Is is acceptable to use cox proportional hazard regression when time-to-event is a discrete, numeric variable? I recently submitted a paper where I performed a cox proportional hazards regression model modelling the effect of group allocation in a randomised controlled trial on treatment retention. The event was dropping out of the study (non-reversible binary event) and the time to event variable was the week that the participant dropped out, which was a numeric variable with possible values 1-12. These values were discrete, i.e. no fractions of weeks.
The reviewer of my paper has stated that the outcome we used was "not continuous, so they were not an appropriate choice for time to event analysis".
Is the reviewer correct? The answer to this post indeed does suggest the the Cox PH model is not appropriate for discrete data and references Singer and Willett's Applied Longitudinal Data Analysis: Modeling Change and Event Occurence. Now I have read the book and while they do say that if the time-to-event variable is continuous you cannot use discrete-time survival analysis, I cannot find anywhere where they state that the reverse is true: that you cannot use a cox proportional hazards model for a discrete, numeric time-to-event variable. Furthermore the biostatistician we consulted advised us to use a cox regression model.
If the Cox Model is not appropriate for continuous time-to-event data can anyone tell me why?

With only 12 possible event times, the discrete-time approaches are certainly better. You necessarily have many tied event times, which require special handling in Cox models. Therneau and Grambsch discuss this in Section 3.3, the source of the quotes that follow.

The partial likelihood for the Cox model is developed under the assumption of continuous data, but real data sets often contain tied event times.

One can calculate an "exact partial likelihood," but

this method involves an exhaustive enumeration of the possible risk sets at each tied death time, and can require a prohibitive amount of computation time...

Unless you asked for that exact partial likelihood you probably didn't get it.
There are approximations for dealing with ties; the Efron approximation is more accurate than the Breslow approximation. You should check which method was used in your models.
The Efron approximation is

quite accurate unless the proportion of ties and/or the number of tied events relative to the size of the risk set is extremely large. (Emphasis added.)

I suspect that your study is covered by that "unless..." disclaimer, in which case the reviewer is correct that a standard Cox model was not an appropriate choice.
As another answer suggests, interval censoring is one way to deal with such data. For Cox models that requires specialized methods like those provided by the icenReg package. See that package documentation for a helpful introduction to the issues.
With a small number of event times like yours, discrete-time methods are most appropriate and easy to implement. Binary regression with a complementary log-log link is most directly related to continuous-time Cox models; see this page for a brief introduction and links to more extensive explanation.
