Preface
Author
1 Do Storks Bring Babies?
1.1 Karl Pearson and Spurious Correlation
1.2 Jerzy Neyman, Storks, and Babies
1.3 Is Poisson Regression the Solution to the Stork Problem?
1.4 Further Reading
2 Risks and Rates
2.1 What Is a Rate?
2.2 Closed and Open Populations
2.3 Measures of Time
2.4 Numerators for Rates: Counts
2.5 Numerators that May Be Mistaken for Counts
2.6 Prevalence Proportions
2.7 Denominators for Rates: Count Denominators for Incidence Proportions (Risks)
2.8 Denominators for Rates: Person-Time for Incidence Rates
2.9 Rate Numerators and Denominators for Recurrent Events
2.10 Rate Denominators Other than Person-Time
2.11 Different Incidence Rates Tell Different Stories
2.12 Potential Advantages of Incidence Rates Compared With Incidence Proportions (Risks)
2.13 Potential Advantages of Incidence Proportions (Risks) Compared with Incidence Rates
2.14 Limitations of Risks and Rates
2.15 Radioactive Decay: An Example of Exponential Decline
2.16 The Relevance of Exponential Decay to Human Populations
2.17 Relationships Between Rates, Risks, and Hazards
2.18 Further Reading
3 Rate Ratios and Differences
3.1 Estimated Associations and Causal Effects
3.2 Sources of Bias in Estimates of Causal Effect
3.3 Estimation versus Prediction
3.4 Ratios and Differences for Risks and Rates
3.5 Relationships between Measures of Association in a Closed Population
3.6 The Hypothetical TEXCO Study
3.7 Breaking the Rules: Army Data for Companies A and B
3.8 Relationships between Odds Ratios, Risk Ratios, and Rate Ratios in Case-Control Studies
3.9 Symmetry of Measures of Association
3.10 Convergence Problems for Estimating Associations
3.11 Some History Regarding the Choice between Ratios and Differences
3.12 Other Influences on the Choice between Use of Ratios or Differences
3.13 The Data May Sometimes Be Used to Choose between a Ratio of a Difference
4 The Poisson Distribution
4.1 Alpha Particle Radiation
4.2 The Poisson Distribution
4.3 Prussian Soldiers Kicked to Death by Horses
4.4 Variances, Standard Deviations, and Standard Errors for Counts and Rates
4.5 An Example: Mortality from Alzheimer's Disease
4.6 Large Sample P-values for Counts, Rates, and Their Differences using the Wald Statistic
4.7 Comparisons of Rates as Differences versus Ratios
4.8 Large Sampel P-values for Counts, Rates, and Their Differences using the Score Statistic
4.9 Large Sample Confidence Intervals for Counts, Rates, and Their Differences
4.10 Large Sample P-values for Counts, Rates, and Their Ratios
4.11 Large Sample Confidence Intervals for Ratios of Counts and Rates
4.12 A constant Rate Based on More Person-Time Is More Precise
4.13 Exact Methods
4.14 What Is a Poisson Process?
4.15 Simulated Examples
4.16 What If the Data Are Not from a Poisson Process? Part 1, Overdispersion
4.17 What If the Data Are Not from a Poisson Process? Part 2, Underdispersion
4.18 Must Anything Be Rare?
4.19 Bicyclist Deaths in 2010 and 2011
5 Criticism of Incidence Rates
5.1 Florence Nightingale, William Farr, and Hospital Mortality Rates. Debate in 1864
5.2 Florence Nightingale, William Farr, and Hospital Mortality Rates. Debate in 1996—1997
5.3 Criticism of Rates in the British Medical Journal in 1995
5.4 Criticism of Incidence Rates in 2009
6 Stratified Analysis: Standardized Rates
6.1 Why Standardize?
6.2 External Weights from a Standard Population: Direct Standardization
6.3 Comparing Directly Standardized Rates
6.4 Choice of the Standard Influences the Comparison of Standardized Rates
6.5 Standardized Comparisons versus Adjusted Comparisons from Variance-Minimizing Methods
6.6 Stratified Analyses
6.7 Variations on Directly Standardized Rates
6.8 Internal Weights from a Population: Indirect Standardization
6.9 The Standardized Mortality Ratio (SMR)
6.10 Advantages of SMRs Compared with SRRs (Ratios of Directly Standardized Rates)
6.11 Disadvantages of SMRs Compared with SRRs (Ratios of Directly Standardized Rates)
6.12 The Terminology of Direct and Indirect Standardization
6.13 P-values for Directly Standardized Rates
6.14 Confidence Intervals for Directly Standardized Rates
6.15 P-values and CIs for SRRs (Ratios of Directly Standardized Rates)
6.16 Large Sample P-values and CIs for SMRs
