Statistics for Terrified Biologists

HELMUT F. VAN EMDEN, PhD, is an internationally respected entomologist who in the UK has been President of both the Royal Entomological Society and the Association of Applied Biologists. He is currently Emeritus Professor of Horticulture in the School of Agriculture, Policy and Development at the University of Reading, UK, where for 35 years he has taught entomology at Masters level to international students of Crop Protection. He has taught and carried out research on applied entomology in six continents.

• Preface to the second edition xvPreface to the first edition xvii1 How to use this book 1Introduction 1The text of the chapters 1What should you do if you run into trouble? 2Elephants 3The numerical examples in the text 3Boxes 4Spare-time activities 4Executive summaries 5Why go to all that bother? 5The bibliography 72 Introduction 9What are statistics? 9Notation 10Notation for calculating the mean 123 Summarising variation 13Introduction 13Different summaries of variation 14Range 14Total deviation 14Mean deviation 15Variance 16Why n−1? 17Why are the deviations squared? 18The standard deviation 19The next chapter 21Spare-time activities 214 When are sums of squares NOT sums of squares? 23Introduction 23Calculating machines offer a quicker method of calculating the sum of squares 24Added squares 24The correction factor 24Avoid being confused by the term sum of squares 24Summary of the calculator method for calculations as far as the standard deviation 25Spare-time activities 265 The normal distribution 27Introduction 27Frequency distributions 27The normal distribution 28What percentage is a standard deviation worth? 30Are the percentages always the same as these? 30Other similar scales in everyday life 33The standard deviation as an estimate of the frequency of a number occurring in a sample 33From percentage to probability 34Executive Summary 1 – The standard deviation 366 The relevance of the normal distribution to biological data 39To recap 39Is our observed distribution normal? 41Checking for normality 42What can we do about a distribution that clearly is not normal? 42Transformation 42Grouping samples 47Doing nothing! 47How many samples are needed? 47Type 1 and Type 2 errors 48Calculating how many samples are needed 497 Further calculations from the normal distribution 51Introduction 51Is A bigger than B? 52The yardstick for deciding 52The standard error of a difference between two means of three eggs 53Derivation of the standard error of a difference between two means 53Step 1: from variance of single data to variance of means 55Step 2: From variance of single data to variance of differences 57Step 3: The combination of Steps 1 and 2: the standard error of difference between means (s.e.d.m.) 58Recap of the calculation of s.e.d.m. from the variance calculated from the individual values 61The importance of the standard error of differences between means 61Summary of this chapter 62Executive Summary 2 – Standard error of a difference between two means 66Spare-time activities 678 Thet-test 69Introduction 69The principle of the t-test 70The t-test in statistical terms 71Why t? 71Tables of the t-distribution 72The standard t-test 75The procedure 76The actual t-test 81t-test for means associated with unequal variances 81The s.e.d.m. when variances are unequal 82A worked example of the t-test for means associated with unequal variances 85The paired t-test 87Pair when possible 90Executive Summary 3 – The t-test 92Spare-time activities 949 One tail or two? 95Introduction 95Why is the analysis of variance F-test one-tailed? 95The two-tailed F-test 96Howmany tails has the t-test? 98The final conclusion on number of tails 9910 Analysis of variance (ANOVA): what is it? How does it work? 101Introduction 101Sums of squares in ANOVA 102Some ‘made-up’ variation to analyse by ANOVA 102The sum of squares table 104Using ANOVA to sort out the variation in Table C 104Phase 1 104Phase 2 105SqADS: an important acronym 107Back to the sum of squares table 108How well does the analysis reflect the input? 109End phase 109Degrees of freedom in ANOVA 110The completion of the end phase 112The variance ratio 113The relationship between t and F 114Constraints on ANOVA 115Adequate size of experiment 115Equality of variance between treatments 117Testing the homogeneity of variance 117The element of chance: randomisation 118Comparison between treatment means in ANOVA 119The least significant difference 121A caveat about using the LSD 123Executive Summary 4 – The principle of ANOVA 12411 Experimental designs for analysis of variance (ANOVA) 129Introduction 129Fully randomised 130Data for analysis of a fully randomised experiment 131Prelims 132Phase 1 132Phase 2 133End phase 133Randomised blocks 135Data for analysis of a randomised block experiment 137Prelims 138Phase 1 139Phase 2 140End phase 141Incomplete blocks 142Latin square 145Data for the analysis of a Latin square 145Prelims 146Phase 1 150Phase 2 150End phase 151Further comments on the Latin square design 152Split plot 154Types of analysis of variance 154One- and two-way analysis of variance 155Fixed-, random-, and mixed-effects analysis of variance 156Executive Summary 5 – Analysis of a one-way randomised block experiment 158Spare-time activities 15912 Introduction to factorial experiments 163What is a factorial experiment? 