How to compute confidence interval for number to treat in a logistic regression?

Question

I have built a multivariate regression model where $\mathbf x$ is the variable vector. Now I am asked to calculate the confidence interval for the number to treat (NNT) at $\mathbf x=\mathbf {x_0}$ which in turn requires to build the confidence interval for $p=Pr(1|\mathbf x=\mathbf {x_0})$. I'm using JMP which, to my knowledge, shows only point value and standard error of the parameter estimates but not the complete Fisher information. So I'm temped to use $\hat{p}=Pr(1|\mathbf x=\mathbf {x_0},\mathbf {\hat{b}})$ where $\mathbf {\hat{b}}$ is the vector of the ML parameter estimates, and then calculate $\hat{p}(1-\hat{p})/n$, but I doubt it is correct. Which is the simple way to do?

Answer 1

NNT, which has been roundly criticized of late, has to be conditioned on a single covariate setting, as you rightly framed the problem. NNT is the reciprocal of the risk difference, and will vary wildly with other covariate settings (sick patients get more benefit). You can get the confidence interval for a risk difference then take the reciprocal of each confidence limit to get the (surprisingly wide) CL for NNT. I don't have a good reference for the CL of the risk difference but you should be able to find it. It will be a function of all the data along with $x_{0}$. You can also get it with the bootstrap.