6.17 Small Sample P-values and CIs for SMRs
6.18 Standardized Rates Should Not Be used as Regression Outcomes
6.19 Standardization Is Not Always the Best Choice
7 Stratified Analysis: Inverse-Variance and Mantel-Haenszel Methods
7.1 Inverse-variance Methods
7.2 Inverse-Variance Analysis of Rate Ratios
7.3 Inverse-Variance Analysis of Rate Differences
7.4 Choosing between Rate Ratios and Differences
7.5 Mantel-Haenszel Methods
7.6 Mantel-Haenszel Analysis of Rate Ratios
7.7 Mantel-Haenszel Analysis of Rate Differences
7.8 P-values for Stratified Rate Ratios of Differences
7.9 Analysis of Sparse Data
7.10 Maximum-Likelihood Stratified Methods
7.11 Stratified Methods versus Regression
8 Collapsibility and Confoundings
8.1 What Is Collapsibility?
8.2 The British X-Trial: Introducing Variation in Risk
8.3 Rate Ratios and Differences Are Noncollapsible because Exposure Influences Person-Time
8.4 Which Estimate of the Rate Ratio Should We Prefer?
8.5 Behavior of Risk Ratios and Differences
8.6 Hazard Ratios and Odds Ratios
8.7 Comparing Risks with Other Outcome Measures
8.8 The Italian X-Trial: 3-Levels of Risk under No Exposure
8.9 The American X-Cohort Study: 3-Levels of Risk in a Cohort Study
8.10 The Swedish X-Cohort Study: A Collapsible Risk Ratio in Confounded Data
8.11 A Summary of Findings
8.12 A Different View of Collapsibility
8.13 Practical Implications: Avoid Common Outcomes
8.14 Practical Implications: Use Risks or Survival Functions
8.15 Practical Implications: Case-Control Studies
8.16 Practical Implications: Uniform Risk
8.17 Practical Implications: Use All Events
9 Poisson Regression for Rate Ratios
9.1 The Poisson Regression Model for Rate Ratios
9.2 A Short Comparison with Ordinary Linear Regression
9.3 A Poisson Model without Variables
9.4 A Poisson Regression Model with One Explanatory Variable
9.5 The Iteration Log
9.6 The Header Information above the Table of Estimates
9.7 Using a Generalized Linear Model to Estimate Rate Ratios
9.8 A Regression Example: Studying Rates over Time
9.9 An Alternative Parameterization for Poisson Models: A Regression Trick
9.10 Further Comments about Person-Time
9.11 A Short Summary
10 Poisson Regression for Rate Differences
10.1 A Regression Model for Rate Differences
10.2 Florida and Alaska Cancer Mortality: Regression Models that Fail
10.3 Florida and Alaska Cancer Mortality: Regression Models that Succeed
10.4 A Generalized Linear Model with a Power Link
10.5 A Caution
11 Linear Regression
11.1 Limitations of Ordinary Least Squares Linear Regression
11.2 Florida and Alaska Cancer Mortality Rates
11.3 Weighted Least Squares Linear Regression
11.4 Importance Weights for Weighted Least Squares Linear Regression
11.5 Comparison of Poisson, Weighted Least Squares, and Ordinary Least Squares Regression
11.6 Exposure to a Carcinogen: Ordinary Linear Regression Ignores the Precision of Each Rate
11.7 Differences in Homicide Rates: Simple Averages versus Population-Weighted Averages
11.8 The Place of Ordinary Least Squares Linear Regression for the Analysis of Incidence Rates
11.9 Variance Weighted Least Squares Regression
11.10 Cautions regarding Inverse-Variance Weights
11.11 Why Use Variance Weighted Least Squares?
11.12 A Short Comparison of Weighted Poisson Regression, Variance Weighted Least Squares, and Weighted Linear Regression
11.13 Problems When Age-Standardized Rates are Used as Outcomes
11.14 Ratios and Spurious Correlation
11.15 Linear Regression with In (Rate) as the Outcome
11.16 Predicting Negative Rates
11.17 Summary
12 Model Fit
12.1 Tabular and Graphic Displays
12.2 Goodness of Fit Tests: Deviance and Pearson Statistics
12.3 A Conditional Moment Chi-Squared Test of Fit
12.4 Limitations of Goodness-of-Fit Statistics
12.5 Measures of Dispersion
12.6 Robust Variance Estimator as a Test of Fit
12.7 Comparing Models using the Deviance
12.8 Comparing Models using Akaike and Bayesian Information Criterion
12.9 Example 1: Using Stata's Generalized Linear Model Command to Decide between a Rate Ratio or a Rate Difference Model for the Randomized Controlled Trial of Exercise and Falls
12.10 Example 2: A Rate Ratio of a Rate Difference Model for Hypothetical Data Regarding the Association between Fall Rates and Age
