Anders Källén, Department of Biostatistics, AstraZeneca R&D, Sweden.

"Overall, the book is well-written . . . The topics are presented in a logical progression as is the level of their mathematical difficulty. Any biostatistician will find this a valuable complement to his/her favorite biostatistics textbook." (Journal of Biopharmaceutical Statistics, 2012)

• Preface ix1 Statistics and medical science 11.1 Introduction 11.2 On the nature of science 31.3 How the scientific method uses statistics 51.4 Finding an outcome variable to assess your hypothesis 71.5 How we draw medical conclusions from statistical results 81.6 A few words about probabilities 131.7 The need for honesty: the multiplicity issue 161.8 Prespecification and p-value history 191.9 Adaptive designs: controlling the risks in an experiment 211.10 The elusive concept of probability 231.11 Comments and further reading 26References 272 Observational studies and the need for clinical trials 292.1 Introduction 292.2 Investigations of medical interventions and risk factors 292.3 Observational studies and confounders 332.4 The experimental study 392.5 Population risks and individual risks 422.6 Confounders, Simpson’s paradox and stratification 442.7 On incidence and prevalence in epidemiology 512.8 Comments and further reading 53References 543 Study design and the bias issue 573.1 Introduction 573.2 What bias is all about 583.3 The need for a representative sample: on selection bias 583.4 Group comparability and randomization 613.5 Information bias in a cohort study 653.6 The study, or placebo, effect 683.7 The curse of missing values 703.8 Approaches to data analysis: avoiding self-inflicted bias 753.9 On meta-analysis and publication bias 793.10 Comments and further reading 81References 824 The anatomy of a statistical test 854.1 Introduction 854.2 Statistical tests, medical diagnosis and Roman law 854.3 The risks with medical diagnosis 874.3.1 Medical diagnosis based on a single test 874.3.2 Bayes’ theorem and the use and misuse of screening tests 894.4 The law: a non-quantitative analogue 914.5 Risks in statistical testing 934.5.1 Does tonsillectomy increase the risk of Hodgkin’s lymphoma? 934.5.2 General discussion about statistical tests 984.6 Making statements about a binomial parameter 1014.6.1 The frequentist approach 1014.6.2 The Bayesian approach 1044.7 The bell-shaped error distribution 1094.8 Comments and further reading 112References 1134.A Appendix: The evolution of the central limit theorem 1155 Learning about parameters, and some notes on planning 1195.1 Introduction 1195.2 Test statistics described by parameters 1205.3 How we describe our knowledge about a parameter from an experiment 1225.4 Statistical analysis of two proportions 1275.4.1 Some ways to compare two proportions 1275.4.2 Analysis of the group difference 1305.5 Adjusting for confounders in the analysis 1335.6 The power curve of an experiment 1385.7 Some confusing aspects of power calculations 1435.8 Comments and further reading 145References 1455.A Appendix: Some technical comments 1465.A.1 The non-central hypergeometric distribution and 2 × 2 tables 1465.A.2 The gamma and χ2 distributions 1476 Empirical distribution functions 1496.1 Introduction 1496.2 How to describe the distribution of a sample 1496.3 Describing the sample: descriptive statistics 1536.4 Population distribution parameters 1566.5 Confidence in the CDF and its parameters 1586.6 Analysis of paired data 1626.7 Bootstrapping 1636.8 Meta-analysis and heterogeneity 1666.9 Comments and further reading 169References 1706.A Appendix: Some technical comments 1716.A.1 The extended family of the univariate Gaussian distributions 1716.A.2 The Wiener process and its bridge 1736.A.3 Confidence regions for the CDF and the Kolmogorov–Smirnov test 1747 Correlation and regression in bivariate distributions 1777.1 Introduction 1777.2 Bivariate distributions and correlation 1787.3 On baseline corrections and other covariates 1837.4 Bivariate Gaussian distributions 1867.5 Regression to the mean 1897.6 Statistical analysis of bivariate Gaussian data 1957.7 Simultaneous analysis of two binomial proportions 1997.8 Comments and further reading 203References 2037.A Appendix: