# In a GLM, are the Maximum likelihood estimators for the regression coefficients always normally distributed?

I'm doing a Poisson regression and using the confint function in R to generate confidence intervals for my regression coefficient. These result in different intervals than when I would calculate them manually using the normal approximation, i.e. the lower bound given by $$\hat{\beta} - 1.96*\hat{\sigma}_\beta/\sqrt(n)$$ and the upper bound by $$\hat{\beta} + 1.96*\hat{\sigma}_\beta/\sqrt(n)$$.

If the beta's were normally distributed, I would expect that using the confint function or the manual calculation has to yield the same result.

• What link function are you using for your Poisson regression? If you are using the log link, then your confidence interval construction has to reflect that, since you are modeling the log expecteded value of your count response variable as a function of the predictors. May 15 '18 at 1:15
• I am using the log link function indeed
– WBM
May 15 '18 at 9:17
• May 16 '18 at 16:57
• Generally, im glm's you cannot expect more than asymptotic normality of the betas. There are often better confidence intervals available than the approximate normal based (on asymptotic theory), like those based on profile likelihood. See stats.stackexchange.com/questions/9833/… or stats.stackexchange.com/questions/260885/… Jun 29 '18 at 10:33

The glm in R doesn't have a $$\sigma$$ or $$\hat{\sigma}$$ for Poisson regression. So one must wonder what $$\sigma$$ you are using to manually calculate these values. If you are calculating this value as one would do in a linear regression model, by calculating the unstandardized residual using the predicted and observed responses, this would not be correct due to the heteroscedasticity of the residual.
If you are using the vcov function and using the "bottom right variance" (of the regression parameter) to calculate these CI bounds, this will give the same result as the confint function in R.
The confint function you used calculates the profile likelihood confidence interval. That will coincide with the symmetric confidence interval you expected if the log likelihood function is quadratic. For many glm's that will not be the case, and sometimes the difference can be dramatic.