# 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, 2018 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, 2018 at 19:18
• PNAS is such a strange venue, I can never tell what's going on there. For instance, the peer review track for members of the national academy is explicitly different than it would be for us peons (see here: pnas.org/author-center/member-contributed-submissions). Jul 20 at 13:41

It makes no sense to me, the p-value depends 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 answer this. – Repmat

I agree that this do not make sense. The usual linear regression (identity link function) and Poisson regression (a multiplicative model with log link function) are very different, and have parameters with different interpretations. See my answer here Goodness of fit and which model to choose linear regression or Poisson for a comparison.

If the authors wanted the linear model, with its parameter interpretation, they could have used Poisson regression with identity link function. See OLS vs. Poisson GLM with identity link

• To combine the two regressions makes indeed not much sense. If one likes to use Poisson regression to express significance, then one can just as well use Poisson regression to compute the coefficients... Jul 24 at 21:02
• ... But, reasons for combining these p-values from Poisson regression with least squares regression can be other than logical application of statistics. Possibly a mixed approach is used because a reviewer asked to use a Poisson model or because authors had different viewpoints. This may not be an unlikely explanation. It is interesting to note that the supplemental material contains a table that includes both p-values based on a t-value (the least squares regression) and p-values based on a z-value (the Poisson regression). Jul 24 at 21:02

The justification for this is biological, not statistical. A selection gradient is defined as the partial derivative of relative fitness with respect to a trait assessed at the mean, and can be approximated by the OLS regression coefficient regardless of whether or not the residuals of the fitness measurement are gaussian (see https://www.jstor.org/stable/2408842). It is an estimate of natural selection that can be used to quantitatively predict evolutionary response (if estimates of genetic variance are available). Of course, the SEs and corresponding P values are going to be unreliable if the fitness measure (response variable) is non-gaussian, and it is a relatively common practice to re-fit a more (statistically) appropriate model to assess statistical significance. Thus, the authors report these estimates because they are a biologically meaningful parameter, comparable to the same parameter measured in other studies (e.g., the strength of selection in other populations)

• The "biological" justification does not make sense. This presents results that are non-comparable as if they regarded the same thing, it's simply misleading. If you want to "re-fit a more (statistically) appropriate model" report the model. If you want to make a conclusion using linear regression report it. If you want to use both models, report both. It's like in a human anatomy textbook you've shown a diagram of a human doby, but described a fish, because "it's simpler to describe".
– Tim
Jul 20 at 13:44
• Welcome to cv, and thank you for taking time to answer! I understand that you describe the reason for fitting two models (Gaussian and Poisson) as common practice, and this is exactly answering the question :"why are they doing it" :-) I don't understand @Tim's comment as critique of your answer, but rather as a critique of that approach.
– Ute
Jul 20 at 15:23