Some technical comments 2057.A.1 The regression to the mode equation 2057.A.2 Analysis of data from the multivariate Gaussian distribution 2067.A.3 On the geometric approach to univariate confidence limits 2078 How to compare the outcome in two groups 2098.1 Introduction 2098.2 Simple models that compare two distributions 2108.3 Comparison done the horizontal way 2128.4 Analysis done the vertical way 2168.5 Some ways to compute p-values 2248.6 The discrete Wilcoxon test 2268.7 The two-period crossover trial 2298.8 Multivariate analysis and analysis of covariance 2328.9 Comments and further reading 239References 2408.A Appendix: About U-statistics 2419 Least squares, linear models and beyond 2459.1 Introduction 2459.2 The purpose of mathematical models 2469.3 Different ways to do least squares 2509.4 Logistic regression, with variations 2529.5 The two-step modeling approach 2579.6 The effect of missing covariates 2609.7 The exponential family of distributions 2639.8 Generalized linear models 2699.9 Comments and further reading 270References 27010 Analysis of dose response 27310.1 Introduction 27310.2 Dose–response relationship 27410.3 Relative dose potency and therapeutic ratio 27810.4 Subject-specific and population averaged dose response 27910.5 Estimation of the population averaged dose–response relationship 28110.6 Estimating subject-specific dose responses 28510.7 Comments and further reading 288References 28811 Hazards and censored data 28911.1 Introduction 28911.2 Censored observations: incomplete knowledge 29011.3 Hazard models from a population perspective 29111.4 The impact of competing risks 29611.5 Heterogeneity in survival analysis 30011.6 Recurrent events and frailty 30411.7 The principles behind the analysis of censored data 30611.8 The Kaplan–Meier estimator of the CDF 30911.9 Comments and further reading 312References 31311.A Appendix: On the large-sample approximations of counting processes 31412 From the log-rank test to the Cox proportional hazards model 31712.1 Introduction 31712.2 Comparing hazards between two groups 31812.3 Nonparametric tests for hazards 31912.4 Parameter estimation in hazard models 32412.5 The accelerated failure time model 32812.6 The Cox proportional hazards model 33112.7 On omitted covariates and stratification in the log-rank test 33612.8 Comments and further reading 338References 33912.A Appendix: Comments on interval-censored data 34113 Remarks on some estimation methods 34313.1 Introduction 34313.2 Estimating equations and the robust variance estimate 34413.3 From maximum likelihood theory to generalized estimating equations 35113.4 The analysis of recurrent events 35513.5 Defining and estimating mixed effects models 36013.6 Comments and further reading 366References 36713.A Appendix: Formulas for first-order bias 368Index 371

Anders Källén, Department of Biostatistics, AstraZeneca R&D, Sweden.

"Overall, the book is well-written . . . The topics are presented in a logical progression as is the level of their mathematical difficulty. Any biostatistician will find this a valuable complement to his/her favorite biostatistics textbook." (Journal of Biopharmaceutical Statistics, 2012)

• Preface ix1 Statistics and medical science 11.1 Introduction 11.2 On the nature of science 31.3 How the scientific method uses statistics 51.4 Finding an outcome variable to assess your hypothesis 71.5 How we draw medical conclusions from statistical results 81.6 A few words about probabilities 131.7 The need for honesty: the multiplicity issue 161.8 Prespecification and p-value history 191.9 Adaptive designs: controlling the risks in an experiment 211.10 The elusive concept of probability 231.11 Comments and further reading 26References 272 Observational studies and the need for clinical trials 292.1 Introduction 292.2 Investigations of medical interventions and risk factors 292.3 Observational studies and confounders 332.4 The experimental study 392.5 Population risks and individual risks 422.6 Confounders, Simpson’s paradox and stratification 442.7 On incidence and prevalence in epidemiology 512.8 Comments and further reading 53References 543 Study design and the bias issue 573.1 Introduction 573.2 What bias is all about 583.3 