Ordinal outcomes in survival analysis

Frank Harrell has a very interesting idea in his talk “Ways to Make Randomized Clinical Trials More Statistically Efficient”, using random effects ordinal logistic models to analyze an ordinal survival outcome (for example “healthy, MI, death”). Instead of, say, dichotomizing and running a regular Cox.

It’s a very logical extension, but I’ve had a hard time finding any papers where the method is used or suggested. And it would be great to have a good reference before I’ll start to apply it myself. Could anyone help me out?

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Accounting for intra-patient correlation using random effects turns out to not fit correlation patterns very well when data are collected serially. Markov models, where you condition on outcomes in previous time periods as if they were ordinary covariates, works quite well for longitudinal ordinal outcomes. The wake-up call for me was running an example where the efficiency of the treatment effect increased as you removed time points from the mixed effect model, whereas for a first-order Markov model the efficiency decreased as it should.

The Markov approach is used to do detailed analyses of the ORCHID and VIOLET 2 studies - see the new links for these in https://hbiostat.org/proj/covid19. The case study for VIOLET 2 also contains random effect analyses. You’ll see that the variance of the random effects needed to be huge to allow for a high intra-patient correlation. This makes the model unstable, takes longer to converge, and means that the effective number of parameters in the model is very large.

The generality of the serial ordinal outcome approach is quite impressive IMHO as you can handle the usual time-to-single-event analysis, regular longitudinal analysis of continuous outcomes, and recurrent event analysis within one framework. One design consideration is that you have to measure the ordinal outcomes (those that are not fatal) also at baseline to start the Markov process.

You’ll see in the case studies that the Markov approach uses only standard frequentist or Bayesian models, and if using Bayes it is easy to uncondition on previous outcome states when you’re finished, to get state occupancy probabilities. For the case of a terminating event (absorbing state) such as death, this is equivalent to a cumulative incidence curve. So from Markov models you can get far more than transition probabilities. The ORCHID case study shows how to get a posterior distribution for the mean time that a patient is in either of two states.

I haven’t seen this applied in clinical trials but there are many papers on ordinal longitudinal outcome modeling, for example here.

The video for the talk Ulf mentioned is here.

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Thanks a lot! I’ve wondered why one couldn’t use random effects logistic models instead of Cox, apart from the computational complexity for large datasets, but the strong serial correlation seems a convincing answer.

My second thought was to use some kind of conditional ordinal logistic model (stratifying on time, similar to Cox being time-stratified Poisson), but it’s not implemented in Stata, and there seems to be some controversy about this in the literature. Anyway, I feel I’m out of my depth experimenting with this stuff, so Markov models seems a great suggestion. I’ll try it out.

The links I provided have a lot of details, plus complete R code. One cool thing about Markov models is that they use only standard software, up until the point where you need to estimate unconditional probabilities. Code for doing the latter is provided. Fortunately that is all done after the fit. Some simulations I’m doing also show promising result when measurement times are irregular.

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