Preface xvii

List of abbreviations xix

1 Statistical measures 1

1.1 Introduction 1

1.2 Mean, mode and median 2

1.3 Variance and standard deviation 3

1.4 Quartiles, deciles and percentiles 4

1.5 Skewness and kurtosis 5

1.6 Frequency distributions 6

1.7 Covariance and correlation 7

1.8 Joint frequency distribution 9

1.9 Linear transformation of the observations 10

1.10 Linear combinations of two sets of observations 10

Exercises 11

2 Probability, random variable, expected value and variance 14

2.1 Introduction 14

2.2 Events and probabilities 14

2.3 Mutually exclusive events 15

2.4 Independent and dependent events 15

2.5 Addition of probabilities 16

2.6 Bayes’ theorem 16

2.7 Random variables and probability distributions 17

2.8 Expected value, variance and standard deviation 17

2.9 Moments of a distribution 18

Exercises 18

3 Odds ratios, relative risk, sensitivity, specificity and the ROC curve 19

3.1 Introduction 19

3.2 Odds ratio 19

3.3 Relative risk 20

3.4 Sensitivity and specificity 21

3.5 The receiver operating characteristic (ROC) curve 22

Exercises 22

4 Probability distributions, expectations, variances and correlation 24

4.1 Introduction 24

4.2 Probability distribution of a discrete random variable 25

4.3 Discrete distributions 25

4.3.1 Uniform distribution 25

4.3.2 Binomial distribution 26

4.3.3 Multinomial distribution 27

4.3.4 Poisson distribution 27

4.3.5 Hypergeometric distribution 28

4.4 Continuous distributions 29

4.4.1 Uniform distribution of a continuous variable 29

4.4.2 Normal distribution 29

4.4.3 Normal approximation to the binomial distribution 30

4.4.4 Gamma distribution 31

4.4.5 Exponential distribution 32

4.4.6 Chisquare distribution 33

4.4.7 Weibull distribution 34

4.4.8 Student’s t, and F distributions 34

4.5 Joint distribution of two discrete random variables 34

4.5.1 Conditional distributions, means and variances 35

4.5.2 Unconditional expectations and variances 36

4.6 Bivariate normal distribution 37

Exercises 38

Appendix A4 38

A4.1 Expected values and standard deviations of the distributions 38

A4.2 Covariance and Correlation of the Numbers of Successes X and Failures (n – X) of the Binomial Random Variable 39

