Statistics

Preface xi Chapter 1 Fundamentals 1 Everything Varies 2 Significance 3 Good and Bad Hypotheses 3 Null Hypotheses 3 p Values 3 Interpretation 4 Model Choice 4 Statistical Modelling 5 Maximum Likelihood 6 Experimental Design 7 The Principle of Parsimony (Occam's Razor) 8 Observation, Theory and Experiment 8 Controls 8 Replication: It's the ns that Justify the Means 8 How Many Replicates? 9 Power 9 Randomization 10 Strong Inference 14 Weak Inference 14 How Long to Go On? 14 Pseudoreplication 15 Initial Conditions 16 Orthogonal Designs and Non-Orthogonal Observational Data 16 Aliasing 16 Multiple Comparisons 17 Summary of Statistical Models in R 18 Organizing Your Work 19 Housekeeping within R 20 References 22 Further Reading 22 Chapter 2 Dataframes 23 Selecting Parts of a Dataframe: Subscripts 26 Sorting 27 Summarizing the Content of Dataframes 29 Summarizing by Explanatory Variables 30 First Things First: Get to Know Your Data 31 Relationships 34 Looking for Interactions between Continuous Variables 36 Graphics to Help with Multiple Regression 39 Interactions Involving Categorical Variables 39 Further Reading 41 Chapter 3 Central Tendency 42 Further Reading 49 Chapter 4 Variance 50 Degrees of Freedom 53 Variance 53 Variance: A Worked Example 55 Variance and Sample Size 58 Using Variance 59 A Measure of Unreliability 60 Confidence Intervals 61 Bootstrap 62 Non-constant Variance: Heteroscedasticity 65 Further Reading 65 Chapter 5 Single Samples 66 Data Summary in the One-Sample Case 66 The Normal Distribution 70 Calculations Using z of the Normal Distribution 76 Plots for Testing Normality of Single Samples 79 Inference in the One-Sample Case 81 Bootstrap in Hypothesis Testing with Single Samples 81 Student's t Distribution 82 Higher-Order Moments of a Distribution 83 Skew 84 Kurtosis 86 Reference 87 Further Reading 87 Chapter 6 Two Samples 88 Comparing Two Variances 88 Comparing Two Means 90 Student's t Test 91 Wilcoxon Rank-Sum Test 95 Tests on Paired Samples 97 The Binomial Test 98 Binomial Tests to Compare Two Proportions 100 Chi-Squared Contingency Tables 100 Fisher's Exact Test 105 Correlation and Covariance 108 Correlation and the Variance of Differences between Variables 110 Scale-Dependent Correlations 112 Reference 113 Further Reading 113 Chapter 7 Regression 114 Linear Regression 116 Linear Regression in R 117 Calculations Involved in Linear Regression 122 Partitioning Sums of Squares in Regression: SSY = SSR + SSE 125 Measuring the Degree of Fit, r2 133 Model Checking 134 Transformation 135 Polynomial Regression 140 Non-Linear Regression 142 Generalized Additive Models 146 Influence 148 Further Reading 149 Chapter 8 Analysis of Variance 150 One-Way ANOVA 150 Shortcut Formulas 157 Effect Sizes 159 Plots for Interpreting One-Way ANOVA 162 Factorial Experiments 168 Pseudoreplication: Nested Designs and Split Plots 173 Split-Plot Experiments 174 Random Effects and Nested Designs 176 Fixed or Random Effects? 177 Removing the Pseudoreplication 178 Analysis of Longitudinal Data 178 Derived Variable Analysis 179 Dealing with Pseudoreplication 179 Variance Components Analysis (VCA) 183 References 184 Further Reading 184 Chapter 9 Analysis of Covariance 185 Further Reading 192 Chapter 10 Multiple Regression 193 The Steps Involved in Model Simplification 195 Caveats 196 Order of Deletion 196 Carrying Out a Multiple Regression 197 A Trickier Example 203 Further Reading 211 Chapter 11 Contrasts 212 Contrast Coefficients 213 An Example of Contrasts in R 214 A Priori Contrasts 215 Treatment Contrasts 216 Model Simplification by Stepwise Deletion 218 Contrast Sums of Squares by Hand 222 The Three Kinds of Contrasts Compared 224 Reference 225 Further Reading 225 Chapter 12 Other Response Variables 226 Introduction to Generalized Linear Models 228 The Error Structure 229 The Linear Predictor 229 Fitted Values 230 A General Measure of Variability 230 The Link Function 231 Canonical Link Functions 232 Akaike's Information Criterion (AIC) as a Measure of the Fit of a Model 233 Further Reading 233 Chapter 13 Count Data 234 A Regression with Poisson Errors 234 Analysis of Deviance with Count Data 237 The Danger of Contingency Tables 244 Analysis of Covariance with Count Data 247 Frequency Distributions 250 Further Reading 255 Chapter 14 Proportion Data 256 Analyses of Data on One and Two Proportions 257 Averages of Proportions 257 Count Data on Proportions 257 Odds 259 Overdispersion and Hypothesis Testing 260 Applications 261 Logistic Regression with Binomial Errors 261 Proportion Data with Categorical Explanatory Variables 264 Analysis of Covariance with Binomial Data 269 Further Reading 272 Chapter 15 Binary Response Variable 273 Incidence Functions 275 ANCOVA with a Binary Response Variable 279 Further Reading 284 Chapter 16 Death and Failure Data 285 Survival Analysis with Censoring 287 Further Reading 290 Appendix Essentials of the R Language 291 R as a Calculator 291 Built-in Functions 292 Numbers with Exponents 294 Modulo and Integer Quotients 294 Assignment 295 Rounding 295 Infinity and Things that Are Not a Number (NaN) 296 Missing Values (NA) 297 Operators 298 Creating a Vector 298 Named Elements within Vectors 299 Vector Functions 299 Summary Information from Vectors by Groups 300 Subscripts and Indices 301 Working with Vectors and Logical Subscripts 301 Addresses within Vectors 304 Trimming Vectors Using Negative Subscripts 304 Logical Arithmetic 305 Repeats 305 Generate Factor Levels 306 Generating Regular Sequences of Numbers 306 Matrices 307 Character Strings 309 Writing Functions in R 310 Arithmetic Mean of a Single Sample 310 Median of a Single Sample 310 Loops and Repeats 311 The ifelse Function 312 Evaluating Functions with apply 312 Testing for Equality 313 Testing and Coercing in R 314 Dates and Times in R 315 Calculations with Dates and Times 319 Understanding the Structure of an R Object Using str 320 Reference 322 Further Reading 322 Index 323