Clinical Medicine

John R. Bradley, CBE MA DM FRCP, Consultant Physician and Honorary Professor of Experimental Medicine, University of Cambridge School of Clinical Medicine, Cambridge University Hospitals, Cambridge Mark Gurnell, MA (MedEd) PhD FAcadMEd FRCP, Clinical SubDean, Senior Lecturer and Honorary Consultant Physician, University of Cambridge School of Clinical Medicine, Cambridge University Hospitals, Cambridge Diana F. Wood, MA MD FRCP, Director of Medical Education, Clinical Dean and Honorary Consultant Physician, University of Cambridge School of Clinical Medicine, Cambridge University Hospitals, Cambridge

• 1 Data and Case Studies 11.1 Case Study: Flight Delays 11.2 Case Study: BirthWeights of Babies 21.3 Case Study: Verizon Repair Times 31.4 Case Study: Iowa Recidivism 41.5 Sampling 51.6 Parameters and Statistics 61.7 Case Study: General Social Survey 71.8 Sample Surveys 81.9 Case Study: Beer and HotWings 91.10 Case Study: Black Spruce Seedlings 101.11 Studies 101.12 Google Interview Question: Mobile Ads Optimization 12Exercises 162 Exploratory Data Analysis 212.1 Basic Plots 212.2 Numeric Summaries 252.2.1 Center 252.2.2 Spread 262.2.3 Shape 272.3 Boxplots 282.4 Quantiles and Normal Quantile Plots 292.5 Empirical Cumulative Distribution Functions 352.6 Scatter Plots 382.7 Skewness and Kurtosis 403 Introduction to Hypothesis Testing: Permutation Tests 473.1 Introduction to Hypothesis Testing 473.2 Hypotheses 483.3 Permutation Tests 503.3.1 Implementation Issues 553.3.2 One-sided and Two-sided Tests 613.3.3 Other Statistics 623.3.4 Assumptions 643.3.5 Remark on Terminology 683.4 Matched Pairs 68Exercises 704 Sampling Distributions 754.1 Sampling Distributions 754.2 Calculating Sampling Distributions 804.3 The Central LimitTheorem 844.3.1 CLT for Binomial Data 864.3.2 Continuity Correction for Discrete Random Variables 894.3.3 Accuracy of the Central Limit Theorem∗ 914.3.4 CLT for SamplingWithout Replacement 92Exercises 935 Introduction to Confidence Intervals: The Bootstrap 1035.1 Introduction to the Bootstrap 1035.2 The Plug-in Principle 1105.2.1 Estimating the Population Distribution 1125.2.2 How Useful Is the Bootstrap Distribution? 1135.3 Bootstrap Percentile Intervals 1185.4 Two-Sample Bootstrap 1195.4.1 Matched Pairs 1245.5 Other Statistics 1285.6 Bias 1315.7 Monte Carlo Sampling: The “Second Bootstrap Principle” 1345.8 Accuracy of Bootstrap Distributions 1355.8.1 Sample Mean: Large Sample Size 1355.8.2 Sample Mean: Small Sample Size 1375.8.3 Sample Median 1385.8.4 Mean–Variance Relationship 1385.9 HowMany Bootstrap Samples Are Needed? 140Exercises 1416 Estimation 1496.1 Maximum Likelihood Estimation 1496.1.1 Maximum Likelihood for Discrete Distributions 1506.1.2 Maximum Likelihood for Continuous Distributions 1536.1.3 Maximum Likelihood for Multiple Parameters 1576.2 Method of Moments 1616.3 Properties of Estimators 1636.3.1 Unbiasedness 1646.3.2 Efficiency 1676.3.3 Mean Square Error 1716.3.4 Consistency 1736.3.5 Transformation Invariance∗ 1756.3.6 Asymptotic Normality of MLE∗ 1776.4 Statistical Practice 1786.4.1 Are You Asking the Right Question? 1796.4.2 Weights 179Exercises 1807 More Confidence Intervals 1877.1 Confidence Intervals for Means 1877.1.1 Confidence Intervals for a Mean, Variance Known 1877.1.2 Confidence Intervals for a Mean, Variance Unknown 1927.1.3 Confidence Intervals for a Difference in Means 1987.1.4 Matched Pairs, Revisited 2047.2 Confidence Intervals in General 2047.2.1 Location and Scale Parameters∗ 2087.3 One-sided Confidence Intervals 2127.4 Confidence Intervals for Proportions 2147.4.1 Agresti–Coull Intervals for a Proportion 2177.4.2 Confidence Intervals for a Difference of Proportions 2187.5 Bootstrap Confidence Intervals 2197.5.1 t Confidence Intervals Using Bootstrap Standard Errors 2197.5.2 Bootstrap t Confidence Intervals 2207.5.3 Comparing