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.