163Interaction: what does it mean biologically? 165If there is no interaction 167What if there IS interaction? 167How about a biological example? 168Measuring any interaction between factors is often the main/only purpose of an experiment 170How does a factorial experiment change the form of the analysis of variance? 171Degrees of freedom for interactions 171The similarity between the residual in Phase 2 and the interaction in Phase 3 172Sums of squares for interactions 17213 2-Factor factorial experiments 175Introduction 175An example of a 2-factor experiment 175Analysis of the 2-factor experiment 176Prelims 176Phase 1 177Phase 2 177End phase (of Phase 2) 178Phase 3 179End phase (of Phase 3) 183Two important things to remember about factorials before tackling the next chapter 185Analysis of factorial experiments with unequal replication 185Executive Summary 6 – Analysis of a 2-factor randomised block experiment 188Spare-time activity 19014 Factorial experiments with more than two factors – leave this out if you wish! 191Introduction 191Different ‘orders’ of interaction 191Example of a 4-factor experiment 192Prelims 194Phase 1 196Phase 2 196Phase 3 197To the end phase 205Spare-time activity 21415 Factorial experiments with split plots 217Introduction 217Deriving the split plot design from the randomised block design 218Degrees of freedom in a split plot analysis 221Main plots 221Sub-plots 222Numerical example of a split plot experiment and its analysis 224Calculating the sums of squares 225End phase 229Comparison of split plot and randomised block experiments 229Uses of split plot designs 233Spare-time activity 23516 The t-test in the analysis of variance 237Introduction 237Brief recap of relevant earlier sections of this book 238Least significant difference test 239Multiple range tests 240Operating the multiple range test 242Testing differences between means 246My rules for testing differences between means 246Presentation of the results of tests of differences between means 247The results of the experiments analysed by analysis of variance in Chapters 11–15 249Fully randomised design (p. 131) 250Randomised block experiment (p. 137) 251Latin square design (p. 146) 2532-Factor experiment (p. 176) 2554-Factor experiment (p. 195) 257Split plot experiment (p. 224) 259Some final advice 261Spare-time activities 26117 Linear regression and correlation 263Introduction 263Cause and effect 264Other traps waiting for you to fall into 264Extrapolating beyond the range of your data 264Is a straight line appropriate? 265The distribution of variability 268Regression 268Independent and dependent variables 272The regression coefficient (b) 272Calculating the regression coefficient (b) 275The regression equation 281A worked example on some real data 282The data 282Calculating the regression coefficient (b), i.e. the slope of the regression line 282Calculating the intercept (a) 284Drawing the regression line 285Testing the significance of the slope (b) of the regression 286How well do the points fit the line? The coefficient of determination (r2) 290Correlation 291Derivation of the correlation coefficient (r) 291An example of correlation 292Is there a correlation line? 293Extensions of regression analysis 296Nonlinear regression 297Multiple linear regression 298Multiple nonlinear regression 300Executive Summary – Linear regression 301Spare time activities 30318 Analysis of covariance (ANCOVA) 305Introduction 305A worked example of ANCOVA 307Data: cholesterol levels of subjects given different diets 307Data: ages of subjects in experiment 308Regression of cholesterol level on age 309The structure of the ANCOVA table 312Total sum of squares 313Residual sum of squares 314Corrected means 316Test for significant difference between means 316Executive Summary 8 – Analysis of covariance (ANCOVA) 319Spare-time activity 32019 Chi-square tests 323Introduction 323When not and where not to use 𝜒 2 324The problem of low frequencies 325Yates’ correction for continuity 325The 𝜒 2 test for goodness of fit 326The case of more than two classes 328𝜒 2 with heterogeneity 331Heterogeneity 𝜒 2 Analysis with ‘Covariance’ 333Association (or contingency) 𝜒 2 3352 × 2 contingency table 336Fisher’s exact test for a 2 × 2 table 338Larger contingency tables 340Interpretation of contingency tables 341Spare-time activities 34320 Nonparametric methods (what are they?) 345Disclaimer 345Introduction 346Advantages and disadvantages of parametric and nonparametric methods 347Where nonparametric methods score 347Where parametric methods score 349Some ways data are organised for nonparametric tests 349The sign test 350The Kruskal–Wallis analysis of ranks 350Kendall’s rank correlation coefficient 352The main nonparametric methods that are available 353Analysis of two replicated treatments as in the t-test (Chapter 8) 353Analysis of more than two replicated treatments as in the analysis of variance (Chapter 11) 354Correlation of two variables (Chapter 17) 354Appendix A How many replicates? 355Appendix B Statistical tables 365Appendix C Solutions to spare-time activities 373Appendix D Bibliography 393Index 397