Choosing and Using Statistics

Preface xiii The third edition xiv How to use this book xiv Packages used xv Example data xv Acknowledgements for the first edition xv Acknowledgements for the second edition xv Acknowledgements for the third edition xvi 1 Eight steps to successful data analysis 1 2 The basics 2 Observations 2 Hypothesis testing 2 P-values 3 Sampling 3 Experiments 4 Statistics 4 Descriptive statistics 5 Tests of difference 5 Tests of relationships 5 Tests for data investigation 6 3 Choosing a test: a key 7 Remember: eight steps to successful data analysis 7 The art of choosing a test 7 A key to assist in your choice of statistical test 8 4 Hypothesis testing, sampling and experimental design 23 Hypothesis testing 23 Acceptable errors 23 P-values 24 Sampling 25 Choice of sample unit 25 Number of sample units 26 Positioning of sample units to achieve a random sample 26 Timing of sampling 27 Experimental design 27 Control 28 Procedural controls 28 Temporal control 28 Experimental control 29 Statistical control 29 Some standard experimental designs 29 5 Statistics, variables and distributions 32 What are statistics? 32 Types of statistics 33 Descriptive statistics 33 Parametric statistics 33 Non-parametric statistics 33 What is a variable? 33 Types of variables or scales of measurement 34 Measurement variables 34 Continuous variables 34 Discrete variables 35 How accurate do I need to be? 35 Ranked variables 35 Attributes 35 Derived variables 36 Types of distribution 36 Discrete distributions 36 The Poisson distribution 36 The binomial distribution 37 The negative binomial distribution 39 The hypergeometric distribution 39 Continuous distributions 40 The rectangular distribution 40 The normal distribution 40 The standardized normal distribution 40 Convergence of a Poisson distribution to a normal distribution 41 Sampling distributions and the 'central limit theorem' 41 Describing the normal distribution further 41 Skewness 41 Kurtosis 43 Is a distribution normal? 43 Transformations 43 An example 44 The angular transformation 44 The logit transformation 45 The t-distribution 46 Confidence intervals 47 The chi-square distribution 47 The exponential distribution 47 Non-parametric 'distributions' 48 Ranking, quartiles and the interquartile range 48 Box and whisker plots 48 6 Descriptive and presentational techniques 49 General advice 49 Displaying data: summarizing a single variable 49 Box and whisker plot (box plot) 49 Displaying data: showing the distribution of a single variable 50 Bar chart: for discrete data 50 Histogram: for continuous data 51 Pie chart: for categorical data or attribute data 52 Descriptive statistics 52 Statistics of location or position 52 Arithmetic mean 53 Geometric mean 53 Harmonic mean 53 Median 53 Mode 53 Statistics of distribution, dispersion or spread 55 Range 55 Interquartile range 55 Variance 55 Standard deviation (SD) 55 Standard error (SE) 56 Confidence intervals (CI) or confidence limits 56 Coefficient of variation 56 Other summary statistics 56 Skewness 57 Kurtosis 57 Using the computer packages 57 General 57 Displaying data: summarizing two or more variables 62 Box and whisker plots (box plots) 62 Error bars and confidence intervals 63 Displaying data: comparing two variables 63 Associations 63 Scatterplots 64 Multiple scatterplots 64 Trends, predictions and time series 65 Lines 65 Fitted lines 67 Confidence intervals 67 Displaying data: comparing more than two variables 68 Associations 68 Three-dimensional scatterplots 68 Multiple trends, time series and predictions 69 Multiple fitted lines 69 Surfaces 70 7 The tests 1: tests to look at differences 72 Do frequency distributions differ? 72 Questions 72 G-test 72 An example 73 Chi-square test 75 An example 76 Kolmogorov–Smirnov test 86 An example 87 Anderson–Darling test 89 Shapiro–Wilk test 90 Graphical tests for normality 90 Do the observations from two groups differ? 92 Paired data 92 Paired t-test 92 Wilcoxon signed ranks test 96 Sign test 99 Unpaired data 103 t-test 103 One-way ANOVA 111 Mann–Whitney U 119 Do the observations from more than two groups differ? 123 Repeated measures 123 Friedman test (for repeated measures) 123 Repeated-measures ANOVA 127 Independent samples 128 One-way ANOVA 129 Post hoc testing: after one-way ANOVA 138 Kruskal–Wallis test 142 Post hoc testing: after the Kruskal–Wallis test 145 There are two independent ways of classifying the data 145 One observation for each factor combination (no replication) 146 Friedman test 146 Two-way ANOVA (without replication) 152 More than one observation for each factor combination (with replication) 160 Interaction 160 Two-way ANOVA (with replication) 163 An example 164 Scheirer–Ray–Hare test 175 An example 175 There are more than two independent ways to classify the data 182 Multifactorial testing 182 Three-way ANOVA (without replication) 183 Three-way ANOVA (with replication) 184 An example 184 Multiway ANOVA 191 Not all classifications are independent 192 Non-independent factors 192 Nested factors 192 Random or fixed factors 193 Nested or hierarchical designs 193 Two-level nested-design ANOVA 193 An example 193 8 The tests 2: tests to look at relationships 199 Is there a correlation or association between two variables? 199 Observations assigned to categories 199 Chi-square test of association 199 An example 200 Cramér coefficient of association 208 Phi coefficient of association 209 Observations assigned a value 209 'Standard' correlation (Pearson's product-moment correlation) 210 An example 210 Spearman's rank-order correlation 214 An example 215 Kendall rank-order correlation 218 An example 218 Regression 219 An example 220 Is there a cause-and-effect relationship between two variables? 220 Questions 220 'Standard' linear regression 221 Prediction 221 Interpreting r2 222 Comparison of regression and correlation 222 Residuals 222 Confidence intervals 222 Prediction interval 223 An example 223 Kendall robust line-fit method 230 Logistic regression 230 An example 231 Model II regression 235 Polynomial, cubic and quadratic regression 235 Tests for more than two variables 236 Tests of association 236 Questions 236 Correlation 236 Partial correlation 237 Kendall partial rank-order correlation 237 Cause(s) and effect(s) 237 Questions 237 Regression 237 Analysis of covariance (ANCOVA) 238 Multiple regression 242 Stepwise regression 242 Path analysis 243 9 The tests 3: tests for data exploration 244 Types of data 244 Observation, inspection and plotting 244 Principal component analysis (PCA) and factor analysis 244 An example 245 Canonical variate analysis 251 Discriminant function analysis 251 An example 251 Multivariate analysis of variance (MANOVA) 256 An example 256 Multivariate analysis of covariance (MANCOVA) 259 Cluster analysis 259 DECORANA and TWINSPAN 263 Symbols and letters used in statistics 264 Greek letters 264 Symbols 264 Upper-case letters 265 Lower-case letters 266 Glossary 267 Assumptions of the tests 282 What if the assumptions are violated? 284 Hints and tips 285 Using a computer 285 Sampling 286 Statistics 286 Displaying the data 287 A table of statistical tests 289 Index 291