I'm looking at data where all of the measurements are only available with quarterly time points, e.g. 2005-Q1, 2005-Q2, ..., 2016-Q4, 2017-Q1. This means that the event-of-interest (e.g. death) falls into some quarter, but so does the event of inclusion in the study (participant starts treatment), as well as the measurements of some possible confounding variables (if they're not constant like gender).
I know that quarterly data should be treated as interval-censored, meaning I can either impute them to make them right-censored using midpoints or left or right time points, but this introduces some bias to the estimation. Alternatively I could use a parametric model (e.g. Weibull).
But I was thinking, since the inclusion of participants is also only known to happen at some time point within a quarter, does it really make sense to go through the hassle of interval-censored survival analysis?
I have to treatment groups that I want to compare, and a reasonable assumption is that at least the time points of inclusion within a quarter should be identically distributed for the two groups. If the survival time of one group is just 1 or 2 months better than the other group, I could not infer it with a standard right-censored model on quarterly data, but is there any way that interval-censored analysis could change that?
Furthermore, is there a "test" of some kind to assess if modelling with interval-censoring is preferable?