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 continuous 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.

 A: This is much more complicated than you might think. The underlying drug dose-effect probably is sigmoidal in its rising phase. I say "probably is" because that is almost always what pharmacologists find when they are able to clearly document a dose-response curve.
All drugs are toxic at some large enough dose, so the top of the curve might well fall downwards after the dose gets large. However, note that many dose-reponse curves are only explored up to the peak and so the downwards going portions are not documented! It's important to know that the biological mechanism of the upwards going portion of the curve is often different from that of the downwards going portion because the toxic effects of most therapeutic drugs is different from the beneficial effect.
In some trials the protocol allows the dose of the drug to be adjusted by the practitioner in response to how the patient is doing, and that often means that the sickest patients —those who die most frequently— get the highest doses. It is easy to incorrectly ascribe excess deaths to the drug when they are really due to the disease severity.
Given that your data are from an RCT the participants who died might have died because of too much drug or too little drug, or because of progression of their underlying disease independently of the dose of drug given. The huge range of doses used in the study (something like 200-fold) certainly allows the possibility that some participants died as a result of getting too much drug, but I do not see much trace of that in the data.
(Are you sure that the doses are correctly labelled as "log(dose)". Often concentrations are given as -logarithms, and that would make the high dose end of the data at the left rather than the right.)
