Poisson Regression

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Poisson Regression

• Advanced Epidemiologic Methods II
• Spring 2002
• 4/2/02

2
Objectives

• To be able to analyze grouped person-time data in
  a systematic manner by simultaneously adjusting
  for multiple independent variables using Poisson
  regression modeling
• To understand the assumptions of the Poisson
  regression model
• To understand the limitations of the Poisson
  regression model

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Objectives, contd

• To examine stratum-specific rate ratios and
  age-adjusted summary rate ratios using different
  approaches for summarization.
• To be able to detect the presence of potentially
  important confounding and/or effect measure
  modification
• To be able to compare and contrast study findings
  based on external and internal standards or
  referent populations

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What is Poisson regression?

• Based on the Poisson Distribution
• f(y) µy exp(-µ)/y! Where y0,1,8
• mean E(Y) µ
• variance s2(Y) µ
• Used for rare, independent events
• Derived similarly to linear regression (which is
  based on the normal distribution), but uses a
  link function to correct for the non-linearity of
  Poisson assumptions

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Whats Poisson Regression used for?

• Modeling dependant variables that describe count
  data and a set of explanatory variables
• In other words, use for grouped/ aggregate data
  NOT individual data
• Examples
• Temporal or spatial modeling of disease incidence
  (rate and risk ratios)
• Colony counts with varying experimental
  conditions
• Equipment failures under varying conditions of use

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Whats different about Poisson?

• Differences between Poisson and linear
  regression
• Dependant variable not continuous, but discrete
• Dependant variable and errors not distributed
  normally before transformation, but as Poisson
• Relationship between independent and dependant
  variables is not linear
• Requires a link function to transform the
  non-linear model to a linear one

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Derivation of Poisson

• GLMs (including Poisson) have form
• E(Yi) µ ß0 ß1xi
• Use a link function to convert unsolvable
  non-linear function to solvable linear function
  (like logistic uses the logit function)
• Link function for Poisson usually ln
• General log-linear model looks like
• ln µ ß0 ß1xi

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Applying log-linear Poisson regression to
epidemiology

• Rate models
• Based on Poisson process, which assumes
  independent events distributed as Poisson with
  mean µ ? t
• MLE of rate is events/person-time or
• ? µ /t
• Suggests Poisson model
• ln ? ln (µ /t) ß0 ß1x

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Almost finished with the math

• ln ? ln (µ /t) ß0 ß1x
• ln µ ln t ß0 ß1x
• ln µ ß0 ß1x ln t
• where ln t is the offset
• To estimate relative rates
• In exposed ln ? e b0 b11 b0 b1
• In unexposed ln ? u b0 b10 b0
• ln ? e / ln ? u ln ? e - ln ? u b0 b1 - b0
  b1
• RR ? e / ? u exp(b1)
• 95 CI expb1 1.96se(b1)

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Other miscellaneous info

• Poisson generally solved with ML estimation
• Can be used to look at relative risks (data from
  surveillance) and to adjust SMRs for confounders
• Also for proportional hazards modeling with count
  data

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Assumptions of Poisson model

• Rates are log-linear (changes linearly with
  increasing exposure)
• Interactions on multiplicative scale
• At each level of each covariate, the number of
  cases has variance to its mean
• Observations are independent

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Violations of assumptions

• Overdispersion
• Estimated variance of Y gt mean, std deviations
  and p values underestimated
• AKA extra-Poisson variation
• Use negative binomial instead
• Plot residuals vs mean to evaluate
• Non-independence of events
• Because of residual confounding
• May need to use autocorrelation/ autoregressive
  model

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Alternatives to Poisson regression

• Mantel-Haenszel
• SMRs/ SIRs
• Cox proportional hazards model (if have
  individual data)

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Summary

• Poisson regression is not a very robust model
  (compared to Cox, logistic, etc.)
• If you have individual data, you should use these
  other models
• However, Poisson may be only option if you have
  count data and want to adjust for multiple
  confounders

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Project 8 data

• Population Vietnam veterans living in Texas
  between 1995-97
• Outcome prostate cancer incidence rate
• Exposure of interest military service in
  Vietnam during 1962-75
• See background info

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Project 8- stata code

• For Poisson regression, reporting RRs
• poisson depvar varlist, exposure(varname) irr
• For M-H, SIR, direct standardization
• ir case_var exp_var time_var, by(varname)
  istandard estandard

Direct
SIR
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Project 8 assignment hints

• 2 versions of dataset
• vietnam-sir.dta
• vietnam-poisson.dta
• Use sir to calculate SIRs, directly adjusted
  RRs, pooled RR, M-H summary RR, and Poisson
  regression using the external comparison
• Use poisson to run Poisson regression using only
  internal comparisons