Poisson_Reg_example

Poisson_Reg_example - 136 Poisson Regression Analysis 13 Poisson Regression Analysis We have so far considered situations where the outcome

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Poisson Regression Analysis 136 13. Poisson Regression Analysis We have so far considered situations where the outcome variable is numeric and Normally distributed, or binary. In clinical work one often encounters situations where the outcome variable is numeric, but in the form of counts. Often it is a count of rare events such as the number of new cases of lung cancer occurring in a population over a certain period of time. The aim of regression analysis in such instances is to model the dependent variable Y as the estimate of outcome using some or all of the explanatory variables (in mathematical terminology estimating the outcome as a function of some explanatory variables. When the response variable had a Normal distribution we found that its mean could be linked to a set of explanatory variables using a linear function like Y = β 0 + β 1 X 1 + β 2 X 2 …….+ β k X k. In the case of binary regression the fact that probability lies between 0-1 imposes a constraint. The normality assumption of multiple linear regression is lost, and so also is the assumption of constant variance. Without these assumptions the F and t tests have no basis. The solution was to use the logistic transformation of the probability p or logit p, such that log e ( p /1− p ) = β 0 + β 1 Χ 1 + β 2 Χ 2 ……. β n Χ n. The β coefficients could now be interpreted as increasing or decreasing the log odds of an event, and exp β (the odds multiplier) could be used as the odds ratio for a unit increase or decrease in the explanatory variable. In survival analysis we used the natural logarithm of the hazard ratio, that is log e h(t)/h 0 (t) = β 0 + β 1 X 1 + …. .+ β n X n When the response variable is in the form of a count we face a yet different constraint. Counts are all positive integers and for rare events the Poisson distribution (rather than the Normal) is more appropriate since the Poisson mean > 0. So the logarithm of the response variable is linked to a linear function of explanatory variables such that log e (Y) = β 0 + β 1 Χ 1 + β 2 Χ 2 etc. and so Y = ( e β0 ) ( e β1Χ1 ) ( e β 2 Χ 2 ) . . etc. In other words, the typical Poisson regression model expresses the log outcome rate as a linear function of a set of predictors. Assumptions in Poisson Regression The assumptions include: 1. Logarithm of the disease rate changes linearly with equal increment increases in the exposure variable. 2. Changes in the rate from combined effects of different exposures or risk factors are multiplicative. 3. At each level of the covariates the number of cases has variance equal to the mean. 4. Observations are independent. Methods to identify violations of assumption (3) i.e. to determine whether variances are too large or too small include plots of residuals versus the mean at different levels of the predictor variable. Recall that in the case of normal linear regression, diagnostics of the model used plots of residuals against fits (fitted values). This means that the same diagnostics can be used in the case of Poisson Regression.
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This note was uploaded on 02/15/2012 for the course GEO 6938 taught by Professor Staff during the Summer '08 term at University of Florida.

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Poisson_Reg_example - 136 Poisson Regression Analysis 13 Poisson Regression Analysis We have so far considered situations where the outcome

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