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I have time-to-event survival data (ie., start, end, fate [death or censor] for each known individual). I am looking to model survival for a population of animals that are released onto a new landscape using Cox Proportional Hazard Models where I include additional demographic predictors (sex, age class) and potentially time-varying environmental predictors (temperature, precipitation).

However, beyond modeling survival, I am most interested in estimating when a break point might occur in estimated survival rates relative to the start of study. That is, we hypothesize there will be a period of acclimation where survival is lower - but that once animals acclimate to their new landscape, survival will increase. What is the best way to estimate the unknown duration of an acclimation period (i.e. what is the best way to estimate when an unknown breakpoint in baseline hazards occurs)?

I've seen this done (with code provided even! - citation at bottom) for a Bayesian formatting of a band-resight Cormack-Jolly-Seber model - but I'm not sure how to adapt it to a time-to-event analysis. These authors treated acclimation period as parameter in their model and their full model included a logit link function between two survival models - one with and one without an acclimation effect depending on if during or after the simultaneously estimated acclimation period.

Any suggestions would be SO appreciated. I'm in semi-early stages but feeling overwhelmed.

(Armstrong, D. P., Le Coeur, C., Thorne, J. M., Panfylova, J., Lovegrove, T. G., Frost, P. G., & Ewen, J. G. (2017). Using Bayesian mark-recapture modelling to quantify the strength and duration of post-release effects in reintroduced populations. Biological Conservation, 215, 39-45.)

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Identifying change-points in time for survival curves, including for Cox proportional-hazards (PH) models, can pose problems as outlined on this page. Questions to consider are whether you think that the change-point is independent of the predictor values and how abrupt a change in baseline survival you expect from acclimatization.

If the change-point in time is independent of predictor values, then simple examination of the baseline hazard/survival estimated from the Cox model might be all that you need. In that case your scenario would mean a rapid drop in survival at early times followed by a much flatter survival-versus-time baseline curve. (Survival can't "increase" with time, it can just stop decreasing so quickly as it had been.) The baseline survival-over-time function is necessarily discrete in a Cox model, but you might be able to submit those discrete baseline survival values to generic change-point software noted in the page linked from the first paragraph to get a change-point, and estimate variability by repeating the modeling on multiple bootstrap samples of the data. As Frank Harrell noted in a comment on that linked page, however: "The sample size required for reliable estimation of such change points is quite large."

If the acclimatization is gradual (which would seem to make sense), you might consider a smooth parametric model. A Weibull model, for example, can fit a situation with a continually decreasing hazard, consistent with acclimatization, in a PH setting. That, and other accelerated failure time models used in non-PH settings (e.g., log-normal, log-logistic) can have nice interpretations in terms of shrinking or stretching the time axis as a function of predictor values, allowing acclimatization rate correspondingly to depend on predictors.

A Bayesian model combining what you've already found in the literature with a Bayesian survival model sounds appealing, but I have no experience with that. There are a few threads about Bayesian survival on Cross Validated, and a brief Wikipedia page links to a couple of references.

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