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I want to know if it is not only possible to handle left-truncation of data in a survival model, but to know/predict the hazard function (probability of event) before the start point, i.e. the point of truncation.

I have species observations for many sampling sites (locations) and years. For each site-year combination there is a start point of the sampling. I want to model/predict the first observation time of a particular species, across all sites and years. I have read that the hazard function only extends after the start point, but I am interested in how the species might have occurred before the sampling started to see what the sampling potentially missed.

Shifting the starting point earlier is not enough, because the model has to know that the data could only occur after the first sampling time.

Is survival analysis appropriate for this? If so, how can I get the hazard function before the truncation point? If not, do you know of any approaches to this type of problem?

Thanks in advance.

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For left-truncated observations, you have no information about cases that had values less than the corresponding left-truncation limit. As Klein and Moeschberger explain in Section 3.4, it's the same principle for all types of observations, for example whether the truncation is "only particles big enough to be seen based on the resolution of the microscope are observed" or "subjects who experience the event of interest prior to the truncation time are not observed." Thus "inference for truncated data is restricted to conditional estimation," conditional upon the observation passing the left-truncation limit.

Perhaps there are additional data of a different type that might be available, based on your understanding of your subject matter, that could be used to address your question. If you are limited to left-truncated observations, however, there are no tricks for getting information about what might have happened in cases that weren't observed because of that truncation.

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  • $\begingroup$ I don't have information about those first occurrences in years when the first occurrance of the species is equal to the first sampling (other than it being at least as early as the first sampling). However, in other years and/or other sites, some or all of the first occurrances were after the first sampling, i.e. not censored. So I should be able to build a model based on environmental covariates that allows me to extrapolate the true time of the first occurances of the censored cases, even though they were not observed. Right? $\endgroup$ Aug 18, 2023 at 9:51
  • $\begingroup$ @MatthewHorn a lot depends on the details of your sampling scheme. For example, does finding no member of the species at your actual first sampling time mean that you couldn't possibly have found a member of the species at some earlier (hypothetical) sampling time? That would seem only to be the case if your "sampling" was actually an exhaustive search to find all members, and if no members of the species ever leave the sampling area once present there. Otherwise you still have a problem that there might have been a member present earlier even if your didn't find it in your first sample. $\endgroup$
    – EdM
    Aug 18, 2023 at 14:34
  • $\begingroup$ Well yes, that's precisely what I want to find out. Would we have found the species if we had sampled earlier? What is the probability of occurrance or predicted occurrance of that species earlier than the first sampling time? And the "search" is trapping the moths and then volunteers identify and count them. If we take certain species we can have a pretty good guarantee that they would be identified if trapped. Of course we can't know whether there were moths in the vicinity that weren't traps. Certainly some nearby moths didn't enter the trap. $\endgroup$ Aug 21, 2023 at 8:57
  • $\begingroup$ @MatthewHorn if you don't have data for earlier sampling dates, there's no completely reliable way to estimate what might have been found if had you sampled earlier. If you have a model that includes environmental covariates associated with occurrence you might be able to extrapolate to earlier times for which you have those covariate values, but that's risky. For example, extrapolation of associations with temperature that work above freezing might not work so well at temperatures below freezing. Furthermore, annual rhythms are likely in such data. $\endgroup$
    – EdM
    Aug 21, 2023 at 13:38
  • $\begingroup$ Ok, well that is my thought as well. I think extrapolation here is fine as long as the covariate space is sufficiently explored. This kind of extrapolation is the usual in species distribution models. But there you would be doing it in time and space, but without an element of truncation. It's just predicting before the truncation point is what I was wondering about in the context of survival models, but I don't see why in principle it's not possible. I could also just use, e.g. a binomial or negative binomial glm to predict presence or abundance at those times before the truncation point. $\endgroup$ Aug 23, 2023 at 4:35

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