Analysis of Incidence Rates: 1st Edition (Hardback) book cover

Analysis of Incidence Rates

1st Edition

By Peter Cummings

Chapman and Hall/CRC

478 pages | 63 B/W Illus.

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Hardback: 9780367152062
pub: 2019-05-07
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This book clarifies what incidence rates are, reviews their advantages and limitations, promotes understanding of analytic methods, describes what can be done in modern software, provides practical suggestions for analyses, and points out problems and pitfalls. Stata commands that can recreate the calculations, tables, and figures are provided for each chapter.

Table of Contents

  1. Do Storks Bring Babies?
  2. Karl Pearson and spurious correlation

    Jerzy Neyman, storks, and babies

    Is Poisson regression the solution to the stork problem?

    Further reading

  3. Risks and Rates
  4. What is a rate?

    Closed and open populations

    Measures of time

    Numerators for rates: counts

    Numerators that may be mistaken for counts

    Prevalence proportions

    Denominators for rates: count denominators for incidence proportions (risks)

    Denominators for rates: person-time for incidence rates

    Rate numerators and denominators for recurrent events

    Rate denominators other than person-time

    Different incidence rates tell different stories

    Potential advantages of incidence rates compared with incidence proportions (risks)

    Potential advantages of incidence proportions (risks) compared with incidence rates

    Limitations of risks and rates

    Radioactive decay: an example of exponential decline

    The relevance of exponential decay to human populations

    Relationships between rates, risks, and hazards

    Further reading

  5. Rate Ratios and Differences
  6. Estimated associations and causal effects

    Sources of bias in estimates of causal effect

    Estimation versus prediction

    Ratios and differences for risks and rates

    Relationships between measures of association in a closed population

    The hypothetical TEXCO study

    Breaking the rules: Army data for Companies A and B

    Relationships between odds ratios, risk ratios, and rate ratios in case-control studies

    Symmetry of measures of association

    Convergence problems for estimating associations

    Some history regarding the choice between ratios and differences

    Other influences on the choice between use of ratios or differences

    The data may sometimes be used to choose between a ratio or a difference

  7. The Poisson Distribution
  8. Alpha particle radiation

    The Poisson distribution

    Prussian soldiers kicked to death by horses

    Variances, standard deviations, and standard errors for counts and rates

    An example: mortality from Alzheimer’s disease

    Large sample P-values for counts, rates, and their differences using the Wald statistic

    Comparisons of rates as differences versus ratios

    Large sample P-values for counts, rates, and their differences using the score statistic

    Large sample confidence intervals for counts, rates, and their differences

    Large sample P-values for counts, rates, and their ratios

    Large sample confidence intervals for ratios of counts and rates

    A constant rate based on more person-time is more precise

    Exact methods

    What is a Poisson process?

    Simulated examples

    What if the data are not from a Poisson process? Part , overdispersion

    What if the data are not from a Poisson process? Part , underdispersion

    Must anything be rare?

    Bicyclist deaths in and

  9. Criticism of Incidence Rates
  10. Florence Nightingale, William Farr, and hospital mortality rates Debate in

    Florence Nightingale, William Farr, and hospital mortality rates Debate in -

    Criticism of rates in the British Medical Journal in

    Criticism of incidence rates in

  11. Stratified Analysis: Standardized Rates
  12. Why standardize?

    External weights from a standard population: direct standardization

    Comparing directly standardized rates

    Choice of the standard influences the comparison of standardized rates

    Standardized comparisons versus adjusted comparisons from variance-minimizing methods

    Stratified analyses

    Variations on directly standardized rates

    Internal weights from a population: indirect standardization

    The standardized mortality ratio (SMR)

    Advantages of SMRs compared with SRRs (ratios of directly standardized rates)

    Disadvantages of SMRs compared with SRRs (ratios of directly standardized rates)

    The terminology of direct and indirect standardization

    P-values for directly standardized rates

    Confidence intervals for directly standardized rates

    P-values and confidence intervals for SRRs (ratios of directly standardized rates)

    Large sample P-values and confidence intervals for SMRs

    Small sample P-values and confidence intervals for SMRs

    Standardized rates should not be used as regression outcomes

    Standardization is not always the best choice

  13. Stratified Analysis: Inverse-variance and Mantel-Haenszel Methods
  14. Inverse-variance methods

    Inverse-variance analysis of rate ratios

    Inverse-variance analysis of rate differences

    Choosing between rate ratios and differences

    Mantel-Haenszel methods

    Mantel-Haenszel analysis of rate ratios

    Mantel-Haenszel analysis of rate differences

    P-values for stratified rate ratios or differences

    Analysis of sparse data

    Maximum-likelihood stratified methods

    Stratified methods versus regression

  15. Collapsibility and Confounding
  16. What is collapsibility?

    The British X-Trial: introducing variation in risk

    Rate ratios and differences are noncollapsible because exposure influences person-time

    Which estimate of the rate ratio should we prefer?

