Analysis of Incidence Rates 

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Comment from the Stata technical groupPeter Cumming's Analysis of Incidence Rates is a fundamental reference for those who want to understand the concepts related to incidence rates and use the resources and techniques available in the 21st century. The classical reference for this topic, Breslow and Day's Statistical Methods in Cancer Research, was cutting edge when written more than 30 years ago, and it's still an important theoretical reference. However, since then, there have been many methodological advances that came with the availability of fast and powerful computers. For example, Breslow and Day recommend an ingenious manual approximation to the estimates from a Poisson regression, which today would be less flexible and less accurate than obtaining the maximum likelihood estimates with a specific sofware, but it's still present in the current literature. Cumming's approach takes advantage of the resources available today and shows how to implement them using Stata. The first part of the book explains theoretical concepts associated with risk ratios in an intuitive way, without relying too heavily on mathematical formulas. The book tackles the old confusion between odds ratios, rate ratios, and risk ratios in case—control analysis and explains direct and indirect standardizations, among other fundamental concepts. Then, the book takes advantage of the poisson Stata command and other Stata features to show how to implement them in practice and discusses applied topics like how to adjust for covariates, marginal methods, and even Bayesian analysis. The process is illustrated with numerous applied examples, using both real datasets and simulated data. Dofiles to reproduce these examples are available online. This book will be very helpful to epidemiologists, social scientists, and any researchers interested in the concept of incidence rates. Reference: Breslow, N.E., and N.E. Day. 1987. Statistical Methods in Cancer Research. Volume IIThe Design and Analysis of Cohort Studies. Lyon, France: International Agency for Research on Cancer.  
Table of contentsView table of contents >> 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: PersonTime for Incidence Rates 2.9 Rate Numerators and Denominators for Recurrent Events 2.10 Rate Denominators Other than PersonTime 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 CaseControl 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 Pvalues for Counts, Rates, and Their Differences using the Wald Statistic 4.7 Comparisons of Rates as Differences versus Ratios 4.8 Large Sampel Pvalues 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 Pvalues 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 PersonTime 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 VarianceMinimizing 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 Pvalues for Directly Standardized Rates 6.14 Confidence Intervals for Directly Standardized Rates 6.15 Pvalues and CIs for SRRs (Ratios of Directly Standardized Rates) 6.16 Large Sample Pvalues and CIs for SMRs 6.17 Small Sample Pvalues 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: InverseVariance and MantelHaenszel Methods
7.1 Inversevariance Methods
7.2 InverseVariance Analysis of Rate Ratios 7.3 InverseVariance Analysis of Rate Differences 7.4 Choosing between Rate Ratios and Differences 7.5 MantelHaenszel Methods 7.6 MantelHaenszel Analysis of Rate Ratios 7.7 MantelHaenszel Analysis of Rate Differences 7.8 Pvalues for Stratified Rate Ratios of Differences 7.9 Analysis of Sparse Data 7.10 MaximumLikelihood Stratified Methods 7.11 Stratified Methods versus Regression 8 Collapsibility and Confoundings
8.1 What Is Collapsibility?
8.2 The British XTrial: Introducing Variation in Risk 8.3 Rate Ratios and Differences Are Noncollapsible because Exposure Influences PersonTime 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 XTrial: 3Levels of Risk under No Exposure 8.9 The American XCohort Study: 3Levels of Risk in a Cohort Study 8.10 The Swedish XCohort 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: CaseControl 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 PersonTime 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 PopulationWeighted 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 InverseVariance 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 AgeStandardized 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 ChiSquared Test of Fit 12.4 Limitations of GoodnessofFit 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 BiasCorrected Percentile Method 13.12 The Bootstrap BiasCorrected and Accelerated Method 13.13 The BootstrapT 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 Pvalues 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 PersonTime in the Homicide Data 17.4 Letting PersonTime 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, VarianceAdjusted 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 PopulationAveraged Estimates (GEE) 19.7 Conditional Poisson Regression, a FixedEffects 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 10State Water Treatment Data 19.12 A Published Study 20 Matched Data
20.1 Matching in CaseControl 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 TwoStage 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 RoystonParmar Model 25.17 The Life Expectancy Difference or Ratio 25.18 Recurrent or Multiple Events 25.19 A Short Summary Bibliography
Index