The need for a representative sample: on selection bias 583.4 Group comparability and randomization 613.5 Information bias in a cohort study 653.6 The study, or placebo, effect 683.7 The curse of missing values 703.8 Approaches to data analysis: avoiding self-inflicted bias 753.9 On meta-analysis and publication bias 793.10 Comments and further reading 81References 824 The anatomy of a statistical test 854.1 Introduction 854.2 Statistical tests, medical diagnosis and Roman law 854.3 The risks with medical diagnosis 874.3.1 Medical diagnosis based on a single test 874.3.2 Bayes’ theorem and the use and misuse of screening tests 894.4 The law: a non-quantitative analogue 914.5 Risks in statistical testing 934.5.1 Does tonsillectomy increase the risk of Hodgkin’s lymphoma? 934.5.2 General discussion about statistical tests 984.6 Making statements about a binomial parameter 1014.6.1 The frequentist approach 1014.6.2 The Bayesian approach 1044.7 The bell-shaped error distribution 1094.8 Comments and further reading 112References 1134.A Appendix: The evolution of the central limit theorem 1155 Learning about parameters, and some notes on planning 1195.1 Introduction 1195.2 Test statistics described by parameters 1205.3 How we describe our knowledge about a parameter from an experiment 1225.4 Statistical analysis of two proportions 1275.4.1 Some ways to compare two proportions 1275.4.2 Analysis of the group difference 1305.5 Adjusting for confounders in the analysis 1335.6 The power curve of an experiment 1385.7 Some confusing aspects of power calculations 1435.8 Comments and further reading 145References 1455.A Appendix: Some technical comments 1465.A.1 The non-central hypergeometric distribution and 2 × 2 tables 1465.A.2 The gamma and χ2 distributions 1476 Empirical distribution functions 1496.1 Introduction 1496.2 How to describe the distribution of a sample 1496.3 Describing the sample: descriptive statistics 1536.4 Population distribution parameters 1566.5 Confidence in the CDF and its parameters 1586.6 Analysis of paired data 1626.7 Bootstrapping 1636.8 Meta-analysis and heterogeneity 1666.9 Comments and further reading 169References 1706.A Appendix: Some technical comments 1716.A.1 The extended family of the univariate Gaussian distributions 1716.A.2 The Wiener process and its bridge 1736.A.3 Confidence regions for the CDF and the Kolmogorov–Smirnov test 1747 Correlation and regression in bivariate distributions 1777.1 Introduction 1777.2 Bivariate distributions and correlation 1787.3 On baseline corrections and other covariates 1837.4 Bivariate Gaussian distributions 1867.5 Regression to the mean 1897.6 Statistical analysis of bivariate Gaussian data 1957.7 Simultaneous analysis of two binomial proportions 1997.8 Comments and further reading 203References 2037.A Appendix: Some technical comments 2057.A.1 The regression to the mode equation 2057.A.2 Analysis of data from the multivariate Gaussian distribution 2067.A.3 On the geometric approach to univariate confidence limits 2078 How to compare the outcome in two groups 2098.1 Introduction 2098.2 Simple models that compare two distributions 2108.3 Comparison done the horizontal way 2128.4 Analysis done the vertical way 2168.5 Some ways to compute p-values 2248.6 The discrete Wilcoxon test 2268.7 The two-period crossover trial 2298.8 Multivariate analysis and analysis of covariance 2328.9 Comments and further reading 239References 2408.A Appendix: About U-statistics 2419 Least squares, linear models and beyond 2459.1 Introduction 2459.2 The purpose of mathematical models 2469.3 Different ways to do least squares 2509.4 Logistic regression, with variations 2529.5 The two-step modeling approach 2579.6 The effect of missing covariates 2609.7 The exponential family of distributions 2639.8 Generalized linear models 2699.9 Comments and further reading 270References 27010 Analysis of dose response 27310.1 Introduction 27310.2 Dose–response relationship 27410.3 Relative dose potency and therapeutic ratio 27810.4 Subject-specific and population averaged dose response 27910.5 Estimation of the population averaged dose–response relationship 28110.6 Estimating subject-specific dose responses 28510.7 Comments and further