# How does GLM coefficient standard error transfer to prediction variance and relative risk variance?

How does a coefficient's standard error in a GLM transfer to what you do with that coefficient?

1. For example, for a logistic regression model, $$e^{\beta_1}$$ is the odds ratio. If the standard error on $$\beta_1$$ is 2, what is it for the odds ratio? If you calculate the confidence interval for $$\beta_1$$, can you just exponentiate it to get the confidence interval for the odds ratio?

2. Another example. How could you calculate the variance for the estimated mean given certain covariates in logistic regression, $$e^{\beta_0+\beta_1x_1+\beta_2x_2}$$? I think this is obtained via the variance function $$\frac{\pi_i(1-\pi_i)}{n_i}$$?

• If you're looking specifically for odds ratio from logistic regression, this question is most relevant. stats.stackexchange.com/questions/163824/… To get a confidence interval the main approaches are to use the Delta Method, Bootstrap, or a transformation of the confidence interval for Beta. All of those generalize to the multivariate case naturally. Searching the site for those topics should get you well on your way May 7 at 22:40