Choosing and Using Statistics

Calvin Dytham has a wide range of research interests in ecology and evolutionary biology and is especially interested in the impacts of dispersal and the arrangement of individuals in space on ecological and evolutionary processes.  He is a Reader in the Department of Biology at the University of York, UK, and has been teaching statistics to undergraduate and postgraduate students since 1994.

"Written in a concise and direct style, this book presents a selection of some of the most widely used statistical tests and data exploration techniques." (Biological Conservation, 1 March 2012) "Written in a concise and direct style, this book presents a selection of some of the most widely used statistical tests and data exploration techniques ... In general, this book is a very good primer for students with no statistical expertise." (Biological Conservation Reviews, 2011)"This book makes everything so easy. Complicated tests are effortlessly condensed, and the instructions are almost too easy to follow. Diagrams and sample data sets are used frequently so you can practise using tests before applying them to your own data sets, whilst the logical layout guides you toward the correct test for both your data, and what you want to prove (or disprove)." (Animals & Men, February 2011)

• Preface xiii The third edition xivHow to use this book xivPackages used xvExample data xvAcknowledgements for the first edition xvAcknowledgements for the second edition xvAcknowledgements for the third edition xvi1 Eight steps to successful data analysis 12 The basics 2Observations 2Hypothesis testing 2P-values 3Sampling 3Experiments 4Statistics 4Descriptive statistics 5Tests of difference 5Tests of relationships 5Tests for data investigation 63 Choosing a test: a key 7Remember: eight steps to successful data analysis 7The art of choosing a test 7A key to assist in your choice of statistical test 84 Hypothesis testing, sampling and experimental design 23Hypothesis testing 23Acceptable errors 23P-values 24Sampling 25Choice of sample unit 25Number of sample units 26Positioning of sample units to achieve a random sample 26Timing of sampling 27Experimental design 27Control 28Procedural controls 28Temporal control 28Experimental control 29Statistical control 29Some standard experimental designs 295 Statistics, variables and distributions 32What are statistics? 32Types of statistics 33Descriptive statistics 33Parametric statistics 33Non-parametric statistics 33What is a variable? 33Types of variables or scales of measurement 34Measurement variables 34Continuous variables 34Discrete variables 35How accurate do I need to be? 35Ranked variables 35Attributes 35Derived variables 36Types of distribution 36Discrete distributions 36The Poisson distribution 36The binomial distribution 37The negative binomial distribution 39The hypergeometric distribution 39Continuous distributions 40The rectangular distribution 40The normal distribution 40The standardized normal distribution 40Convergence of a Poisson distribution to a normal distribution 41Sampling distributions and the 'central limit theorem' 41Describing the normal distribution further 41Skewness 41Kurtosis 43Is a distribution normal? 43Transformations 43An example 44The angular transformation 44The logit transformation 45The t-distribution 46Confidence intervals 47The chi-square distribution 47The exponential distribution 47Non-parametric 'distributions' 48Ranking, quartiles and the interquartile range 48Box and whisker plots 486 Descriptive and presentational techniques 49General advice 49Displaying data: summarizing a single variable 49Box and whisker plot (box plot) 49Displaying data: showing the distribution of a single variable 50Bar chart: for discrete data 50Histogram: for continuous data 51Pie chart: for categorical data or attribute data 52Descriptive statistics 52Statistics of location or position 52Arithmetic mean 53Geometric mean 53Harmonic mean 53Median 53Mode 53Statistics of distribution, dispersion or spread 55Range 55Interquartile range 55Variance 55Standard deviation (SD) 55Standard error (SE) 56Confidence intervals (CI) or confidence limits 56Coefficient of variation 56Other summary statistics 56Skewness 57Kurtosis 57Using the computer packages 57General 57Displaying data: summarizing two or more variables 62Box and whisker plots (box plots) 62Error bars and confidence intervals 63Displaying data: comparing two variables 63Associations 63Scatterplots 64Multiple scatterplots 64Trends, predictions and time series 65Lines 65Fitted lines 67Confidence intervals 67Displaying data: comparing more than two variables 68Associations 68Three-dimensional scatterplots 68Multiple trends, time series and predictions 69Multiple fitted lines 69Surfaces 707 The tests 1: tests to look at differences 72Do frequency distributions differ? 72Questions 72G-test 72An example 73Chi-square test 75An example 76Kolmogorov–Smirnov test 86An example 87Anderson–Darling test 89Shapiro–Wilk test 90Graphical tests for normality 90Do the observations from two groups differ? 92Paired data 92Paired t-test 92Wilcoxon signed ranks test 96Sign test 99Unpaired data 103t-test 103One-way ANOVA 111Mann–Whitney U 119Do the observations from more than two groups differ? 123Repeated measures 123Friedman test (for repeated measures) 123Repeated-measures ANOVA 127Independent samples 128One-way ANOVA 129Post hoc testing: after one-way ANOVA 138Kruskal–Wallis test 142Post hoc testing: after the Kruskal–Wallis test 145There are two independent ways of classifying the data 145One observation for each factor combination (no replication) 146Friedman test 146Two-way ANOVA (without replication) 152More than one observation for each factor combination (withreplication) 160Interaction 160Two-way ANOVA (with replication) 163An example 164Scheirer–Ray–Hare test 175An example 175There are more than two independent ways to classify the data 182Multifactorial testing 182Three-way ANOVA (without replication) 183Three-way ANOVA (with replication) 184An example 184Multiway ANOVA 191Not all classifications are independent 192Non-independent factors 192Nested factors 192Random or fixed factors 193Nested or hierarchical designs 193Two-level nested-design ANOVA 193An example 1938 The tests 2: tests to look at relationships 199Is there a correlation or association between two variables? 199Observations assigned to categories 199Chi-square test of association 199An example 200Cramér coefficient of association 208Phi coefficient of association 209Observations assigned a value 209'Standard' correlation (Pearson's product-moment correlation) 210An example 210Spearman's rank-order correlation 214An example 215Kendall rank-order correlation 218An example 218Regression 219An example 220Is there a cause-and-effect relationship between two variables? 220Questions 220'Standard' linear regression 221Prediction 221Interpreting r2 222Comparison of regression and correlation 222Residuals 222Confidence intervals 222Prediction interval 223An example 223Kendall robust line-fit method 230Logistic regression 230An example 231Model II regression 235Polynomial, cubic and quadratic regression 235Tests for more than two variables 236Tests of association 236Questions 236Correlation 236Partial correlation 237Kendall partial rank-order correlation 237Cause(s) and effect(s) 237Questions 237Regression 237Analysis of covariance (ANCOVA) 238Multiple regression 242Stepwise regression 242Path analysis 2439 The tests 3: tests for data exploration 244Types of data 244Observation, inspection and plotting 244Principal component analysis (PCA) and factor analysis 244An example 245Canonical variate analysis 251Discriminant function analysis 251An example 251Multivariate analysis of variance (MANOVA) 256An example 256Multivariate analysis of covariance (MANCOVA) 259Cluster analysis 259DECORANA and TWINSPAN 263Symbols and letters used in statistics 264Greek letters 264Symbols 264Upper-case letters 265Lower-case letters 266Glossary 267Assumptions of the tests 282What if the assumptions are violated? 284Hints and tips 285Using a computer 285Sampling 286Statistics 286Displaying the data 287A table of statistical tests 289Index 291