reading 288References 28811 Hazards and censored data 28911.1 Introduction 28911.2 Censored observations: incomplete knowledge 29011.3 Hazard models from a population perspective 29111.4 The impact of competing risks 29611.5 Heterogeneity in survival analysis 30011.6 Recurrent events and frailty 30411.7 The principles behind the analysis of censored data 30611.8 The Kaplan–Meier estimator of the CDF 30911.9 Comments and further reading 312References 31311.A Appendix: On the large-sample approximations of counting processes 31412 From the log-rank test to the Cox proportional hazards model 31712.1 Introduction 31712.2 Comparing hazards between two groups 31812.3 Nonparametric tests for hazards 31912.4 Parameter estimation in hazard models 32412.5 The accelerated failure time model 32812.6 The Cox proportional hazards model 33112.7 On omitted covariates and stratification in the log-rank test 33612.8 Comments and further reading 338References 33912.A Appendix: Comments on interval-censored data 34113 Remarks on some estimation methods 34313.1 Introduction 34313.2 Estimating equations and the robust variance estimate 34413.3 From maximum likelihood theory to generalized estimating equations 35113.4 The analysis of recurrent events 35513.5 Defining and estimating mixed effects models 36013.6 Comments and further reading 366References 36713.A Appendix: Formulas for first-order bias 368Index 371

Anders Källén, Department of Biostatistics, AstraZeneca R&D, Sweden.

"Overall, the book is well-written . . . The topics are presented in a logical progression as is the level of their mathematical difficulty. Any biostatistician will find this a valuable complement to his/her favorite biostatistics textbook." (Journal of Biopharmaceutical Statistics, 2012)

• Preface ix1 Statistics and medical science 11.1 Introduction 11.2 On the nature of science 31.3 How the scientific method uses statistics 51.4 Finding an outcome variable to assess your hypothesis 71.5 How we draw medical conclusions from statistical results 81.6 A few words about probabilities 131.7 The need for honesty: the multiplicity issue 161.8 Prespecification and p-value history 191.9 Adaptive designs: controlling the risks in an experiment 211.10 The elusive concept of probability 231.11 Comments and further reading 26References 272 Observational studies and the need for clinical trials 292.1 Introduction 292.2 Investigations of medical interventions and risk factors 292.3 Observational studies and confounders 332.4 The experimental study 392.5 Population risks and individual risks 422.6 Confounders, Simpson’s paradox and stratification 442.7 On incidence and prevalence in epidemiology 512.8 Comments and further reading 53References 543 Study design and the bias issue 573.1 Introduction 573.2 What bias is all about 583.3 The need for a representative sample: on selection bias 583.4 Group comparability and randomization 613.5 Information bias in a cohort study 653.6 The study, or placebo, effect 683.7 The curse of missing values 703.8 Approaches to data analysis: avoiding self-inflicted bias 753.9 On meta-analysis and publication bias 793.10 Comments and further reading 81References 824 The anatomy of a statistical test 854.1 Introduction 854.2 Statistical tests, medical diagnosis and Roman law 854.3 The risks with medical diagnosis 874.3.1 Medical diagnosis based on a single test 874.3.2 Bayes’ theorem and the use and misuse of screening tests 894.4 The law: a non-quantitative analogue 914.5 Risks in statistical testing 934.5.1 Does tonsillectomy increase the risk of Hodgkin’s lymphoma? 