# Survival analysis: continuous vs discrete time

I am confused about how to decide whether to treat time as continuous or discrete in survival analysis. Specifically, I want to use survival analysis to identify child- and household-level variables that have the largest discrepancy in their impact on boys' versus girls' survival (up to age 5). I have a dataset of child ages (in months) along with an indicator for whether the child is alive, the age at death (in months), and other child- and household-level variables.

Since time is recorded in months and all children are under age 5, there are many tied survival times (often at half-year intervals: 0mos, 6mos, 12mos, etc). Based on what I have read about survival analysis, having many tied survival times makes me think I should be treating time as discrete. However, I have read several other studies where survival time is in, for example, person-years (and so surely there are tied survival times) and continuous-time methods like Cox proportional hazards are used.

What are the criteria I should use to decide whether to treat time as continuous or discrete? For my data and question, using some continuous-time model (Cox, Weibull, etc) makes intuitive sense to me, but the discrete nature of my data and the amount of tied survival times seem to suggest otherwise.

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The choice of the survival model should be guided by the underlying phenomenon. In this case it appears to be continuous, even if the data is collected in a somewhat discrete manner. A resolution of one month would be just fine over a 5-year period. However, the large number of ties at 6 and 12 months makes one wonder wether you really have a 1-month precision (the ties at 0 are expected - that's a special value where relatively lot of deaths actually happen). I am not quite sure what you can do about that as this most likely reflects after-the-fact rounding rather than interval censoring.

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I suspect if you use continuous time models you will want to use interval censoring, reflecting the fact that you don't know the exact time of failure, just an interval in which the failure ocurred. If you fit parametric regression models with interval censoring using maximum likelihhod the tied survival times is not an issue IIRC.

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There will be tied survival times in most analysis, but big, clear chunks of ties at particular events is troubling. I would think long and hard about the study itself, how its collecting data, etc.

Because, outside of some methodological needs to use one type of time or the other, how you model survival should depend on whether or not the underlying process is discrete or continuous in the world.

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If you have covariates that vary over time for the some individuals (e.g. family income may vary in your example over the lifetime of a child), survival models (parametric and the cox model) require you to slice up the data into discrete intervals defined by the varying covariates.

I found this pdf of lecture notes by German Rodriguez helpful.

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