# Joint Models for Longitudinal and Time-to-Event Data vs Survival analysis with a time dependent covariate

I need some help to understand the difference between survival analysis (cox regression) with time-dependent covariates and Joint Models for Longitudinal and Time-to-Event Data (JM package in R). Could you please explain the difference between these two methods by a practical example.

Even though a longitudinal marker may be endogenous, I think that modeling it as a time-dependent covariate in a Cox model may be meaningful. Though without a causal interpretation, estimating how a marker trajectory predicts instantaneous risk can still be useful.

When the longitudinal marker and outcome events (including recurrent events) are desired to be combined into an overall time-dependent outcome severity (single ordinal time-dependent outcome variable), Markov ordinal state transition models are natural choices that yield models that are easier to interpret than the others mentioned here.

State transition models handle absorbing states such as death extremely elegantly.

• Dear Prof. Harrell. I am so thankful for your time and advice. Commented Jun 5 at 7:09

I'm not an expert (actually, I just started reading recently about it for my own research), but from my point of view, joint modelling can be used to model multiple longitudinal outcomes or a combination between longitudinal and survival outcomes, in situations where a 'standard' survival analysis falls short.

Let's say you want to see how the longitudinal evolutions of a biomarker affects the hazard of developing some disease. Here you link a longitudinal outcome with a survival outcome. Why not do that with an (extended) Cox model? Well, one reason is that the Cox model assumes external (or exogenous) covariates, i.e. the covariate value at time $$t$$ is not affected by the occurrence of an event at time $$u$$, where $$t > u$$. However, if the event you are studying is 'death', you can't measure your biomarker anymore when the patient has experienced the event, so here we are dealing with what is called an internal (or endogenous) covariate, which depends on the occurrence of past events! This cannot be incorporated in the (extended) cox model.

How to deal with this? This is where joint modelling comes into the picture, we describe the longitudinal evolution of this biomarker with a separate model and then link it to the model for the survival outcome (e.g. by using shared random effects that capture their association). (Source: This presentation)

Joint modelling in general is thus employed when modelling multiple longitudinal outcomes, or mixtures between longitudinal and survival outcomes. Note thus that although one of the most common examples is modelling a longitudinal outcome with a survival outcome, you can also just couple a longitudinal with another longitudinal one, or even extend it to multiple outcomes at once.

All this is much better described in books and papers by Dimitris Rizopoulos for example. Please correct me if I missed some crucial points!

• thank you so much for your time and feedback. Commented Jun 5 at 7:10
• I wish you had answered my challenge to the exogeneity assumption above. Cox models have been used for both internal and external time-dependent covariates ever since its invention in 1972. Commented Jun 5 at 11:47
• Dear Prof. Harell, I am afraid I lack the history and knowledge to argue with your experience :) In the introductory courses and articles I have followed and read, I got the impression that the usual cox models are better suited for external covariates, whereas internal covariates can create a bias, and joint modelling is often suggested for handling with these external covariates. I'm not saying it is always wrong to do it or impossible to do it, I even think it is indeed something that should be considered, but I think there are other alternatives that might be better suited maybe. Commented Jun 5 at 13:51