Comparing Different Types of Survival Models

Question

Suppose there is a system in which individuals are measured only at discrete time points (e.g. blood pressure is measured once every year) and these individuals have the ability to transition between multiple states (e.g. disease free, disease stage 1, disease stage 2, death by disease of interest , death by comorbidity, lost to follow up etc.) until one of the absorbing states is reached or the period of study is over. The goal of the analysis is to understand what how different cohorts of patients and patient characteristics contribute to the transitions between these states.

I am trying to understand what types of Survival Models are generally used for this type of problem.

At first I thought that perhaps the competing risk model might be suitable seeing as there are "competing absorption states" (e.g. death by disease of interest vs death by comorbidity) - but I am not sure if a multistate/multistate-survival model is better suited for these types of problems.

Another approach that I have been considering is using several multinomial logistic regression. For example, if there are "n" states and "k" absorbing states (i.e. "n - k" non-absorbing states):

• Isolate all rows of data in "state 1" and create a multinomial logistic regression model where the outcomes are "state 1, state 2, state 3... state n"
• Repeat this process and create "n - k" multinomial logistic regression models

Thus, as a recap:

• Approach 1: Discrete Competing Risks Model
• Approach 2: Discrete Multistate Survival Model
• Approach 3: Multinomial Logistic Regression

Can someone please comment which of these approaches (or perhaps some other approach) is generally used for these types of Discrete Time problems?

Thanks!

Answer 1

A multinomial regression at a single evaluation time for a multi-state survival problem has the same limitation that a single binomial regression has for single-event survival models: it doesn't nicely handle censored survival times prior to the single evaluation time. Treating "censored" (loss to follow up) as a separate state poses an interesting problem: if censoring can be predicted by the same covariates that are associated with outcome, then censoring is presumably informative and the standard assumption of non-informative censoring in survival analysis doesn't hold. You would have to do more extensive modeling.

I don't see that separate single multinomial regressions based on each of the non-absorbing states, as suggested in the question, would be appropriate.

You can, however, analyze discrete-time survival data as a series of binomial, ordinal, or multinomial regressions at each of the observation times. The data need to be formatted properly, so that only those who are still at risk are included in the regression at an observation time. Censoring is then handled by the omission of individuals without an event time from regressions at times after the censoring time. Tutz and Schmid describe the rationale. The discSurv package in R provides tools for formatting data in a way that allows application of standard regression methods to discrete-time data.

Strict competing-risks models assume that all non-initial states are absorbing states. If you are modeling multiple disease stages or back-and-forth transitions among states, a competing-risks model won't work. Tutz and Schmid only seem to describe multinomial models for competing risks scenarios; I don't think that they cover back-and-forth transitions between states at all.

That leaves multi-state models, which can take several forms.* You might treat such a model as a series of multinomial models in time, as in the example you cite in a comment, but if back-and-forth transitions are allowed between states (unlike the unidirectional transitions in that example) that could become messy. You might as well use Markov-type transition models that are designed for such data.

The R "Multi-state models and competing risks" vignette is a good introduction. For panel data like you seem to have (same individuals followed over a set of discrete time periods), Section 6 of the vignette recommends the msm package. I don't regularly deal with discrete-time survival data and I don't have experience with that package, however, so I can't provide an informed opinion about its suitability.

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*Even a simple standard survival model can be considered "multi-state" if you allow for two states (initial state and absorbing event state) to be called "multi." To that extent, some of this discussion is terminological rather than substantive.

Answer 2

Offering a different perspective on the matter, this is a common issue in cancer clinical trials. The typical endpoint is PFS or progression-free survival - it is a composite endpoint of death and disease progression. Disease progression is surveilled by conducting periodic scans. So with a 3 (or 6) month interval between scans, if disease progression is detected, we have no idea at which exact timepoint the cancer grew - as early as the day after the previous scan to the day prior to the current scan. This interval censoring problem is well explored.

What's desired in this case is a consistent association measure to make inference that the survival is better in one group - provided the proportional hazards assumption is met. So this motivates the hazard ratio from a Cox proportional hazards model. What we also desire is a measure of person-time follow-up so that we can approximate the population-based incidence density by dividing events to follow-up.

In analyzing the data, the event timepoint is, for PFS events, date of death or cancer scan showing disease progression. For controls, they are censored back to latest evaluable scan, and subjects can be censored if they miss a certain number of scans because the interval censoring becomes intolerable.

You can do some relatively simple simulations to show that the extent of bias is relatively small and the power of the statistical test is acceptable and, most favorably, the analysis model is easy to specify, fit, replicate, and understand.