5 Means, standard errors and confidence limits 40

5.1 Introduction 40

5.2 Expectation, variance and standard error (S.E.) of the sample mean 41

5.3 Estimation of the variance and standard error 42

5.4 Confidence limits for the mean 43

5.5 Estimator and confidence limits for the difference of two means 44

5.6 Approximate confidence limits for the difference of two means 46

5.6.1 Large samples 46

5.6.2 Welch-Aspin approximation (1949, 1956) 46

5.6.3 Cochran’s approximation (1964) 46

5.7 Matched samples and paired comparisons 47

5.8 Confidence limits for the variance 48

5.9 Confidence limits for the ratio of two variances 49

5.10 Least squares and maximum likelihood methods of estimation 49

Exercises 51

Appendix A5 52

A5.1 Tschebycheff’s inequality 52

A5.2 Mean square error 53

6 Proportions, odds ratios and relative risks: Estimation and confidence limits 54

6.1 Introduction 54

6.2 A single proportion 54

6.3 Confidence limits for the proportion 55

6.4 Difference of two proportions or percentages 56

6.5 Combining proportions from independent samples 56

6.6 More than two classes or categories 57

6.7 Odds ratio 58

6.8 Relative risk 59

Exercises 59

Appendix A6 60

A6. 1 Approximation to the variance of lnp 1 60

7 Tests of hypotheses: Means and variances 62

7.1 Introduction 62

7.2 Principle steps for the tests of a hypothesis 63

7.2.1 Null and alternate hypotheses 63

7.2.2 Decision rule, test statistic and the Type I & II errors 63

7.2.3 Significance level and critical region 64

7.2.4 The p-value 64

7.2.5 Power of the test and the sample size 65

7.3 Right-sided alternative, test statistic and critical region 65

7.3.1 The p-value 66

7.3.2 Power of the test 66

7.3.3 Sample size required for specified power 67

7.3.4 Right-sided alternative and estimated variance 68

7.3.5 Power of the test with estimated variance 69

7.4 Left-sided alternative and the critical region 69

7.4.1 The p-value 70

7.4.2 Power of the test 70

7.4.3 Sample size for specified power 71

7.4.4 Left-sided alternative with estimated variance 71

7.5 Two-sided alternative, critical region and the p-value 72

7.5.1 Power of the test 73

7.5.2 Sample size for specified power 74

7.5.3 Two-sided alternative and estimated variance 74

7.6 Difference between two means: Variances known 75

7.6.1 Difference between two means: Variances estimated 76

7.7 Matched samples and paired comparison 77

7.8 Test for the variance 77

7.9 Test for the equality of two variances 78

7.10 Homogeneity of variances 79

Exercises 80

8 Tests of hypotheses: Proportions and percentages 82

8.1 A single proportion 82

8.2 Right-sided alternative 82

8.2.1 Critical region 83

8.2.2 The p-value 84

8.2.3 Power of the test 84

8.2.4 Sample size for specified power 84

8.3 Left-sided alternative 85

8.3.1 Critical region 85

8.3.2 The p-value 86

8.3.3 Power of the test 86

8.3.4 Sample size for specified power 86

8.4 Two-sided alternative 87

8.4.1 Critical region 87

8.4.2 The p-value 88

8.4.3 Power of the test 88

8.4.4 Sample size for specified power 89

8.5 Difference of two proportions 90

8.5.1 Right-sided alternative: Critical region and p-value 90

8.5.2 Right-sided alternative: Power and sample size 91

8.5.3 Left-sided alternative: Critical region and p-value 92

8.5.4 Left-sided alternative: Power and sample size 93

8.5.5 Two-sided alternative: Critical region and p-value 93

8.5.6 Power and sample size 94

8.6 Specified difference of two proportions 95

8.7 Equality of two or more proportions 95

8.8 A common proportion 96

Exercises 97

9 The Chisquare statistic 99

9.1 Introduction 99

9.2 The test statistic 99

9.2.1 A single proportion 100

9.2.2 Specified proportions 100

9.3 Test of goodness of fit 101

9.4 Test of Independence: (r X C) Classification 101

9.5 Test of independence: (2x2) classification 104

9.5.1 Fisher’s exact test of independence 105

9.5.2 Mantel-Hanszeltest statistic 106

Exercises 107

Appendix A9 109

A9.1 Derivations of 9.4(a) 109

A9.2 Equality of the proportions 109

10 Regression and correlation 110

10.1 Introduction 110

10.2 The regression model: One independent variable 110

10.2.1 Least squares estimation of the regression 112

10.2.2 Properties of the estimators 113

10.2.3 ANOVA (Analysis of Variance) for the significance of the regression 114

10.2.4 Tests of hypotheses, confidence limits and prediction intervals 116

10.3 Regression on two independent variables 118

10.3.1 Properties of the estimators 120

10.3.2 ANOVA for the significance of the regression 121

10.3.3 Tests of hypotheses, confidence limits and prediction intervals 122

10.4 Multiple regression: The least squares estimation 124

10.4.1 ANOVA for the significance of the regression 126

10.4.2 Tests of hypotheses, confidence limits and prediction intervals 127

10.4.3 Multiple correlation, adjusted R 2 and partial correlation 128

