# How to deal with noisy observation in Survival Analysis

I'm new to Survival Analysis. Usually in survival analysis, we want to model the survival function progress w.r.t time. This is normally done through Cox model, or KM-model within a specific time horizon

However, this assume that we know exactly when the hazard event occurred. If we want to model a system which we can only observed the state periodically, but not continuously. Then modeled using traditional cox model, this model will be biased as we don't know the exact time when the hazard event occurred

For example, we want to model a survival function on a system with multiple agents which can be either alive, or dead, but we can only observed at a specific time interval. Then we know the amount of alive, and dead agents at time step t-1, and t. But we can't model how the survival function had progress between that interval

So in general, how does this kind of observational noise be deal with such that the model is not biased?

We never have access to the exact moment an event occurred: even if we record time to event in days, we do not discern between time of day and therefore the time intervals are days. However, with contextual reasoning, you might say the difference of less than a day is negligible for a specific analysis (although it might not be: for example hours of survival in an intensive care unit). It is all about what time unit to you is acceptable.

If you believe that the interval you measured is sufficient to capture differences, then there might not be a problem. If you believe it is not sufficient, then your data is lacking and a model might not solve that.

I am not aware of any way to model specific occurrence of the events within your time intervals, but if any such methods exist, they are likely assumption-heavy and unlikely to give reliable results without meeting these assumptions.

• So we have to accept that the survival model only temporal accurate up to our fixed time interval? We can assume how the underlying function works, but it heavily depends on our assumption, and did not generalize. That is a bit sad, but I guess it is a sensible answer :-/ May 9 at 11:29

... we know the amount of alive, and dead agents at time step t-1, and t. But we can't model how the survival function had progress between that interval

What you describe is a discrete-time survival model, for which there are well defined methods of analysis. Obviously the details of the survival function within each time period can't be determined, but there is no need to resign yourself to "biased" results that depend "heavily" on assumptions if your primary interest is in the associations between predictor variables and outcome.

Such survival models can be handled as binomial regressions with "long-form" "person-period" data. For each time period there is a separate row of data for each at-risk individual, specifying the time period, the individual's covariate values in place for that period, and whether or not the individual experienced the event during that period. An individual no longer at risk after a last observation time has no data rows after the last observation time, which handles right-censoring in a very straightforward way. Search this site for pages discussing discrete-time survival; this page is one of many, with some links to further reading.

In particular, if you use a complementary log-log link in the binomial regression, the results are those of a "grouped proportional hazards model." You get regression coefficients for associations of covariates with outcome just as you do for a Cox proportional hazards regression in continuous time (which, as another answer notes, is seldom truly "continuous" in practice, anyway), under the same proportional hazards assumption. See this page for an outline of the approach, with a link to a detailed text on discrete-time survival models.