# Can an interval censored survival model be used to predict an event in a future interval?

Suppose I have an interval censored survival model in which each interval is a month long period, and there are twelve intervals (i.e. one year total). Also suppose that this model has been trained and tested with good results. Now I collect new data in real-time, meaning that I collect data for each month sequentially. However, now I want to be able to predict in which interval an event will occur BEFORE I actually collect the data for that interval. For example, I would like to collect data for January and get the probability of event occurrence for February, March, April... etc. It seems like this would be a commonly desired output for interval survival models, but I can't seem to find a straight-forward answer.

UPDATE There was a discussion about future predictions and survival analysis here. It seems that the general consensus was "no", you can't predict future events with only current data. However, because this post was six years ago (maybe there have been new advances in survival analysis techniques?), and because it seems there was uncertainty in the original answers, I would like to reopen this discussion.

Additional Information Per @AdamO 's request, I will try to make the example more specific/exact, although I do want the answers to be applicable to a general case. Suppose I am collecting data on the time-to-death (# of days) of annual plants in a grassland ecology. All of the independent variables are continuous numeric and time-dependent. For clarity, let's say we have three independent variables: 1) The ratio of the soil temperature of the topsoil(10cm) with that of the subsoil (100cm). 2) The average soil moisture across the whole soil profile (0-100cm depth). 3) The variability (let's say coefficient of variation [cv]) of rainfall during the interval. Data for all of the three independent variables are collected on a daily time step and averaged for the whole interval.

Lastly, let's say that we don't know when the plants germinated, so we don't know when growth for each interval started. All we know is that in January (our first interval, see initial explanation), all the plants have been growing for at least a week or two.

Hope this makes the scenario a bit more clear.

• can you describe more carefully exactly how the data are collected and the nature of the interval censoring? Perhaps including examples or descriptions of the outcome measure? – AdamO Mar 28 '17 at 18:44
• Sure, as long as there isn't a secular shift that makes the new data now governed by a different model than the old data used to train the model. Just like any other issue with extrapolation... – gammer Mar 29 '17 at 3:36

## 1 Answer

Yes, you absolutely can use a model to predict outcome conditional on survival up to that given time. It doesn't specifically need to be a model for interval-censoring, you just need to be able to get the model for survival curve.

Suppose I have a survival curve for a subject, i.e.

$P(T > t | x) = S(t)$

If $t_c$ time has passed and that the event has not occurred for that subject, this results in a truncated distribution in which we know $t > t_c$. At that point, our updated survival curve is

$S(t| t > t_c) = \begin{cases} 1 & \text{if$t \leq t_c$} \\ S(t) / S(t_c) &\text{if$t > t_c$}\\ \end{cases}$

Given the survival curve, we can generate whatever variable we want from it for our estimate, such as the median.

So once you've fit your survival curve model, you can get these estimates regardless of whether you used interval censored data or not.