# 12.3 - Log-binomial Regression

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If modeling a risk ratio instead of an odds ratio and the risk ratio is not well-estimated by the odds-ratio (recall in rare diseases, the OR estimates the RR), SAS PROC GENMOD can be used for estimation and inference. (Skinner, Li, Hertzmark and Speigelman, 2012)  PROC GENMOD can also be used for Poisson regression.

#### Poisson regression

If the outcome variable follows a Poisson distribution, then Poisson regression is useful.  For example: counts of  relatively rare events, e.g., the number of cancer cases over a defined period in a cohort of subjects.

Discrete counts represent information collected over time (days, years), in space (volume for bacteria counts) or for a population. Poisson regression allows modeling a rate, e.g., incidence rates of cancer,  as a function of some covariates.

Example of a Poisson Regression Model

Assume that the number of cancer cases has a Poisson probability distribution and that its mean, μi, is related to the factors race and sex for observation i by:

\begin{align} ln(\mu_{i})& =ln(N_{i})+x_{i}\beta \\ & = ln(N_{i})+\beta_{0}+ race_{i}(1)\beta_{1}+ race_{i}(2)\beta_{2}+ race_{i}(3)\beta_{3}+ sex_{i}(1)\beta_{4}+ sex_{i}(2)\beta_{5}\\ \end{align}

where racei(j) = 1 if race = j, and 0 if racej

The logarithm of  N is used as an offset, that is, a regression variable with a constant coefficient of 1 for each observation.

The Poisson regression model assumes:
1. The logarithm of the cancer rate changes linearly with equal increment increases in the exposure variables, the race and sex indicators.
2. Changes in the cancer rate from combined effects of race and sex (exposure 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.

The Research Methods II monographs from the Journal of Tropical Pediatrics provides a worked example of   Poisson regression analysis.