I have been running some binomial logistic regressions in R on a data set and I realised that the p-values of the estimated coefficients are not computed based upon a Normal distribution. For e.g. I have the following result from the glm() function:
Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -1.6127 0.1124 -14.347 < 2e-16 *** relevel(Sex, "M")F -0.3126 0.1410 -2.216 0.0267 * relevel(Feed, "Bottle")Both -0.1725 0.2056 -0.839 0.4013 relevel(Feed, "Bottle")Breast -0.6693 0.1530 -4.374 1.22e-05 ***
I previously thought that the beta hats (estimated coefficients) are asymptotically normal and the p-value should be calculated using the normal distribution. But it seems from the p-values here that a t-distribution is used to compute them (I think).
I know that the asymptotic condition does not hold when sample size is small but if so, what is the process that causes the estimator to be distributed with a Students distribution? And how do I find out the degrees of freedom for such a distribution?