Introductory biostatistics for the health sciences : modern applications including bootstrap

1. What is Statistics? How is it Applied in the Health Sciences? 1 1.1 Definitions of Statistics and Statisticians 2 1.2 Why Study Statistics? 3 1.3 Types of Studies 8 1.3.1 Surveys and Cross-Sectional Studies 9 1.3.2 Retrospective Studies 10 1.3.3 Prospective Studies 10 1.3.4 Experimental Studies and Quality Control 10 1.3.5 Clinical Trials 12 1.3.6 Epidemiological Studies 14 1.3.7 Pharmacoeconomic Studies and Quality of Life 16 1.4 Exercises 18 1.5 Additional Reading 19 2. Defining Populations and Selecting Samples 22 2.1 What are Populations and Samples? 22 2.2 Why Select a Sample? 23 2.3 How Samples Can be Selected 25 2.3.1 Simple Random Sampling 25 2.3.2 Convenience Sampling 25 2.3.3 Systematic Sampling 26 2.3.4 Stratified Random Sampling 28 2.3.5 Cluster Sampling 28 2.3.6 Bootstrap Sampling 29 viii CONTENTS 2.4 How to Select a Simple Random Sample 29 2.5 How to Select a Bootstrap Sample 39 2.6 Why Does Random Sampling Work? 41 2.7 Exercises 41 2.8 Additional Reading 45 3. Systematic Organization and Display of Data 46 3.1 Types of Data 46 3.1.1 Qualitative 47 3.1.2 Quantitative 47 3.2 Frequency Tables and Histograms 48 3.3 Graphical Methods 51 3.3.1 Frequency Histograms 51 3.3.2 Frequency Polygons 53 3.3.3 Cumulative Frequency Polygon 54 3.3.4 Stem-and-Leaf Diagrams 56 3.3.5 Box-and-Whisker Plots 58 3.3.6 Bar Charts and Pie Charts 61 3.1 Exercises 63 3.1 Additional Reading 67 4. Summary Statistics 68 4.1 Measures of Central Tendency 61 4.1.1 The Arithmetic Mean 68 4.1.2 The Median 70 4.1.3 The Mode 73 4.1.4 The Geometric Mean 73 4.1.5 The Harmonic Mean 74 4.1.6 Which Measure Should You Use? 75 4.2 Measures of Dispersion 76 4.2.1 Range 78 4.2.2 Mean Absolute Deviation 78 4.2.3 Population Variance and Standard Deviation 79 4.2.4 Sample Variance and Standard Deviation 82 4.2.5 Calculating the Variance and Standard Deviation from Group Data 84 4.3 Coefficient of Variation (CV) and Coefficient of Dispersion (CD) 85 4.4 Exercises 88 4.5 Additional Reading 91 CONTENTS ix 5. Basic Probability 92 5.1 What is Probability? 92 5.2 Elementary Sets as Events and Their Complements 95 5.3 Independent and Disjoint Events 95 5.4 Probability Rules 98 5.5 Permutations and Combinations 100 5.6 Probability Distributions 103 5.7 The Binomial Distribution 109 5.8 The Monty Hall Problem 110 5.9 A Quality Assurance Problem 113 5.10 Exercises 115 5.11 Additional Reading 120 6. The Normal Distribution 121 6.1 The Importance of the Normal Distribution in Statistics 121 6.2 Properties of Normal Distributions 122 6.3 Tabulating Areas under the Standard Normal Distribution 124 6.4 Exercises 129 6.5 Additional Reading 132 7. Sampling Distributions for Means 133 7.1 Population Distributions and the Distribution of Sample Averages from the Population 133 7.2 The Central Limit Theorem 141 7.3 Standard Error of the Mean 143 7.4 Z Distribution Obtained When Standard Deviation Is Known 144 7.5 Student’s t Distribution Obtained When Standard Deviation Is Unknown 144 7.6 Assumptions Required for t Distribution 147 7.7 Exercises 147 7.8 Additional Reading 149 8. Estimating Population Means 150 8.1 Estimation Versus Hypothesis Testing 150 8.2 Point Estimates 151 8.3 Confidence Intervals 153 8.4 Confidence Intervals for a Single Population Mean 154 8.5 Z and t Statistics for Two Independent Samples 159 8.6 Confidence Intervals for the Difference between Means from Two Independent Samples (Variance Known) 161 8.7 Confidence Intervals for the Difference between Means from Two Independent Samples (Variance Unknown) 161 8.8 Bootstrap Principle 166 8.9 Bootstrap Percentile Method Confidence Intervals 167 8.10 Sample Size Determination for Confidence Intervals 176 8.11 Exercises 179 8.12 Additional Reading 181 9. Tests of Hypotheses 182 9.1 Terminology 182 9.2 Neyman–Pearson Test Formulation 183 9.3 Test of a Mean (Single Sample, Population Variance Known) 186 9.4 Test of a Mean (Single sample, Population Variance Unknown) 187 9.5 One-Tailed Versus Two-Tailed Tests 188 9.6 p-Values 191 9.7 Type I and Type II Errors 191 9.8 The Power Function 192 9.9 Two-Sample t Test (Independent Samples with a Common Variance) 193 9.10 Paired t Test 195 9.11 Relationship between Confidence Intervals and Hypothesis Tests 199 9.12 Bootstrap Percentile Method Test 200 9.13 Sample Size Determination for Hypothesis Tests 201 9.14 Sensitivity and Specificity in Medical Diagnosis 202 9.15 Meta-Analysis 204 9.16 Bayesian Methods 207 9.17 Group Sequential Methods 209 9.18 Missing Data and Imputation 210 9.19 Exercises 212 9.20 Additional Reading 215 10. Inferences Regarding Proportions 217 10.1 Why Are Proportions Important? 