Statistics for Terrified Biologists

Makes mathematical and statistical analysis understandable to even the least math-minded biology student

This unique textbook aims to demystify statistical formulae for the average biology student. Written in a lively and engaging style, Statistics for Terrified Biologists, 2^nd Edition draws on the author’s 30 years of lecturing experience to teach statistical methods to even the most guarded of biology students. It presents basic methods using straightforward, jargon-free language. Students are taught to use simple formulae and how to interpret what is being measured with each test and statistic, while at the same time learning to recognize overall patterns and guiding principles. Complemented by simple examples and useful case studies, this is an ideal statistics resource tool for undergraduate biology and environmental science students who lack confidence in their mathematical abilities. 

Statistics for Terrified Biologists presents readers with the basic foundations of parametric statistics, the t-test, analysis of variance, linear regression and chi-square, and guides them to important extensions of these techniques. It introduces them to non-parametric tests, and includes a checklist of non-parametric methods linked to their parametric counterparts. The book also provides many end-of-chapter summaries and additional exercises to help readers understand and practice what they’ve learned.

• Presented in a clear and easy-to-understand style
• Makes statistics tangible and enjoyable for even the most hesitant student
• Features multiple formulas to facilitate comprehension
• Written by of the foremost entomologists of his generation

This second edition of Statistics for Terrified Biologists is an invaluable guide that will be of great benefit to pre-health and biology undergraduate students.

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 xv

Preface to the first edition xvii

1 How to use this book 1

Introduction 1

The text of the chapters 1

What should you do if you run into trouble? 2

Elephants 3

The numerical examples in the text 3

Boxes 4

Spare-time activities 4

Executive summaries 5

Why go to all that bother? 5

The bibliography 7

2 Introduction 9

What are statistics? 9

Notation 10

Notation for calculating the mean 12

3 Summarising variation 13

Introduction 13

Different summaries of variation 14

Range 14

Total deviation 14

Mean deviation 15

Variance 16

Why n−1? 17

Why are the deviations squared? 18

The standard deviation 19

The next chapter 21

Spare-time activities 21

4 When are sums of squares NOT sums of squares? 23

Introduction 23

Calculating machines offer a quicker method of calculating the sum of squares 24

Added squares 24

The correction factor 24

Avoid being confused by the term sum of squares 24

Summary of the calculator method for calculations as far as the standard deviation 25

Spare-time activities 26

5 The normal distribution 27

Introduction 27

Frequency distributions 27

The normal distribution 28

What percentage is a standard deviation worth? 30

Are the percentages always the same as these? 30

Other similar scales in everyday life 33

The standard deviation as an estimate of the frequency of a number occurring in a sample 33

From percentage to probability 34

Executive Summary 1 – The standard deviation 36

6 The relevance of the normal distribution to biological data 39

To recap 39

Is our observed distribution normal? 41

Checking for normality 42

What can we do about a distribution that clearly is not normal? 42

Transformation 42

Grouping samples 47

Doing nothing! 47

How many samples are needed? 47

Type 1 and Type 2 errors 48

Calculating how many samples are needed 49

7 Further calculations from the normal distribution 51

Introduction 51

Is A bigger than B? 52

The yardstick for deciding 52

The standard error of a difference between two means of three eggs 53

Derivation of the standard error of a difference between two means 53

Step 1: from variance of single data to variance of means 55

Step 2: From variance of single data to variance of differences 57

Step 3: The combination of Steps 1 and 2: the standard error of difference between means (s.e.d.m.) 58

Recap of the calculation of s.e.d.m. from the variance calculated from the individual values 61

The importance of the standard error of differences between means 61

Summary of this chapter 62

Executive Summary 2 – Standard error of a difference between two means 66

Spare-time activities 67

8 Thet-test 69

Introduction 69

The principle of the t-test 70

The t-test in statistical terms 71

Why t? 71

Tables of the t-distribution 72

The standard t-test 75

The procedure 76

The actual t-test 81

t-test for means associated with unequal variances 81

The s.e.d.m. when variances are unequal 82

A worked example of the t-test for means associated with unequal variances 85

The paired t-test 87

Pair when possible 90

Executive Summary 3 – The t-test 92

Spare-time activities 94

9 One tail or two? 95

Introduction 95

Why is the analysis of variance F-test one-tailed? 95

The two-tailed F-test 96

Howmany tails has the t-test? 98

The final conclusion on number of tails 99

10 Analysis of variance (ANOVA): what is it? How does it work? 101

Introduction 101

Sums of squares in ANOVA 102

Some ‘made-up’ variation to analyse by ANOVA 102

The sum of squares table 104

Using ANOVA to sort out the variation in Table C 104

Phase 1 104

Phase 2 105

SqADS: an important acronym 107

Back to the sum of squares table 108

How well does the analysis reflect the input? 109

End phase 109

Degrees of freedom in ANOVA 110

The completion of the end phase 112

The variance ratio 113

The relationship between t and F 114

Constraints on ANOVA 115

Adequate size of experiment 115

Equality of variance between treatments 117

Testing the homogeneity of variance 117

The element of chance: randomisation 118

Comparison between treatment means in ANOVA 119

The least significant difference 121

A caveat about using the LSD 123

Executive Summary 4 – The principle of ANOVA 124

11 Experimental designs for analysis of variance (ANOVA) 129

Introduction 129

Fully randomised 130

Data for analysis of a fully randomised experiment 131

Prelims 132

Phase 1 132

Phase 2 133

End phase 133

Randomised blocks 135

Data for analysis of a randomised block experiment 137

Prelims 138

Phase 1 139

Phase 2 140

End phase 141

Incomplete blocks 142

Latin square 145

Data for the analysis of a Latin square 145

Prelims 146

Phase 1 150

Phase 2 150

End phase 151

Further comments on the Latin square design 152

Split plot 154

Types of analysis of variance 154

One- and two-way analysis of variance 155

Fixed-, random-, and mixed-effects analysis of variance 156

Executive Summary 5 – Analysis of a one-way randomised block experiment 158

Spare-time activities 159

12 Introduction to factorial experiments 163

What is a factorial experiment? 163

Interaction: what does it mean biologically? 165

If there is no interaction 167

What if there IS interaction? 167

How about a biological example? 168

Measuring any interaction between factors is often the main/only purpose of an experiment 170

