks Further reading Interesting real research

Epilogue: life after discovering statistics Troubleshooting R Glossary Appendix

A.1. Table of the standard normal distribution A.2. Critical values of the t-distribution A.3. Critical values of the F-distribution A.4. Critical values of the chi-square distribution

References Index Functions in R Packages in R

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