I'm trying to run a Response Surface Analysis in SAS, but this is only possible with a continuous outcome, whereas my outcome variable is binary. I got the advice to first run a logistic regression and save the predicted probabilities. I could then run the Response Surface Analysis with the predicted probabilities as the continuous outcome variable.

However, the significance test of the Response Surface Analysis do not make any sense and my R² is extremely high. Someone else told me that I should plot a ROC curve and look at the AUC, but I have the feeling that there is more going on then simply adjusting my threshold. Unfortunately I can't seem to find any information on this subject online. Does somebody have any idea how to properly use the predicted probabilities of a logistic regression as the outcome variable of a linear regression?

Best wishes, Alexander


Imagine you run a linear regression, take the predicted values of the dependent variable, and run a linear regression of the predicted values on the same explanatory variables you used to generate the predictions. Your $R^2$ will obviously be 1. If you use a non-linear regression like logit to generate the predicted values and then try to fit them with a linear regression, it won't quite be 1, but it should still be extremely high. The only source of error is your linear approximation.

There is no "proper" way to do this -- any standard errors or analysis of model fit you generate will be plain wrong. You may still find it worthwhile to run the response surface analysis on the predicted probabilities if your goal is something like finding the surface maximum, but the significance test will not be meaningful.

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