How does a factorial experiment change the form of the analysis of variance? 171

Degrees of freedom for interactions 171

The similarity between the residual in Phase 2 and the interaction in Phase 3 172

Sums of squares for interactions 172

13 2-Factor factorial experiments 175

Introduction 175

An example of a 2-factor experiment 175

Analysis of the 2-factor experiment 176

Prelims 176

Phase 1 177

Phase 2 177

End phase (of Phase 2) 178

Phase 3 179

End phase (of Phase 3) 183

Two important things to remember about factorials before tackling the next chapter 185

Analysis of factorial experiments with unequal replication 185

Executive Summary 6 – Analysis of a 2-factor randomised block experiment 188

Spare-time activity 190

14 Factorial experiments with more than two factors – leave this out if you wish! 191

Introduction 191

Different ‘orders’ of interaction 191

Example of a 4-factor experiment 192

Prelims 194

Phase 1 196

Phase 2 196

Phase 3 197

To the end phase 205

Spare-time activity 214

15 Factorial experiments with split plots 217

Introduction 217

Deriving the split plot design from the randomised block design 218

Degrees of freedom in a split plot analysis 221

Main plots 221

Sub-plots 222

Numerical example of a split plot experiment and its analysis 224

Calculating the sums of squares 225

End phase 229

Comparison of split plot and randomised block experiments 229

Uses of split plot designs 233

Spare-time activity 235

16 The t-test in the analysis of variance 237

Introduction 237

Brief recap of relevant earlier sections of this book 238

Least significant difference test 239

Multiple range tests 240

Operating the multiple range test 242

Testing differences between means 246

My rules for testing differences between means 246

Presentation of the results of tests of differences between means 247

The results of the experiments analysed by analysis of variance in Chapters 11–15 249

Fully randomised design (p. 131) 250

Randomised block experiment (p. 137) 251

Latin square design (p. 146) 253

2-Factor experiment (p. 176) 255

4-Factor experiment (p. 195) 257

Split plot experiment (p. 224) 259

Some final advice 261

Spare-time activities 261

17 Linear regression and correlation 263

Introduction 263

Cause and effect 264

Other traps waiting for you to fall into 264

Extrapolating beyond the range of your data 264

Is a straight line appropriate? 265

The distribution of variability 268

Regression 268

Independent and dependent variables 272

The regression coefficient (b) 272

Calculating the regression coefficient (b) 275

The regression equation 281

A worked example on some real data 282

The data 282

Calculating the regression coefficient (b), i.e. the slope of the regression line 282

Calculating the intercept (a) 284

Drawing the regression line 285

Testing the significance of the slope (b) of the regression 286

How well do the points fit the line? The coefficient of determination (r²) 290

Correlation 291

Derivation of the correlation coefficient (r) 291

An example of correlation 292

Is there a correlation line? 293

Extensions of regression analysis 296

Nonlinear regression 297

Multiple linear regression 298

Multiple nonlinear regression 300

Executive Summary – Linear regression 301

Spare time activities 303

18 Analysis of covariance (ANCOVA) 305

Introduction 305

A worked example of ANCOVA 307

Data: cholesterol levels of subjects given different diets 307

Data: ages of subjects in experiment 308

Regression of cholesterol level on age 309

The structure of the ANCOVA table 312

Total sum of squares 313

Residual sum of squares 314

Corrected means 316

Test for significant difference between means 316

Executive Summary 8 – Analysis of covariance (ANCOVA) 319

Spare-time activity 320

19 Chi-square tests 323

Introduction 323

When not and where not to use 𝜒 ² 324

The problem of low frequencies 325

Yates’ correction for continuity 325

The 𝜒 ² test for goodness of fit 326

The case of more than two classes 328

𝜒 ² with heterogeneity 331

Heterogeneity 𝜒 ² Analysis with ‘Covariance’ 333

Association (or contingency) 𝜒 ² 335

2 × 2 contingency table 336

Fisher’s exact test for a 2 × 2 table 338

Larger contingency tables 340

Interpretation of contingency tables 341

Spare-time activities 343

20 Nonparametric methods (what are they?) 345

Disclaimer 345

Introduction 346

Advantages and disadvantages of parametric and nonparametric methods 347

Where nonparametric methods score 347

Where parametric methods score 349

Some ways data are organised for nonparametric tests 349

The sign test 350

The Kruskal–Wallis analysis of ranks 350

Kendall’s rank correlation coefficient 352

The main nonparametric methods that are available 353

Analysis of two replicated treatments as in the t-test (Chapter 8) 353

Analysis of more than two replicated treatments as in the analysis of variance (Chapter 11) 354

Correlation of two variables (Chapter 17) 354

Appendix A How many replicates? 355

Appendix B Statistical tables 365

Appendix C Solutions to spare-time activities 373

Appendix D Bibliography 393

Index 397

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