Mathematical connection between hazard functions for e.g. Cox models and expected values in GLMs I intuitively understand why the hazard function is interesting to link to patient attributes, as explained in this answer. However, is there a mathematical reason this is done? I'm particularly familiar with generalized linear models, where the expected value is modeled as being link linear. Is there a connection between the expected value in exponential dispersion families and the hazard function? perhaps through the log partition function somehow?

Just to clarify, can we rephrase the question as follows: "Why is the conditional hazard used as the response in the Cox model?" In which case, I would say that the answer is that this is an assumption about the data: that the impact of the predictors is to act multiplicatively on the hazard function. In other words, the hazard ratio is constant across all values of the predictors.
As an example, say we were analyzing survival time T, and included group status (X) as a predictor. Then we make the assumption that there is a baseline hazard function that describes how the hazard changes with time, T. The impact of the predictor X is to scale up or down the entire hazard
function. It could look something like this:

Here are some details from Computer Age Statistical Inference, p144

In summary, in most GLMs, we start with the assumption that the impact of predictors is to act additively on the conditional mean (or on a value that is connected to the mean via a link function). However, in the Cox model, we start with the assumption that the impact of the predictors is to act multiplicatively on the baseline hazard.
As a side note, the Cox model is popular because it allows us to estimate hazard ratios even though we never actually specify the form of the baseline hazard function.
