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Ch13. Poisson Regression Analysis

Ch13. Poisson Regression Analysis. Poisson Regression Analysis is used when the outcome variable comprises counts, usually of rather rare events e.g. number of cases of cancer over a defined period in a cohort of subjects.

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Ch13. Poisson Regression Analysis

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  1. Ch13. Poisson Regression Analysis Poisson Regression Analysis is used when the outcome variable comprises counts, usually of rather rare events e.g. number of cases of cancer over a defined period in a cohort of subjects. In the case of multiple linear regression the outcome variable is continuous numeric. We model the data by linking the mean of the outcome variable to a linear function Y = β0 + β1X1 + β2X2+….+βnXn When the outcome variable is binary we use a logistic transformation of the probability P of the outcome occurring i.e. log(p/1-p) to a linear function Log(p/1-p) = β0 + β1X1 + β2X2 + …. +βnXn When the response variable is in the form of a discrete number e.g. a count we model the data by linking the logarithm of the outcome variable to a linear function Log (Y) = β0 + β1X1 + β2X2 + … +βnXn

  2. Assumptions in Poisson Regression • Logarithm of the disease rate changes linearly with equal increments in the exposure variable. • Changes in the rate from combined effects of different exposures or risk factors are multiplicative. • At each level of the covariates the number of cases have variance equal to the mean. • Observations are independent.

  3. Example of Poisson Regression Births by caesarean section are said to be more frequent in private (fee paying) hospitals than in public hospitals. To test this hypothesis data about total number of births and the number of caesarean sections performed were collected from the records of 4 private hospitals and 16 public hospitals. The analysis is shown in the following slides

  4. Analysis in Poisson Regression Link function: Log Fitted terms: Constant, BIRTHS mean deviance d.f. deviance deviance ratio • Regression 1 63.575488916 63.575488916 63.58 • Residual 18 36.414789139 2.023043841 • Total 19 99.990278055 5.262646213 Estimates of regression coefficients estimate s.e. t(*) Constant 2.132 0.102 20.95 BIRTHS 0.0004405 0.0000540 8.17 Caesareans = ( e ) 2.132 x ( e )0.00044

  5. Analysis - 2 Link function: Log • Fitted terms: Constant, BIRTHS, HOSP_TYP mean deviance d.f. deviance deviance ratio • Regression 2 81.951077606 40.975538803 40.98 • Residual 17 18.039200449 1.061129438 • Total 19 99.990278055 5.262646213 Estimates of regression coefficients estimate s.e. t(*) Constant 1.351 0.249 5.43 BIRTHS 0.0003261 0.0000603 5.41 HOSP_TYP 1.045 0.272 3.84 Caesarean = ( e )1.351 x ( e )0.00032 x ( e )1.045 when HOSP-TYP =1

  6. Analysis - 3 Checking the fit of the model by plotting deviance against fitted values

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