# Is the Distribution of Survival Times always IID?

I was reading about an approach to Survival Analysis called "First Hitting Time Models" (threshold regression): https://www.jstatsoft.org/article/view/v066i08 , Can Survival Models model the time at which a random variable will first pass a certain point? , Modelling Time to Events

It took me a long time to understand, but I finally think I understood the following. First Hitting Time models offer the following advantage: If the true probability density of survival times (conditional on the covariates) is a Brownian Motion with Drift (i.e. a stochastic process, non IID), First Hitting Time models might be able to offer an advantage. But both AFT and First Hitting Time Models can both explicitly model "first passage time" - the only thing that they differ in, is the underlying distributional assumptions they make about the distribution of survival times.

This makes me think about the following point: In the real world, under what kinds of situations would the Survival Probability Distribution Function not be IID?

In all parametric cases of Survival Analysis I have come across, the probability distribution of survival times is always assigned a classic distribution (e.g. Weibull, Exponential, Log Normal, etc.). When we look at this problem from the semi-parametric approach, the Cox-PH model is used which does not explicitly make an assumption about the distribution of survival times, but nonetheless I don't think its treating hazard as a non-IID.

In the real world, what kinds of situations could arise such that the probability density of Survival Times would need to be non IID? Something in which the probability of survival at the current time will influence the probability of survival at future times? In such cases where we assign the survival probabilities as non-IID, will the survival function still be monotonic and strictly decreasing?

• "IID" is not a meaningful property of a "Survival Probability Distribution Function." It refers to a collection of random variables, not a function per se. Are you trying to ask about the form of a single survival function or about collections of observations? If it's the latter, then isn't the proper question why there would be anything in common about any collection of observations?
– whuber
Mar 1 at 15:34
• whuber I think you are correct and I did not ask the question properly. I think it will be better to ask: under what situations could survival times naturally be a stochastic process? Mar 1 at 16:34
• Are you asking in what scenarios might survival times be non-independent? You could imagine a single cause of all the events, like the survival times of people involved in a particular plane crash. Or about scenarios in which survival times would be non-identically distributed? This would happen if you have a changing population - the overall lifetime survival distribution is different for women than men in most places, for example. Mar 1 at 17:41

## 1 Answer

will the survival function still be monotonic and strictly decreasing?

With at most one absorbing event per individual, a survival function $$S(t)$$ is just the complement of a cumulative probability distribution function of event times, $$F(t)=\int_0^t f(\tau) d\tau$$ if time starts from 0. That is, $$S(t)=1-F(t)$$. $$S(t)$$ is thus necessarily monotonic decreasing (not necessarily strictly decreasing; Kaplan-Meier and Cox survival curves are flat between event times). That's true even if the event times are determined by some latent stochastic process that might be randomly increasing or decreasing over time. Once an event has happened in typical survival analysis, it has happened. There's no going back, no resurrection.

There are multi-state "survival" models that allow for reversible transitions among states (e.g., Markov-type models), but those don't seem to be what you're asking about. In that situation the probability of being in any one state can go up and down as a function of time, subject to the constraint that the sum of probabilities among all possible states at any one time equals 1.

Something in which the probability of survival at the current time will influence the probability of survival at future times?

In the context of a monotonic decreasing survival curve, framing this in terms of survival probabilities isn't very helpful. You might be thinking about the hazard of an event: the probability of an event given that one hasn't yet happened. For a continuous-time model, $$h(t)=f(t)/S(t)$$. Whether that's increasing or decreasing or jumping around as a function of time is determined by the probability density function $$f(t)$$.

In the real world, what kinds of situations could arise such that the probability density sampling of Survival Times would need to be considered non IID? (Edits shown; I hope that editing captures what you're asking about.)

There are situations in which you need to account for potential correlations among survival outcomes. For example, survival analysis based on results from multiple institutions might need to account for within-institution correlations in outcomes beyond the predictors included explicitly in the model. Another is in the extension of single-event survival analysis to the possibility of multiple events within an individual. Then within-individual correlations need to be taken into account.