Statistical Methodologies with Medical Applications
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- Published: 09 December 2016
- ISBN: 9781119258490
- Author(s): Poduri S.R.S. Rao
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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.
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
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