217 10.2 Mean and Standard Deviation for the Binomial Distribution 218 10.3 Normal Approximation to the Binomial 221 10.4 Hypothesis Test for a Single Binomial Proportion 222 10.5 Testing the Difference between Two Proportions 224 x CONTENTS 10.6 Confidence Intervals for Proportions 225 10.7 Sample Size Determination—Confidence Intervals and Hypothesis Tests 227 10.8 Exercises 228 10.9 Additional Reading 229 11. Categorical Data and Chi-Square Tests 231 11.1 Understanding Chi-Square 232 11.2 Chi-Square Distributions and Tables 233 11.3 Testing Independence between Two Variables 233 11.4 Testing for Homogeneity 236 11.5 Testing for Differences between Two Proportions 237 11.6 The Special Case of 2 × 2 Contingency Table 238 11.7 Simpson’s Paradox in the 2 × 2 Table 239 11.8 McNemar’s Test for Correlated Proportions 241 11.9 Relative Risk and Odds Ratios 242 11.10 Goodness of Fit Tests—Fitting Hypothesized Probability Distributions 244 11.11 Limitations to Chi-Square and Exact Alternatives 246 11.12 Exercises 247 11.13 Additional Reading 250 12. Correlation, Linear Regression, and Logistic Regression 251 12.1 Relationships between Two Variables 252 12.2 Uses of Correlation and Regression 252 12.3 The Scatter Diagram 254 12.4 Pearson’s Product Moment Correlation Coefficient and Its Sample Estimate 256 12.5 Testing Hypotheses about the Correlation Coefficient 258 12.6 The Correlation Matrix 259 12.7 Regression Analysis and Least Squares Inference Regarding the Slope and Intercept of a Regression Line 259 12.8 Sensitivity to Outliers, Outlier Rejection, and Robust Regression 264 12.9 Galton and Regression toward the Mean 271 12.10 Multiple Regression 277 12.11 Logistic Regression 283 12.12 Exercises 287 12.13 Additional Reading 293 CONTENTS xi 13. One-Way Analysis of Variance 295 13.1 Purpose of One-Way Analysis of Variance 296 13.2 Decomposing the Variance and Its Meaning 297 13.3 Necessary Assumptions 298 13.4 F Distribution and Applications 298 13.5 Multiple Comparisons 301 13.5.1 General Discussion 301 13.5.2 Tukey’s Honest Signficant Difference (HSD) Test 301 13.6 Exercises 302 13.7 Additional Reading 307 14. Nonparametric Methods 308 14.1 Advantages and Disadvantages of Nonparametric Versus Parametric Methods 308 14.2 Procedures for Ranking Data 309 14.3 Wilcoxon Rank-Sum Test 311 14.4 Wilcoxon Signed-Rank Test 314 14.5 Sign Test 317 14.6 Kruskal–Wallis Test: One-Way ANOVA by Ranks 319 14.7 Spearman’s Rank-Order Correlation Coefficient 322 14.8 Permutation Tests 324 14.8.1 Introducing Permutation Methods 324 14.8.2 Fisher’s Exact Test 327 14.9 Insensitivity of Rank Tests to Outliers 330 14.10 Exercises 331 14.11 Additional Reading 334 15. Analysis of Survival Times 336 15.1 Introduction to Survival Times 336 15.2 Survival Probabilities 338 15.2.1 Introduction 338 15.2.2 Life Tables 339 15.2.3 The Kaplan–Meier Curve 341 15.2.4 Parametric Survival Curves 344 15.2.5 Cure Rate Models 348 15.3 Comparing Two or More Survival Curves—The Log Rank Test 349 15.4 Exercises 352 15.5 Additional Reading 354 xii CONTENTS 16. Software Packages for Statistical Analysis 356 16.1 General-Purpose Packages 356 16.2 Exact Methods 359 16.3 Sample Size Determination 359 16.4 Why You Should Avoid Excel 360 16.5 References 361 Postscript 362 Appendices 363 A Percentage Points, F-Distribution (
= 0.05) 363 B Studentized Range Statistics 364 C Quantiles of the Wilcoxon Signed-Rank Test Statistic 366 D 2 Distribution 368 E Table of the Standard Normal Distribution 370 F Percentage Points, Student’s t Distribution 371 G Answers to Selected Exercises 373