# Why use Poisson regression for p-values for linear regression?

Byars et al.'s paper "Natural selection in a contemporary human population" includes a multiple linear regression of number of children (LRS, lifetime reproductive success) on several variables (mother's total cholesterol, age of of mother at first birth, etc.). Most of the independent variables are continuous (see e.g. table 4 on p. 1790), but LRS is obviously a count variable.

Although the standardized regression coefficients ("selection gradients") reported are from a linear regression, Byars et al. report p-values from a multiple Poisson regression (see e.g. p. 1791 in "Statistical analysis" in the Methods section: "Multiple linear regression was used to infer the strength and direction of selection, and multiple Poisson regression was used to test statistical significance.")

I am wondering why the authors use Poisson regression for calculating the p-value, even though the regression coefficients are from a linear regression. (I'm not sure whether I've given enough information for an answer--or even an educated guess.)

(My present guess is that the answer is something like this: EDIT: If the LRS data was the result of a simple random sampling process from a larger population, it could be assumed to be normally distributed. However, the LRS data are being treated here as values of a conditional r.v. that is the same for each woman in the study, except that they have different conditioning characters. Since the LRS is a count variable in which increments happen at time intervals (>= 9 months), it's more or less reasonable to take this r.v. to be distributed according to a Poisson distribution. Therefore calculating the p-value needs the Poisson regression. I'm not sure whether that makes any sense. Perhaps also, a linear regression makes sense given the selection gradients' role in subsequent calculations. I am just trying to work my way into this part of the paper--I don't have a deep understanding of multiple linear regression, and I only know about Poisson regression from Wikipedia.)

References to relevant books or papers are welcome. I've been looking for sources--whether from biological or statistical literature--that would explain what Byars et al. are doing with the two regressions.

• I makes no sense to me, the p-value dependes on the parameter and the estimated error - these will not be the same across the two models, and therefore you cannot use p-values from one model to infer anything about the other model. At a surface glance this method appears strange, hope someone can answser this Jun 19 '18 at 5:58
• @Repmat, thanks. That confirms my intuition: It sounded strange to me too--but I don't know much. By default I'm inclined to trust that the authors and the PNAS reviewers knew what they were doing. (The corresponding author is well-known and respected in the field, and I know scientists I respect who read it and didn't comment at the time on the oddity of this relationship.)
– Mars
Jun 19 '18 at 19:18