I have been told that one of the main advantages of Semi-Parametric Models (e.g. Survival Analysis, Cox Proportional Hazards Model) is that these models do not require assuming the response variable (e.g. Survival Times) to follow a specific probability distribution, thus allowing for a higher level of flexibility.
This got me to thinking - suppose if the Survival Times are discrete. As an example, suppose that we only have available to us how many years each patient remained in the study.
Initially, I had thought that if we decided to choose a Semi-Parametric Survival Model (e.g. Cox PH), the presence of discrete survival times should not be a problem. As an example, perhaps the true underlying distribution of survival times might follow some discrete probability distribution such as the Poisson Distribution - but since a Semi-Parametric approach (according to my likely flawed understanding) does not require you to specify a specific probability distribution , in theory it should not matter if the true underlying distribution is continuous or discrete.
However, when reading more about this topic (e.g. https://grodri.github.io/glms/notes/c7s6, https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-022-01679-6), it appears that there have been different models specifically developed for this task (i.e. discrete survival times).
This slightly confuses me - if one of the main advantages of Semi-Parametric Models is their ability to avoid using a specific probability distribution, how come you can't just use a common Semi-Parametric Model such as Cox-Ph when faced with a discrete probability distribution?
The only thing which comes to mind are "philosophical reasons". Perhaps the Cox-PH model was specifically designed for continuous survival times - that is, the advantages of the Cox PH model come into effect only provided that the true probability distribution of survival times are continuous. Perhaps the Cox PH model was fundamentally not intended to be used with discrete data. Thus, in the case of discrete data, you would want to choose a probability distribution that is defined in a "discrete sense" - that is, the random variable can not have negative values nor non-integer values. Although when faced with discrete data, you might be able to "trick" the computer into using a continuous distribution and proceed with a Cox-PH model, perhaps some problems relating to inference and interpretation may later arise (e.g. negative hazards and survival probabilities .... but I am not entirely sure about this).
Can someone please comment on this?