Statistical Methodologies with Medical Applications

SRS Rao Poduri, Professor of Statistics, University of Rochester. Since receiving his?Ph.D degree in Statistics in 1965 from Harvard University under the supervision of the eminent professor William G. Cochran, Professor Poduri has?been teaching courses in five or six major areas of statistics to graduate and undergraduate students at the University of Rochester.

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

4.4 Continuous distributions 29

4.5 Joint distribution of two discrete random variables 34

4.6 Bivariate normal distribution 37

Exercises 38

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

5.7 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

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

7 Tests of hypotheses: Means and variances 62

7.1 Introduction 62

7.2 Principle steps for the tests of a hypothesis 63

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

7.4 Left-sided alternative and the critical region 69

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

7.6 Difference between two means: Variances known 75

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

8.3 Left-sided alternative 85

8.4 Two-sided alternative 87

8.5 Difference of two proportions 90

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

9.3 Test of goodness of fit 101

9.4 Test of independence: (r x c) classification 101

9.5 Test of independence: (2x2) classification 104

Exercises 107

10 Regression and correlation 110

10.1 Introduction 110

10.2 The regression model: One independent variable 110

10.3 Regression on two independent variables 118

10.4 Multiple regression: The least squares estimation 124

10.5 Indicator variables 132

10.6 Regression through the origin 135

10.7 Estimation of trends 136

10.8 Logistic regression and the odds ratio 138

10.9 Weighted Least Squares (WLS) estimator 141

10.10 Correlation 142

10.11 Further topics in regression 144

Exercises 148

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

11.9 Analysis of covariance: Unbalanced design 165

11.10 Randomized blocks 168

11.11 Repeated measures design 170

11.12 Latin squares 172

11.13 Cross-over design 174

11.14 Two-way cross-classification 175

11.15 Missing observations in the designs of experiments 184

Exercises 186

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 σ2 α 194

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

Exercises 195

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

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

14.6 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

Solutions to exercises 222

Appendix tables 249

References 261

Index 264