Prediction intervals for the outcome of a logistic regression with binomial response

Question

Suppose we have a logistic regression model:

$$\begin{align} P(y=1\vert\mathbf{x}) &= p \\ \log\left(\frac{p}{1-p}\right) &= \boldsymbol{\beta}\mathbf{x} \end{align}$$

Given a random sample $D=\{\mathbf{X},\mathbf{y}\}$ of size $N$, we can compute confidence intervals for the $\boldsymbol{\beta}$ and correspondingly prediction intervals for $p$, given a certain value $\mathbf{x}^*$ of the predictor vector. This is all very standard and detailed, for example, here.

Suppose instead that I'm interested in a prediction interval for $y$, given $\mathbf{x}^*$. Of course, it doesn't make any sense at all to compute a prediction interval for a single realization of $y$, because $y$ can only take the values 0 and 1, and no value in between. However, if we consider $m$ realizations of $y$ for the same fixed value of $\mathbf{x}^*$ , then this becomes similar (but not identical) to the question of computing a prediction interval for a binomial random variable. This is basically the same situation described by Glen_b in the comments to this answer. Does this question have an answer, apart from the trivial one "use nonparametric bootstrap"?

Answer 1

One way this should work without bootstrapping (which in practice may be the fastest thing tho implement), would be:

1. Assume that a normal approximation for the predicted log-odds ($x \hat{\beta}$) plus/minus its standard error works. Any logistic regression software will provide this.
2. The percentiles of this distribution transform to probabilities via the anti-logit.
3. One can find a (mixture of) beta distribution(s) that approximates the predictive distribution for the probability well.
4. The predictive distribution for the outcome is then a (mixture of) beta-binomial distribution(s with the same mixing weights as used in step 3).

Alternatively, one can "just" integrate out the log-odds from the joint predictive of outcome and log-odds, but I believe that will be a complete mess with no closed form solution.