# Connecting Survival Analysis to Overall Predicted Survival Probabilities

I've got a question on combining survival analysis with predicted survival probabilities that I'm sure someone must have thought about, but I just can't find anything out there.

Imagine I have a good model for predicting how well certain patients will survive a disease and a pretty good model for how likely any given patient is to die $$t$$ days after infection. I want to calculate how likely a certain patient is to survive given that they've survived $$t$$ days so far.

Abusing the standard survival notation I can write a "survival" function (that looks sort of like Kaplan-Meier function) for a patient $$i$$ as

$$S_i(t)=lim_{h \rightarrow \infty} \prod_{j=t}^h s_{i,j}$$

where $$s_{i,j}$$ is the probability that the patient doesn't die on day $$j$$. The overall probability the person survives is then $$S_i(0)=p_i$$. In a perfect world I would know $$s_{i,j}$$ for $$j=1,...,h$$ where $$h$$ was sufficiently large.

Instead, I have a pretty good model for how likely a patient $$i$$ is to survive, i.e., $$p_i$$ and a more global model for the overall survival function $$S(t)$$ that isn't as accurate at an individual patient level. Is there a way to connect these so I can give a probability that patient $$i$$ will survive given that he or she has survived $$t$$ days so far?

I've tried a few things I can post in the comments, but can't seem to guarantee that the probabilities $$s_{i,j}$$ are always in $$[0,1]$$ or that the function $$S_i(t)$$ is monotonically increasing with $$t$$. Any ideas, places to look, or papers to read?

I have previously posted some work on estimating Hazard Rates here.

I use that analysis to render another opinion (actually more like an improved composite estimate) on say the parent's distribution location parameter.

Improving the underlying distribution parameter estimate may lead directly to better survival function opinion (see definitional connection here).

As I did, I suggest simulation verification of your methodology.

It sounds like you have two models, one survival curve for the entire patient population and a second survival model for a subgroup which is considered "subject-specific." You can simply further subgroup each of these models by conditioning on surviving at least as long as $$t$$.

What this shows is that probability is a proportion of many samples. Your understanding of the sampling behavior over many samples gives you confidence in what to expect from the result of a single sample.

It also shows that the sampling frame from which you obtained a given subject is crucial for referencing the correct model. How was a given subject selected? If a subject was selected at random from the entire patient population without regard for a minimum survival time, then referencing a "subject-specific" model will yield the wrong inference. Likewise, if a given subject was selected according to a specific set of covariates including a minimum survival time, then referencing a model for the entire patient population will yield the wrong inference.