10.4.4 Effect of including two or more independent variables and the partial F-test 129

10.4.5 Equality of two or more series of regressions 130

10.5 Indicator variables 132

10.5.1 Separate regressions 132

10.5.2 Regressions with equal slopes 133

10.5.3 Regressions with the same intercepts 134

10.6 Regression through the origin 135

10.7 Estimation of trends 136

10.8 Logistic regression and the odds ratio 138

10.8.1 A single continuous predictor 139

10.8.2 Two continuous predictors 139

10.8.3 A single dichotomous predictor 140

10.9 Weighted Least Squares (WLS) estimator 141

10.10 Correlation 142

10.10.1 Test of the hypothesis that two random variables are uncorrelated 143

10.10.2 Test of the hypothesis that the correlation coefficient takes a specified value 143

10.10.3 Confidence limits for the correlation coefficient 144

10.11 Further topics in regression 144

10.11.1 Linearity of the regression model and the lack of fit test 144

10.11.2 the Assumption That V (ε I X_i)= σ² , Same at Each X_i 146

10.11.3 Missing observations 146

10.11.4 Transformation of the regression model 147

10.11.5 Errors of Measurements of (X_i , Y_i) 147

Exercises 148

Appendix A10 149

A0.1 Square of the Correlation of Y_i and Ŷ_i 149

A10.2 Multiple regression 149

A10.3 Expression for SSR in (10.38) 151

11 Analysis of variance and covariance: Designs of experiments 152

11.1 Introduction 152

11.2 One-way classification: Balanced design 153

11.3 One-way random effects model: Balanced design 155

11.4 Inference for the variance components and the mean 155

11.5 One-way classification: Unbalanced design and fixed effects 157

11.6 Unbalanced one-way classification: Random effects 159

11.7 Intraclass correlation 160

11.8 Analysis of covariance: The balanced design 161

11.8.1 The model and least squares estimation 161

11.8.2 Tests of hypotheses for the slope coefficient and equality of the means 163

11.8.3 Confidence limits for the adjusted means and their differences 164

11.9 Analysis of covariance: Unbalanced design 165

11.9.1 Confidence limits for the adjusted means and the differences of the treatment effects 167

11.10 Randomized blocks 168

11.10.1 Randomized blocks: Random and mixed effects models 170

11.11 Repeated measures design 170

11.12 Latin squares 172

11.12.1 The model and analysis 172

11.13 Cross-over design 174

11.14 Two-way cross-classification 175

11.14.1 Additive model: Balanced design 176

11.14.2 Two-way cross-classification with interaction: Balanced design 178

11.14.3 Two-way cross-classification: Unbalanced additive model 179

11.14.4 Unbalanced cross-classification with interaction 183

11.14.5 Multiplicative interaction and Tukey’s test for nonadditivity 184

11.15 Missing observations in the designs of experiments 184

Exercises 186

Appendix A11 189

A11.1 Variance of σ_α ² in (11.25) from Rao (1997, p. 20) 189

A11.2 The total sum of squares (T_xx , T_yy) and sum of products (T_xy) can be expressed as the within and between components as follows 189

12 Meta-analysis 190

12.1 Introduction 190

12.2 Illustrations of large-scale studies 190

12.3 Fixed effects model for combining the estimates 191

12.4 Random effects model for combining the estimates 193

12.5 Alternative estimators for σ 2 α 194

12.6 Tests of hypotheses and confidence limits for the variance components 194

Exercises 195

Appendix A12 196

13 Survival analysis 197

13.1 Introduction 197

13.2 Survival and hazard functions 198

13.3 Kaplan-Meir product-limit estimator 198

13.4 Standard error of Ŝ(t_m) and confidence limits for S(t_m) 199

13.5 Confidence limits for S(t_m) with the right-censored observations 199

13.6 Log-Rank test for the equality of two survival distributions 201

13.7 Cox’s proportional hazard model 202

Exercises 203

Appendix A13 Expected value and variance of Ŝ(t_m) and confidence limits for S(t_m) 203

14 Nonparametric statistics 205

14.1 Introduction 205

14.2 Spearman’s rank correlation coefficient 205

14.3 The Sign test 206

14.4 Wilcoxon (1945) Matched-pairs Signed-ranks test 208

14.5 Wilcoxon’s test for the equality of the distributions of two non-normal populations with unpaired sample observations 209

14.5.1 Unequal sample sizes 210

14.6 McNemer’s (1955) matched pair test for two proportions 210

14.7 Cochran’s (1950) Q-test for the difference of three or more matched proportions 211

14.8 Kruskal-Wallis one-way ANOVA test by ranks 212

Exercises 213

15 Further topics 215

15.1 Introduction 215

15.2 Bonferroni inequality and the Joint Confidence Region 215

15.3 Least significant difference (LSD) for a pair of treatment effects 217

15.4 Tukey’s studentized range test 217

15.5 Scheffe’s simultaneous confidence intervals 218

15.6 Bootstrap confidence intervals 219

15.7 Transformations for the ANOVA 220

Exercises 221

Appendix A15 221

A5.1 Variance stabilizing transformation 221

Solutions to exercises 222

Appendix tables 249

References 261

Index 264