Longitudinal survival data analysis: predict survival by a repeated measure over time

Here is a mock dataset that aims to answer a question on whether subjects with increasing value between trip (from v0 to v2, 3 visits) will have significant shorter/longer survival time (time). The mock dataset is shown below (can be reproduced in R):

subj = rep(1:50, each=3)
trip = rep(c("v0", "v1", "v2"), 50)
value = round(abs(rnorm(n=150, mean=6, sd=2)), 3)
time = round(rexp(n=150, rate=0.3), 3)
censor = rep(rbinom(n=50, size=1, prob=0.5), each=3)
type = c(rep("type 1", 12), rep("type 2", 6), rep("type 3", 5), rep("type 4", 16), rep("type 5", 11))
dat = data.frame(subj, trip, value, time, censor, type)


There are a total of 50 subjects (subj), each having 3 visits (trip), and for each visit, there is a measurement value taken. Each subject had survival time (time), censoring status (censor, 0=censor, 1=death), and disease type (type). Please advise any well-established statistical methods for such analysis. Ideally such methods can be extended to incorporate additional covariates at the subject level, with R package implementation available. Probably Cox PH model with random effect for disease type (type) may be one way to consider? Thank you for the suggestions.

• IN recent years there have been lots of papers that jointly analyzed longitudinal and survival data using a Bayesian framework. – bdeonovic Mar 23 '17 at 13:20
• I am heartened to see you starting your exploration with simulated data, and have an idea about how to answer your question usefully. But one thing I'll need to understand better is the meaning of these 'disease types'. Are these unrelated diseases, or do they represent grades/levels of severity of a given disease? Also, although I appreciate your aiming for abstraction, it would also help me to know more about the real-world problem in the background. – David C. Norris Mar 28 '17 at 19:20
• @DavidC.Norris Thanks for your feedback. The disease types actually consist of different tumor types, such as lung cancer, breast cancer, ovarian cancer, etc. The real-world problem is to see if any trend (increasing/decreasing) lab parameter value taken at different time points can help predict the survival time, taking into account the heterogeneity of different tumor types (type). – alittleboy Mar 29 '17 at 12:10

Philosophically, two basic approaches present themselves here. One approach is to seek off-the-shelf models that (supposedly) enable you to 'extract' or 'derive' knowledge from your data. I would identify that approach with inductivism. Karl Popper famously called induction "a myth," and I share his view. Inductivism remains, however, the dominant epistemology in the (clinical) medical sciences at this time. Its spirit is nowhere more vividly represented than in the field of 'data science', the basic premise of which (it seems to me) is that data are like underground ore deposits to be 'mined', and that for this we need only to employ the right 'earth-moving equipment' (i.e., techniques or methods).

An alternative spirit is that of 'Strong Inference' described in a famous essay by Platt [1,2]. Here, the focus is on building and testing theories. Notwithstanding the strong 'inductivist' leanings of many Bayesians, I think Bayesian methods (as distinct from 'Bayesianism') make a very suitable basis for pursuing strong inference. If you are interested, I would be glad to assist you in pursuing that course.

Presently, your simulation code generates data having the formal appearance of the sort of data you might collect, but not their informational content. To achieve the latter, you would need to write in some equations (models) of a data generating process (DGP) describing what might be going on 'beneath the surface' in your data.

If you could add such equations, I would propose to show you how to implement your model in JAGS, a fully declarative Bayesian modeling tool that has the lovely property of letting you run the very same model both 'forwards' (from given parameters to simulated data) and 'backwards' (from measured data to inference about unknown parameters).

This approach would – as its first and perhaps most important benefit – enable you to check whether various theories you might consider (as expressed in your models of the DGP) would yield to exploration in different regimes of sample size. If your actual sample sizes are of the order of the $N=50$ individuals of your simulation (split across 5 categorical disease sub-types, no less), then I would not hold out great hopes of learning anything new that isn't already 'obvious' to clinicians treating these types of cancer. But this approach would let you demonstrate such an intuition objectively, at least.

1. Platt JR. Strong Inference: Certain systematic methods of scientific thinking may produce much more rapid progress than others. Science. 1964;146(3642):347-353. doi:10.1126/science.146.3642.347.

2. Fudge DS. Fifty years of J. R. Platt’s strong inference. Journal of Experimental Biology. 2014;217(8):1202-1204. doi:10.1242/jeb.104976.