12.11 A Test of the Model Link
12.12 Residuals, Influence Analysis, and Other Measures
12.13 Adding Model Terms to Improve Fit
12.14 A Caution
12.15 Further Reading
13 Adjusting Standard Errors and Confidence Intervals
13.1 Estimating the Variance without Regression
13.2 Poisson Regression
13.3 Rescaling the Variance using the Pearson Dispersion Statistic
13.4 Robust Variance
13.5 Generalized Estimating Equations
13.6 Using the Robust Variance to Study Length of Hospital Stay
13.7 Computer Intensive Methods
13.8 The Bootstrap Idea
13.9 The Bootstrap Normal Method
13.10 The Bootstrap Percentile Method
13.11 The Bootstrap Bias-Corrected Percentile Method
13.12 The Bootstrap Bias-Corrected and Accelerated Method
13.13 The Bootstrap-T Method
13.14 Which Bootstrap CI Is Best?
13.15 Permutation and Randomization
13.16 Randomization to Nearly Equal Groups
13.17 Better Randomization Using the Randomized Block Design of the Original Study
13.18 A Summary
14 Storks and Babies, Revisited
14.1 Neyman's Approach to His Data
14.2 Using Methods for Incidence Rates
14.3 A Model That uses the Stork/Women Ratio
15 Flexible Treatment of Continuous Variables
15.1 The Problem
15.2 Quadratic Splines
15.3 Fractional Polynomials
15.4 Flexible Adjustment for Time
15.5 Which Method Is Best?
16 Variation in Size of an Association
16.1 An Example: Shoes and Falls
16.2 Problem 1: Using Subgroup P-values for Interpretation
16.3 Problem 2: Failure to Include Main Effect Terms When Interaction Terms Are Used
16.4 Problem 3: Incorrectly Concluding that There Is No Variation in Association
16.5 Problem 4: Interaction May Be Present on a Ratio Scale but Not on a Difference Scale, and Vice Versa
16.6 Problem 5: Failure to Report All Subgroup Estimates in an Evenhanded Manner
17 Negative Binomial Regression
17.1 Negative Binomial Regression Is a Random Effects or Mixed Model
17.2 An Example: Accidents among Workers in a Munitions Factory
17.3 Introducing Equal Person-Time in the Homicide Data
17.4 Letting Person-Time Vary in the Homicide Data
17.5 Estimating a Rate Ratio for the Homicide Data
17.6 Another Example using Hypothetical Data for Five Regions
17.7 Unobserved Heterogeneity
17.8 Observing Heterogeneity in the Shoe Data
17.9 Underdispersion
17.10 A Rate Difference Negative Binomial Regression Model
17.11 Conclusion
18 Clustered Data
18.1 Data from 24 Fictitious Nursing Homes
18.2 Results from 10,000 Data Simulations for the Nursing Homes
18.3 A Single Random Set of Data for the Nursing Homes
18.4 Variance Adjustment Methods
18.5 Generalized Estimating Equations (GEE)
18.6 Mixed Model Methods
18.7 What Do Mixed Models Estimate
18.8 Mixed Models Estimates for the Nursing Home Intervention
18.9 Simulation Results for Some Mixed Models
18.10 Mixed Models Weight Observations Differently that Poisson Regression
18.11 Which Should We Prefer for Clustered Data, Variance-Adjusted or Mixed Models?
18.12 Additional Model Commands for Clustered Data
18.13 Further Reading
19 Longitudinal Data
19.1 Just Use Rates
19.2 Using Rates to Evaluate Governmental Policies
19.3 Study Designs for Governmental Policies
19.4 A Fictitious Water Treatment and U.S. Mortality 1999—2013
19.5 Poisson Regression
19.6 Population-Averaged Estimates (GEE)
19.7 Conditional Poisson Regression, a Fixed-Effects Approach
19.8 Negative Binomial Regression
19.9 Which Method Is Best?
19.10 Water Treatment in Only 10 States
