Statistical Methodologies with Medical Applications

This book presents the methodology and applications of a range of important topics in statistics, and is designed for graduate students in Statistics and  Biostatistics and for medical researchers.  Illustrations and more than ninety exercises with solutions are presented.  They are constructed from the research findings of the medical journals, summary reports of the Centre for Disease Control (CDC)  and the World Health Organization (WHO), and practical situations.  The illustrations and exercises are related to topics such as immunization, obesity, hypertension, lipid levels, diet and exercise, harmful effects of smoking and air pollution, and the benefits of gluten free diet. This book can be recommended for a one or two semester graduate level course for students studying Statistics, Biostatistics, Epidemiology and Health Sciences.  It will also be useful as a companion for medical researchers and research oriented physicians. 

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 4.4 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

About the Author :
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.