# Is it possible to get a prediction interval for logistic regression via a latent variable?

carbocation asked how to compute prediction intervals for logistic regression. The answer was that prediction intervals don't make sense for logistic regression because the response variable only takes two values, so the prediction interval would always be the same.

Carl posed a follow-up question, modifying it to "is there ... a sensible way to compute a prediction interval in log-odds space?", that is, a prediction interval on the linear predictor - or some monotone transform of it. (He wanted to transform the prediction interval from the log-odds scale of the linear predictor to a probability scale.)

The answers appeared to me to fall into two camps:

1. It's still a binary outcome: These basically said that the prediction interval is, by definition, about the distribution of outcome values and as there are only two outcome values the concept of prediction interval either doesn't apply or if it does is essentially useless.
2. Confidence interval: These basically suggested that what Carl wanted was equivalent to the confidence interval on the linear predictor. I don't think this can be right because if you take the confidence interval of a linear regression as an analogue it is about the distribution of the mean value conditional on the predictors, whereas Carl's question is about the distribution of values of individual cases.

I have a suggestion that might be a step towards answering these questions.

[I am asking this as a question because I don't know whether what I am suggesting works (and therefore, whether it constitutes an answer) and I don't have enough reputation points to post this as a comment to the previous questions I have cited. Also, my final points are questions.]

I wonder if Carl's question can be approached as follows:

• Assume the existence of a latent variable that is the true log-odds for each case.
• Make the response for each case be a random draw from the set of binary outcomes with probability determined by the true log-odds.

It is trivially easy to generate simulated data consistent with this conceptual model. If there are predictors of the true log-odds and the true log-odds is observable in the simulated data, then a regression model can be built predicting the true log-odds from the predictors and it is possible to compute prediction intervals for the true log-odds, which will be different from the confidence interval. These prediction intervals appears to be what Carl wants.

However, if the true log-odds are not observable and only the random binary outcome is observable:

1. Is it possible to estimate a model that treats the true log-odds as a latent variable?
2. If so, how does this differ from logistic regression?
3. Is there a standard name for a model of this form?
4. Does binary outcome quantile regression do what we want here? (e.g. http://econpapers.repec.org/article/wlyjapmet/v_3a27_3ay_3a2012_3ai_3a7_3ap_3a1174-1188.htm, https://www.researchgate.net/publication/229676506_Binary_quantile_regression_a_Bayesian_approach_based_on_the_asymmetric_Laplace_distribution)