934.5.2 General discussion about statistical tests 984.6 Making statements about a binomial parameter 1014.6.1 The frequentist approach 1014.6.2 The Bayesian approach 1044.7 The bell-shaped error distribution 1094.8 Comments and further reading 112References 1134.A Appendix: The evolution of the central limit theorem 1155 Learning about parameters, and some notes on planning 1195.1 Introduction 1195.2 Test statistics described by parameters 1205.3 How we describe our knowledge about a parameter from an experiment 1225.4 Statistical analysis of two proportions 1275.4.1 Some ways to compare two proportions 1275.4.2 Analysis of the group difference 1305.5 Adjusting for confounders in the analysis 1335.6 The power curve of an experiment 1385.7 Some confusing aspects of power calculations 1435.8 Comments and further reading 145References 1455.A Appendix: Some technical comments 1465.A.1 The non-central hypergeometric distribution and 2 × 2 tables 1465.A.2 The gamma and χ2 distributions 1476 Empirical distribution functions 1496.1 Introduction 1496.2 How to describe the distribution of a sample 1496.3 Describing the sample: descriptive statistics 1536.4 Population distribution parameters 1566.5 Confidence in the CDF and its parameters 1586.6 Analysis of paired data 1626.7 Bootstrapping 1636.8 Meta-analysis and heterogeneity 1666.9 Comments and further reading 169References 1706.A Appendix: Some technical comments 1716.A.1 The extended family of the univariate Gaussian distributions 1716.A.2 The Wiener process and its bridge 1736.A.3 Confidence regions for the CDF and the Kolmogorov–Smirnov test 1747 Correlation and regression in bivariate distributions 1777.1 Introduction 1777.2 Bivariate distributions and correlation 1787.3 On baseline corrections and other covariates 1837.4 Bivariate Gaussian distributions 1867.5 Regression to the mean 1897.6 Statistical analysis of bivariate Gaussian data 1957.7 Simultaneous analysis of two binomial proportions 1997.8 Comments and further reading 203References 2037.A Appendix: Some technical comments 2057.A.1 The regression to the mode equation 2057.A.2 Analysis of data from the multivariate Gaussian distribution 2067.A.3 On the geometric approach to univariate confidence limits 2078 How to compare the outcome in two groups 2098.1 Introduction 2098.2 Simple models that compare two distributions 2108.3 Comparison done the horizontal way 2128.4 Analysis done the vertical way 2168.5 Some ways to compute p-values 2248.6 The discrete Wilcoxon test 2268.7 The two-period crossover trial 2298.8 Multivariate analysis and analysis of covariance 2328.9 Comments and further reading 239References 2408.A Appendix: About U-statistics 2419 Least squares, linear models and beyond 2459.1 Introduction 2459.2 The purpose of mathematical models 2469.3 Different ways to do least squares 2509.4 Logistic regression, with variations 2529.5 The two-step modeling approach 2579.6 The effect of missing covariates 2609.7 The exponential family of distributions 2639.8 Generalized linear models 2699.9 Comments and further reading 270References 27010 Analysis of dose response 27310.1 Introduction 27310.2 Dose–response relationship 27410.3 Relative dose potency and therapeutic ratio 27810.4 Subject-specific and population averaged dose response 27910.5 Estimation of the population averaged dose–response relationship 28110.6 Estimating subject-specific dose responses 28510.7 Comments and further reading 288References 28811 Hazards and censored data 28911.1 Introduction 28911.2 Censored observations: incomplete knowledge 29011.3 Hazard models from a population perspective 29111.4 The impact of competing risks 29611.5 Heterogeneity in survival analysis 30011.6 Recurrent events and frailty 30411.7 The principles behind the analysis of censored data 30611.8 The Kaplan–Meier estimator of the CDF 30911.9 Comments and further reading 312References 31311.A Appendix: On the large-sample approximations of counting processes 31412 From the log-rank test to the Cox proportional hazards model 31712.1 Introduction 31712.2 Comparing hazards between two groups 31812.3 Nonparametric tests for hazards 31912.4 Parameter estimation in hazard models 32412.5 The accelerated failure time model 32812.6 The Cox proportional hazards model 33112.7 On omitted covariates and stratification in the log-rank test 33612.8 Comments and further reading 338References 33912.A Appendix: Comments on interval-censored data 34113 Remarks on some estimation methods 34313.1 Introduction 34313.2 Estimating equations and the robust variance estimate 34413.3 From maximum likelihood theory to generalized estimating equations 35113.4 The analysis of recurrent events 35513.5 Defining and estimating mixed effects models 36013.6 Comments and further reading 366References 36713.A Appendix: Formulas for first-order bias 368Index 371