    Behavior of risk ratios and differences

    Hazard ratios and odds ratios

    Comparing risks with other outcome measures

    The Italian X-Trial: -levels of risk under no exposure

    The American X-Cohort study: -levels of risk in a cohort study

    The Swedish X-Cohort study: a collapsible risk ratio in confounded data

    A summary of findings

    A different view of collapsibility

    Practical implications: avoid common outcomes

    Practical implications: use risks or survival functions

    Practical implications: case-control studies

    Practical implications: uniform risk

    Practical implications: use all events

  17. Poisson Regression for Rate Ratios
  18. The Poisson regression model for rate ratios

    A short comparison with ordinary linear regression

    A Poisson model without variables

    A Poisson regression model with one explanatory variable

    The iteration log

    The header information above the table of estimates

    Using a generalized linear model to estimate rate ratios

    An alternative parameterization for Poisson models: a regression trick

    Further comments about person-time

    A short summary

  19. Poisson Regression for Rate Differences
  20. A regression model for rate differences

    Florida and Alaska cancer mortality: regression models that fail

    Florida and Alaska cancer mortality: regression models that succeed

    A generalized linear model with a power link

    A caution

  21. Linear Regression
  22. Limitations of ordinary least squares linear regression

    Florida and Alaska cancer mortality rates

    Weighted least squares linear regression

    Importance weights for weighted least squares linear regression

    Comparison of Poisson, weighted least squares, and ordinary least squares regression

    Exposure to a carcinogen: ordinary linear regression ignores the precision of each rate

    Differences in homicide rates: simple averages versus population-weighted averages

    The place of ordinary least squares linear regression for the analysis of incidence rates

    Variance weighted least squares regression

    Cautions regarding inverse-variance weights

    Why use variance weighted least squares?

    A short comparison of weighted Poisson regression, variance weighted least squares, and weighted linear regression

    Problems when age-standardized rates are used as outcomes

    Ratios and spurious correlation

    Linear regression with ln(rate) as the outcome

    Predicting negative rates


  23. Model Fit
  24. Tabular and graphic displays

    Goodness of fit tests: deviance and Pearson statistics

    A conditional moment chi-squared test of fit

    Limitations of goodness-of-fit statistics

    Measures of dispersion

    Robust variance estimator as a test of fit

    Comparing models using the deviance

    Comparing models using Akaike and Bayesian information criterion

    Example : 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

    Example : a rate ratio or a rate difference model for hypothetical data regarding the association between fall rates and age

    A test of the model link

    Residuals, influence analysis, and other measures

    Adding model terms to improve fit

    A caution

  25. Adjusting Standard Errors and Confidence Intervals
  26. Estimating the variance without regression

    Poisson regression

    Rescaling the variance using the Pearson dispersion statistic

    Robust variance

    Generalized Estimating Equations

    Using the robust variance to study length of hospital stay

    Computer intensive methods

    The bootstrap idea

    The bootstrap normal method

    The bootstrap percentile method

    The bootstrap bias-corrected percentile method

    The bootstrap bias-corrected and accelerated method

    The bootstrap-t method

    Which bootstrap CI is best?

    Permutation and Randomization

    Randomization to nearly equal groups

    Better randomization using the randomized block design of the original study

    A summary

  27. Storks and Babies, Revisited
  28. Neyman’s approach to his data

    Using methods for incidence rates

    A model that uses the stork/women ratio

  29. Flexible Treatment of Continuous Variables
  30. The problem

    Quadratic splines

    Fractional polynomials

    Flexible adjustment for time

    Which method is best?

