# Dose-response curve in logistic regression: transformation of dose variable and what to do about non-monotonic relationships?

I'm attempting to develop a logistic regression model to predict a binary outcome (namely death) for an RCT cohort. One of my predictor variables is the empirical dose of the given treatment, which is a continuous log-normally distributed variable (so no fixed dose groups).

It may be assumed a priori is that there will be a type of sigmoidal dose-response curve (if I would have had a continous outcome). The logit link function is of course already sigmoidal, but my understanding of the mathematics involved is too limited to see if that's appropriate then. From what I understand logistic regression doesn't necessarily rely on a normally distributed predictor, so should I even log transform my dose to be normally distributed?

Secondly, I have some evidence from other studies that there may be dose-optimum, i.e. that the dose -response curve is not monotonic but rather bell-shaped. If that should be the case within the dose-range administered in my study, how do I find out? I've tried plotting a simple model with just the dose variable and it doesn't seem bell-shaped, but that hardly sounds like a robust approach. Besides my final model includes additional variables and that makes diagnostics by plot more confusing.

Thanks for your time :-) • Welcome to Cross Validated! Do the answers here, here, or here help? – Scortchi Jun 19 '15 at 13:48
• The usual--and most effective, convincing, and generalizable--way to start off analyzing data in any dose-response study is with some kind of biological model of the response mechanism. This is especially important for complex responses, which might not be easily linearizable or even display u-shaped behaviors such as hormesis. As an example of how a model can guide the analysis, I would direct to you to yet a fourth post at stats.stackexchange.com/a/64039. – whuber Jun 19 '15 at 14:08
• Hi Scortchi and Whuber, thanks for your answers! I've been toying around with the data since and have made some progress in solving it. I've first fit a (restriced cubic) spline as a sort of general exploratory analysis and found that the relationship is indeed monotonic (you would kinda expect if you look at the events vs dose relation above). Later I found that there is indeed an implementation for dose response curves in the drc package in R, so I'll try that next. Advantage is that the biological interpretation of the curves is a lot easier for a beginner like me. – Hgremmels Jun 23 '15 at 10:49