5.7 poisson regression in the analysis of cohort data
- 1. Poisson Regression in the Analysis of Cohort Data adapted from: Szklo, 2014 STATA Technical User’s
- 3. * 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 predictors RATE = exp(b0 + b1X1 + b2X2 + … + bkXk) ln(RATE) = (b0 + b1X1 + b2X2 + … + bkXk)
- 4. * Overview of some linear and log-linear regression models used by epidemiologists • Linear regression y = (b0 + b1X1 + b2X2 + … + bkXk) • Logistic regression ln (odds) = (b0 + b1X1 + b2X2 + … + bkXk) • Poisson regression ln(rate) =(b0 + b1X1 + b2X2 + … + bkXk)
- 5. * 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
- 6. 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) *
- 7. * 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)
- 8. * 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)? –
- 11. * Poisson regression example #2 • Doll and Hill • Research question: Does smoking increase the relative risk of death from coronary disease among male British doctors?