Bootstrap t and Formula t Confidence Intervals 2247.6 Confidence Interval Properties 2267.6.1 Confidence Interval Accuracy 2267.6.2 Confidence Interval Length 2277.6.3 Transformation Invariance 2277.6.4 Ease of Use and Interpretation 2277.6.5 Research Needed 228Exercises 2288 More Hypothesis Testing 2418.1 Hypothesis Tests for Means and Proportions: One Population 2418.1.1 A Single Mean 2418.1.2 One Proportion 2448.2 Bootstrap t-Tests 2468.3 Hypothesis Tests for Means and Proportions: Two Populations 2488.3.1 Comparing Two Means 2488.3.2 Comparing Two Proportions 2518.3.3 Matched Pairs for Proportions 2548.4 Type I and Type II Errors 2558.4.1 Type I Errors 2578.4.2 Type II Errors and Power 2618.4.3 P-Values versus Critical Regions 2668.5 Interpreting Test Results 2678.5.1 P-Values 2678.5.2 On Significance 2688.5.3 Adjustments for Multiple Testing 2698.6 Likelihood Ratio Tests 2718.6.1 Simple Hypotheses and the Neyman–Pearson Lemma 2718.6.2 Likelihood Ratio Tests for Composite Hypotheses 2758.7 Statistical Practice 2798.7.1 More Campaigns with No Clicks and No Conversions 284Exercises 2859 Regression 2979.1 Covariance 2979.2 Correlation 3019.3 Least-Squares Regression 3049.3.1 Regression Toward the Mean 3089.3.2 Variation 3109.3.3 Diagnostics 3119.3.4 Multiple Regression 3179.4 The Simple LinearModel 3179.4.1 Inference for 𝛼 and 𝛽 3229.4.2 Inference for the Response 3269.4.3 Comments about Assumptions for the Linear Model 3309.5 Resampling Correlation and Regression 3329.5.1 Permutation Tests 3359.5.2 Bootstrap Case Study: Bushmeat 3369.6 Logistic Regression 3409.6.1 Inference for Logistic Regression 346Exercises 35010 Categorical Data 35910.1 Independence in Contingency Tables 35910.2 Permutation Test of Independence 36110.3 Chi-square Test of Independence 36510.3.1 Model for Chi-square Test of Independence 36610.3.2 2 × 2 Tables 36810.3.3 Fisher’s Exact Test 37010.3.4 Conditioning 37110.4 Chi-square Test of Homogeneity 37210.5 Goodness-of-fit Tests 37410.5.1 All Parameters Known 37410.5.2 Some Parameters Estimated 37710.6 Chi-square and the Likelihood Ratio∗ 379Exercises 38011 Bayesian Methods 39111.1 Bayes Theorem 39211.2 Binomial Data: Discrete Prior Distributions 39211.3 Binomial Data: Continuous Prior Distributions 40011.4 Continuous Data 40611.5 Sequential Data 409Exercises 41412 One-way ANOVA 41912.1 Comparing Three or More Populations 41912.1.1 The ANOVA F-test 41912.1.2 A Permutation Test Approach 428Exercises 42913 Additional Topics 43313.1 Smoothed Bootstrap 43313.1.1 Kernel Density Estimate 43513.2 Parametric Bootstrap 43713.3 The Delta Method 44113.4 Stratified Sampling 44513.5 Computational Issues in Bayesian Analysis 44613.6 Monte Carlo Integration 44813.7 Importance Sampling 45213.7.1 Ratio Estimate for Importance Sampling 45813.7.2 Importance Sampling in Bayesian Applications 46113.8 The EM Algorithm 46713.8.1 General Background 469Exercises 472Appendix A Review of Probability 477A.1 Basic Probability 477A.2 Mean and Variance 478A.3 The Normal Distribution 480A.4 The Mean of a Sample of RandomVariables 481A.5 Sums of Normal Random Variables 482A.6 The Law of Averages 483A.7 Higher Moments and the Moment-generating Function 484Appendix B Probability Distributions 487B.1 The Bernoulli and Binomial Distributions 487B.2 The Multinomial Distribution 488B.3 The Geometric Distribution 490B.4 The Negative Binomial Distribution 491B.5 The Hypergeometric Distribution 492B.6 The Poisson Distribution 493B.7 The Uniform Distribution 495B.8 The Exponential Distribution 495B.9 The Gamma Distribution 497B.10 The Chi-square Distribution 499B.11 The Student’s t Distribution 502B.12 The Beta Distribution 504B.13 The F Distribution 505Exercises 507Appendix C Distributions Quick Reference 509Solutions to Selected Exercises 513References 525Index 531

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