Poisson Regression in the Analysis of Cohort Data

adapted from: Szklo, 2014

STATA Technical User’s

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Poisson regression• Poisson regression

– permits multiple regression analysis of cohort data with a dichotomous outcome and one or more categorical or continuous variables

– usually used when the outcome is a rate or rate ratio

– especially useful for rare diseases in large populations

– In words: models the magnitude of the rate as an exponential function of a linear combination of covariates and unknown parameters

– Referred to as a “log-linear” model because it is a log transformation of an outcome variable (e.g. a rate) related to a linear equation of predictorsRATE = exp(b0 + b1X1 + b2X2 + … + bkXk)

ln(RATE) = (b0 + b1X1 + b2X2 + … + bkXk)

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Overview of some linear and log-linear regression models

used by epidemiologists• Linear regression

y = (b0 + b1X1 + b2X2 + … + bkXk)

• Logistic regressionln (odds) = (b0 + b1X1 + b2X2 + … + bkXk)

• Poisson regressionln(rate) =(b0 + b1X1 + b2X2 + … + bkXk)

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Poisson regression• Poisson regression--algebraic transformations

RATE = exp(b0 + b1X1 + b2X2 + … + bkXk)take ln of both sides using logarithmic property: ln {exp(a)} = a

ln(RATE) =(b0 + b1X1 + b2X2 + … + bkXk) rewrite “RATE” as “events/person-time”

ln(events / person-time) =(b0 + b1X1 + b2X2 + … + bkXk)use properties of ln to rewrite left side:

ln (a/b) = ln (a) - ln (b)

ln(events) - ln(person-time) =(b0 + b1X1 + b2X2 + … + bkXk) move ln(person-time) to right side of equation

*ln(events) = ln(person-time) + b0 + b1X1 + b2X2 + … + bkXk

combine ln(person-time)--which is a constant--with b0 to form b0

*ln(events) = b0 + b1X1 + b2X2 + … + bkX

Poisson regression

ln(RATEexp

ln(RATEunexp

• Poisson regression

For the exposed (X1 = 1):) =(b0 + b1*1 + b2X2 + … + bkXk)

For the unexposed (X1 = 0)) =(b0 + b1*0 + b2X2 + … + bkXk)

Subtract the two equations:) - ln(RATE ) = b1ln(RATEexp unexp

Rewrite using rule of logarithms:ln(RATEexp / RATEunexp) = ln (RATEexp / RATE unexp) = ln(rate ratio) = b1

Exponeniate both sides:) } =exp { ln(rate ratio) } = exp (b1)

) = rate ratio = exp ( b1)

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Poisson regression

Exponeniate both sides:

exp { ln(RATEexp / RATE ) } =exp { ln(rate ratio) } = exp (b1)

unexp

(RATEexp / RATE ) = rate ratio = exp ( b1)unexp

Comment: The rate ratio (comparing exposed vs. unexposed) is the exponentiation of the coefficient

All of the independent (X) variables in the Poisson regression examples used today are categorical (but the model can include continuous independent (X) variables)

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Poisson regression example #1• ARIC (Atherosclerosis Risk in Communities)

– Washington County cohort– Ages 45-64 at baseline, 1987-1994

• (One) research question:– Does smoking, adjusted for age, gender,

hyptertension and obesity, increase the relative risk of coronary heart disease (compared to non-smokers)?

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Poisson regression example #2• Doll and Hill• Research question: Does smoking increase the relative

risk of death from coronary disease among male British doctors?