# Prediction interval on X with Poisson

What I'm trying to do is predict some biodata based on time. Because the biodata are not normally distributed, I have modeled them with a poisson using a log link function ( E[y] = exp[BX]). What I'm trying to do is estimate the time at which the predicted curve reaches a certain threshold. (i.e., y=1 when x=?). Obtaining a point estimate is fairly straightforward, but I would like to place a confidence interval around that time (e.g., x = 5+-1).

My initial idea was to flip the x and the y and generate prediction intervals as one normally would. However, x is normally distributed while y is poisson, so that's not a good solution. Any other ideas?

• what do you mean by 'the predicted line' ... what line is that? did you fit some kind of model? can you be more specific? – Glen_b Dec 18 '13 at 1:59
• the fitted line. I find the best-fitting line that relates x to y. The fitted line will have the form y=exp(b0 +b1x). Does that help? – dfife Dec 18 '13 at 13:50
• To answer my own question--one option is to bootstrap. – dfife Dec 18 '13 at 13:52
• Hang on -- you're fitting a GLM with log link? I really don't know how a reader is supposed to have known you meant a Poisson GLM with $E(y)=\exp(b_0 +b_1x)$ from "the predicted line". Such a relationship is more usually described as a curve than a line (unless you meant a prediction interval for the linear predictor, $\eta=\log\mu$, but that's a CI not a PI). Could you add the details about your model to your question, please? – Glen_b Dec 18 '13 at 18:38
• When you say that $x$ is normal: the usual assumptions (in regression and GLMs) is that the x-variables are observed without error. If $x$ is, as it sounds, a random variable, then I worry that it might not be the case that x is observed without error (in which case you have all the issues of errors-in-variables). If that's not an issue, then I think you're in the Poisson-regression equivalent of an inverse regression problem. – Glen_b Dec 18 '13 at 23:15