# Survival analysis, repeated measurements, large samples

My basic question

What are some standard time-dependent, repeated measures, full longitudinal survival analysis models?

The more precise question and context is as follows.

Context

Suppose we have $N$ = 1 million samples $\{Y_k(T)\}_{1\leq k\leq N}$ and we wish to predict the longevity of each $Y$ using the following information. Each $Y$ consists of $p$ (e.g. $p=25$) time-dependent covariates $x_1(t), \ldots, x_p(t)$, a binary censor (event) variable $c$ and current age $d(t)$. We would like to model $d(t)$ based on the covariates and censor variable.

Recall that in basic survival models such as Aalen's additive model (AAM) or CoxPH, we traditionally train the model on exactly one snapshot of data, i.e. we fix exactly one $\hat{t}>0$ and restrict the training to the set $\{Y_{k'}(T=\hat{t}\,)\}_{k'}$, for some $k'\leq N$.

Goal

Now, since my data has several time values $x_p(t')$ whenever $d(t)>0$ for $t\geq t'$, my goal is to do survival analysis'' for more than one snapshot. For this goal, the (standard) AAM or CoxPH do not suffice.

Question

What are other models or extensions of AAM/CoxPH that might help achieve the above goal?

Addenda

It would be great if, for example, suggested models account for rates $dx_i/dt$ for each $1\leq i\leq p$.

As a concrete example, if we have two samples $Y_1$ and $Y_2$ and we know that the velocity'' of $x_1$ for $Y_1$ is greater than the corresponding velocity for $Y_2$, then this information should impact the expected duration for $Y_1$ differently than for $Y_2$.

Literature references would be highly appreciated, as well as Python, R, or SAS, though literature with a mathematical tone is favored.

Thanks in advance!

• I would suggest looking into frailty models. One way of interpreting these models are as survival analysis methods that allow for random intercepts. – Cliff AB Oct 10 '15 at 19:48
• @Cliff: thank you - could you suggest frailty models and literature related to my question... I'm obviously new to this field and hoping practitioners' advice may be more useful than naive internet searches. – Quetzalcoatl Oct 10 '15 at 20:07
• I would point you toward the CRAN package frailty pack for software. The following paper will give you a good description of the models used. jstatsoft.org/v47/i04/paper – Cliff AB Oct 10 '15 at 22:29
• Thank you! This package looks like an excellent start. I also very much like the linked paper so far, esp their treatment of the "readmission" data set – Quetzalcoatl Oct 10 '15 at 22:41
• It seems frailty models may not be necessary (e.g. I don't require (intra) random effects). An extended Cox model seems more appropriate: stats.stackexchange.com/questions/143340/… – Quetzalcoatl Oct 23 '15 at 15:25