Table of Contents
Topics for illustrations, examples and exercises xv
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 Xi)= σ2 , Same at Each Xi 146
10.11.3 Missing observations 146
10.11.4 Transformation of the regression model 147
10.11.5 Errors of Measurements of (Xi , Yi) 147
Exercises 148
Appendix A10 149
A0.1 Square of the Correlation of Yi and Ŷi 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 σα 2 in (11.25) from Rao (1997, p. 20) 189
A11.2 The total sum of squares (Txx , Tyy) and sum of products (Txy) 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(tm) and confidence limits for S(tm) 199
13.5 Confidence limits for S(tm) 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(tm) and confidence limits for S(tm) 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
