# Time varying covariates and Interpolation issue

Based on my reading on time-varying survival analysis, I am encountering two different and conflicting sets of advice with regards to time-varying covariates and interpolation.

1. The first advice is to avoid basing covariates on future events, which may introduce bias. As example, suppose a subject has two lab measurements 25 at time 0 and 50 at time 2; using counting process notation, the subject would be entered as two time intervals A and B: A. (0,2] 25 died = 0, and B. (2,5] 50 died = 1. Under one interpolation of the values the subject would have 37.5 for A. Based on the above advice, bias (perhaps small?) may be introduced as the value 37.5 is based on a future event.
2. The second advice is to go ahead and interpolate, and there are some creative methods such as joint mixed models which seem to do this.

Which advice to take? If it depends, on what situations would it be appropriate to prefer one of the other?

• This question has no correct answer. You have two lab measurements at the two ends of a time period. You want to derive the lab measurement in the middle of the interval. There are infinite number of methods to derived it. So if you select A method, I can say you are wrong and you should select B. – user158565 Aug 12 '19 at 19:30
• Can you assume that the time varying process is varying slowly? That is not an unreasonable assumption, and it is the first thing a physicist would check. Essentially nothing can be said about a rapidly varying process with only 2 time points. So, under the condition that the process varies slowly, the two points (with naive priors) center expectations, so the interpolation is equivalently centered. Many conditions for a weak conclusion. But, if that’s all you got, that’s all you can say! – Peter Leopold Aug 13 '19 at 13:16
• @user158565 I think the second part of this question does have a right answer. The first step is clearly outlining the assumptions that either approach takes. The key in any particular problem is assessing whether or not those assumptions are reasonable. This assumption will likely require expert knowledge of the scientific context. Examples that illustrate situations where the assumptions required for interpolation are unreasonable could be helpful in establishing the general principles that one needs to consider in deciding between the two approaches in any given situation. – jsk Aug 15 '19 at 23:02
• @user158565 This part is answerable: "what situations would it be appropriate to prefer one or the other?" – jsk Aug 15 '19 at 23:04
• @PeterLeopold Indeed, I think that you are correct that it will depend on the reasonable of the assumptions. The key though is not just assumptions about the time-varying nature of the process, but the connection between the process and the survival probability. – jsk Aug 15 '19 at 23:07