  31. Judging Variation in Size of an Association
  32. An example: shoes and falls

    Problem : Using subgroup P-values for interpretation

    Problem : Failure to include main effect terms when interaction terms are used

    Problem : Incorrectly concluding that there is no variation in association

    Problem : Interaction may be present on a ratio scale but not on a difference scale, and vice versa

    Problem : Failure to report all subgroup estimates in an evenhanded manner

  33. Negative Binomial Regression
  34. Negative binomial regression is a random effects or mixed model

    An example: accidents among workers in a munitions factory

    Introducing equal person-time in the homicide data

    Letting person-time vary in the homicide data

    Estimating a rate ratio for the homicide data

    Another example using hypothetical data for five regions

    Unobserved heterogeneity

    Observing heterogeneity in the shoe data


    A rate difference negative binomial regression model


  35. Clustered Data
  36. Data from fictitious nursing homes

    Results from , data simulations for the nursing homes

    A single random set of data for the nursing homes

    Variance adjustment methods

    Generalized estimating equations (GEE)

    Mixed model methods

    What do mixed models estimate?

    Mixed model estimates for the nursing home intervention

    Simulation results for some mixed models

    Mixed models weight observations differently than Poisson regression

    Which should we prefer for clustered data, variance-adjusted or mixed models?

    Additional model commands for clustered data

    Further reading

  37. Longitudinal Data
  38. Just use rates

    Using rates to evaluate governmental policies

    Study designs for governmental policies

    A fictitious water treatment and US mortality -

    Poisson regression

    Population-averaged estimates (GEE)

    Conditional Poisson regression, a fixed-effects approach

    Negative binomial regression

    Which method is best?

    Water treatment in only states

    Conditional Poisson regression for the -state water-treatment data

    A published study

  39. Matched Data
  40. Matching in case-control studies

    Matching in randomized controlled trials

    Matching in cohort studies

    Matching to control confounding in some randomized trials and cohort studies

    A benefit of matching; only matched sets with at least one outcome are needed

    Studies designs that match a person to themselves

    A matched analysis can account for matching ratios that are not constant

    Choosing between risks and rates for the crash data in Tables and

    Stratified methods for estimating risk ratios for matched data

    Odds ratios, risk ratios, cell A, and matched data

    Regression analysis of matched data for the odds ratio

    Regression analysis of matched data for the risk ratio

    Matched analysis of rates with one outcome event

    Matched analysis of rates for recurrent events

    The randomized trial of exercise and falls; some problems revealed

    Final words

  41. Marginal Methods
  42. What are margins?

    Converting logistic regression results into risk ratios or risk differences: marginal standardization

    Estimating a rate difference from a rate ratio model

    Death by age and sex: a short example

    Skunk bite data: a long example

    Obtaining the rate difference: crude rates

    Using the robust variance

    Adjusting for age

    Full adjustment for age and sex

    Marginal commands for interactions

    Marginal methods for a continuous variable

    Using a rate difference model to estimate a rate ratio: use the ln scale

  43. Bayesian Methods
  44. Cancer mortality rate in Alaska

    The rate ratio for falling in a trial of exercise

  45. Exact Poisson Regression
  46. A simple example

    A perfectly predicted outcome

    Memory problems

    A caveat

  47. Instrumental Variables
  48. The problem: what does a randomized controlled trial estimate?

    Analysis by treatment received may yield biased estimates of treatment effect

    Using an instrumental variable

    Two-stage linear regression for instrumental variables

    Generalized method of moments

    Generalized method of moments for rates

    What does an instrumental variable analysis estimate?

    There is no free lunch

    Final comments

  49. Hazards

Data for a hypothetical treatment with exponential survival times

Poisson regression and exponential proportional hazards regression

Poisson and Cox proportional hazards regression

Hypothetical data for a rate that changes over time

A piecewise Poisson model

A more flexible Poisson model: quadratic splines

Another flexible Poisson model: restricted cubic splines

Flexibility with fractional polynomials

When should a Poisson model be used? Randomized trial of a terrible treatment

A real randomized trial, the PLCO screening trial

What if events are common?

Cox model or a flexible parametric model?

Collapsibility and survival functions

Relaxing the assumption of proportional hazards in the Cox model

Relaxing the assumption of proportional hazards for the Poisson model

Relaxing proportional hazards for the Royston-Parmar model

The life expectancy difference or ratio

Recurrent or multiple events

A short summary



About the Author

Peter Cummings MD, MPH is Emeritus Professor of Epidemiology, School of Public Health, University of Washington, Seattle, Washington. His primary research interest has been in studies related to injuries, particularly car crashes. He has published articles about the use of case-control and matched-cohort methods for the study of injuries. He has over 100 publications in peer-reviewed journals.

About the Series

Chapman & Hall/CRC Biostatistics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General