19.11 Conditional Poisson Regression for the 10-State Water Treatment Data
19.12 A Published Study
20 Matched Data
20.1 Matching in Case-Control Studies
20.2 Matching in Randomized Controlled Trials
20.3 Matching in Cohort Studies
20.4 Matching to Control Confounding in Some Randomized Trials and Cohort Studies
20.5 A Benefit of Matching; Only Matched Sets with at Least One Outcome Are Needed
20.6 Studies Designs that Match a Person to Themselves
20.7 A Matched Analysis Can Account for Matching Ratios that Are Not Constant
20.8 Choosing between Risks and Rates for the Crash Data in Tables 20.1 and 20.2
20.9 Stratified Methods for Estimating Risk Ratios for Matched Data
20.10 Odds Ratios, Risk Ratios, Cell A, and Matched Data
20.11 Regression Analysis of Matched Data for the Odds Ratio
20.12 Regression Analysis of Matched Data for the Risk Ratio
20.13 Matched Analysis of Rates with One Outcome Event
20.14 Matched Analysis of Rates for Recurrent Events
20.15 The Randomized Trial of Exercise and Falls; Additional Analyses
20.16 Final Words
21 Marginal Methods
21.1 What Are Margins?
21.2 Converting Logistic Regression Results into Risk Ratios or Risk Differences: Marginal Standardization
21.3 Estimating a Rate Difference from a Rate Ratio Model
21.4 Death by Age and Sex: A Short Example
21.5 Skunk Bite Data: A Long Example
21.6 Obtaining the Rate Difference: Crude Rates
21.7 Using the Robust Variance
21.8 Adjusting for Age
21.9 Full Adjustment for Age and Sex
21.10 Marginal Commands for Interactions
21.11 Marginal Methods for a Continuous Variable
21.12 Using a Rate Difference Model to Estimate a Rate Ratio: Use the In Scale
22 Bayesian Methods
22.1 Cancer Mortality Rate in Alaska
22.2 The Rate Ratio for Falling in a Trial of Exercise
23 Exact Poisson Regression
23.1 A Simple Example
23.2 A Perfectly Predicted Outcome
23.3 Memory Problems
23.4 A Caveat
24 Instrumental Variables
24.1 The Problem: What Does a Randomized Controlled Trial Estimate?
24.2 Analysis by Treatment Received May Yield Biased Estimates of Treatment Effect
24.3 Using an Instrumental Variable
24.4 Two-Stage Linear Regression for Instrumental Variables
24.5 Generalized Method of Moments
24.6 Generalized Method of Moments for Rates
24.7 What Does an Instrumental Variable Analysis Estimate?
24.8 There Is No Free Lunch
24.9 Final Comments
25 Hazards
25.1 Data for a Hypothetical Treatment with Exponential Survival Times
25.2 Poisson Regression and Exponential Proportional Hazards Regression
25.3 Poisson and Cox Proportional Hazards Regression
25.4 Hypothetical Data for a Rate that Changes over Time
25.5 A Piecewise Poisson Model
25.6 A More Flexible Poisson Model: Quadratic Splines
25.7 Another Flexible Poisson Model: Restricted Cubic Splines
25.8 Flexibility with Fractional Polynomials
25.9 When Should a Poisson Model Be Used? Randomized Trial of a Terrible Treatment
25.10 A Real Randomized Trial, the PLCO Screening Trial
25.11 What If Events Are Common?
25.12 Cox Model or a Flexible Parametric Model?
25.13 Collapsibility and Survival Functions
25.14 Relaxing the Assumption of Proportional Hazards in the Cox Model
25.15 Relaxing the Assumption of Proportional Hazards for the Poisson Model
25.16 Relaxing Proportional Hazards for the Royston-Parmar Model
25.17 The Life Expectancy Difference or Ratio
25.18 Recurrent or Multiple Events
25.19 A Short Summary
Bibliography
Index
