Joint Longitudinal and Time-to-Event Models

Join Terrence E. Murphy, PhD, MS, Pennsylvania State University College of Medicine, and Kendra Plourde, PhD, Yale School of Medicine, as they discuss joint modeling methods. They also discuss longitudinal models, modeling survival events, collider bias and the many types and useful applications of joint models.

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Terrence E. Murphy, PhD, MS: Well, hello. Today, I have the pleasure of introducing Dr. Kendra Plourde, who wrote a nice module for us called Joint Longitudinal and Time to Event Models. Dr. Plourde is an assistant professor of biostatistics at Yale. She has methodological expertise in correlated data in both observational studies and clinical trials.

Her methodological research interests focus on the design and analysis of observational studies and trials that have correlated data. For example, longitudinal studies, cluster randomized and stepped wedge designs that feature complex sources of clustering and clinical trials with multivariate outcomes.

For collaborative research, [01:00] interests are in the studies of Alzheimer's disease and related dementias and aging. Kendra is also a member of the Yale Center for Analytical Sciences, as well as the Yale Program on Aging, which is the home organization of the Yale Pepper Center. So welcome, Kendra. We're very happy to have you here today.

Kendra Plourde, PhD: Thank you for having me, Terry. 

Terrence E. Murphy, PhD, MS: So I thought I would start with summarizing at a high level what I saw as the takeaways from your module. I will note that on her CV, Kendra received excellence in teaching awards in both her undergraduate and graduate levels. So not surprisingly, these slides are very well organized and easy to read.

And I think it's a, a double bonus that we get to pick her brain a little today as we talk to her. So typically, Kendra, in joint modeling, for one thing, there are many types of [02:00] joint modeling, but the most common, which, which you present to us is the pairing of a longitudinal outcome, which is typically measured over time with linear mixed effects models.

And a survival or time to event model, which is commonly modeled with a Cox model. And oftentimes people are more interested in either the longitudinal or the Cox model, the survival model. If they're more interested in the longitudinal model, one of the major advantages of jointly modeling it with the survival model is that If the survival model, the outcome of that survival model, for instance, death, is associated with the longitudinal model that's being evaluated, for instance, disability or cognitive decline, then the dropout that's caused [03:00] by the survival event, such as death, is informative and therefore likely to inject bias into any inference made on associations between explanatory variables and the longitudinal outcome itself. Okay, so that's that's the big advantage for those who are interested in modeling the longitudinal outcome.

For those with a primary interest in modeling the survival event, oftentimes, they want sophisticated adjustment for time varying covariates, and the joint modeling allows them to do that. I think, as you point out in your slides, time varying covariates, as they change, it often leads to kind of a step function in terms of survival probabilities, and jointly modeling smooths that out, so you get a better adjustment.

There are also cases when the primary interest is in both outcomes. And in that case, it, joint [04:00] modeling is especially informative, and even sometimes when we're interested in mediation, the mediation of some exposure. On the survival outcome, exerting its effect and indirect effect through the longitudinal outcome.

So there's, there's a lot in there, a lot to unpack, but I think increasingly in aging, this is really an important practice. Because so many of the longitudinal things we want to study such as disability, such as cognitive decline, the underlying causal processes are related to things like death or first cancer or first diagnosis of dementia, those sort of things.

So, first of all, I'd like you to. Now comment on this very high level simplistic summary and fill in any [05:00] obvious gaps I left. And let's just hear your take on that. 

Kendra Plourde, PhD: No, I thought that was a very, very good summary of the usefulness of joint models. Very beautiful how you put it and summarized it. I guess I can add a little bit with just when you're looking at the longitudinal outcome, for instance, you gave the example of cognitive decline. If we're interested in modeling cognitive measures over time, as you can imagine, individuals who are experiencing cognitive decline are at a higher risk of death. So another nice way to put it or explain this informative dropout processes, you know, if you have patients who are experiencing cognitive decline, they're at a higher risk of death. Therefore, they're going to drop out of your study sooner, and you won't actually see or be able to measure their cognitive [06:00] decline after they have dropped out of your study. So you, you might get some bias in your estimate because you didn't observe some of the more significant decline in your population.

So that's a one way I always like to try to give an example of what this informative dropout means. And when the, with the time to event outcome of interest, I thought you gave a very good summary of if we're repeatedly measuring, say, blood pressure over time, typically when we fit a Cox model investigators might only use the baseline measure of someone's blood pressure, but we actually measure blood pressure, say, at every clinical visit, and wouldn't it be great to update that blood pressure measure in our Cox model, because our Cox model, otherwise, we'll just assume that their blood pressure stays that exact baseline measure throughout time throughout the whole [07:00] study when we know, in fact, blood pressure can vary quite a bit over time.

In fact, their blood pressure could go way up over time in the study, and we wouldn't be able to capture the fact that higher blood pressure has an increased risk on some event, say a cardiovascular event or death. So those are just some extras bringing it down even to just how I try to talk to you know, clinicians or just everyday people, what I do and why and what a joint model is, because I know it's very complex when you get into the actual nitty gritty of it all so trying to make it even simpler. Otherwise, it's hard to capture it. 

Terrence E. Murphy, PhD, MS: Sure. I think you do a beautiful job of that in your slides. One term that you pointed out, Kendra has a very nice paper. In the journal Epidemiology in 2022, she's the first author. [08:00] It's called "Joint Models for Estimating Determinants of Cognitive Decline in the Presence of Survival Bias."

In this paper, she's modeling cognitive decline after dementia diagnosis. And you do talk about death as the survival model. And you mentioned the term collider. Could you just tell us a little bit about collider? Because I read that and I'm like, Oh, great. Someone's giving me a great concrete example. And I'm thinking, isn't that just a competing risk?

You know, so how about you contrast those two for us? 

Kendra Plourde, PhD: Sure, of course. I wish I had a whiteboard so I could bring out because you really need to think about, you know, we have this exposure, say, smoking, whether someone smokes or not. And then we have our longitudinal outcome of interest. We can just stay with cognition.

So we want to, you know, [09:00] measure cognitive decline or model cognitive decline, and we know that smoking is associated with death. You know, those who smoke higher risk of death. Those who are declining cognitively are also at a higher risk of death. And so we have these arrows coming from smoking to death and coming from cognitive decline to death.

And so these arrows create a collider on death. So collider bias in that way. And the issue here is that when we are interested in looking at the association of smoking and cognitive decline, you know, does smoking have any effect on your cognition over time? The fact that these two different, the exposure, smoking, and our outcome of interest, cognition, are both associated on survival, well, that means that when we run our [10:00] model, if we don't take into account the fact that we have these associations with death, when we fit a linear mixed effects model, for instance, looking at smoking on cognition, we will see there will be a little arrow dotted, connecting smoking to cognitive decline because of that collider.

So it makes there appear to be a relationship when in fact there may not be, and that is what survival bias is. So what a joint model can do, it actually models each of these separate relationships. So we can model smoking on cognition, but then also have a model for death where we take into account the association, not only of smoking on death, but then also on our longitudinal outcome of interest, cognition and death.

So we can actually take into account those associations and relationships and therefore free ourselves of this [11:00] potential bias that we call survival bias. And that is what the paper shows is really, if you model the dropout process or the the model for death correctly you can eliminate all bias that you would have seen otherwise.

And again, that's because we're taking into account those relationships. So we break those arrows and we're free of the survival bias or the collider bias. 

Terrence E. Murphy, PhD, MS: And so, as you state, I think in your slides or in this paper, a telltale sign of this sort of collider bias is when you have this unexpected association that's counterintuitive and often, in fact, it's it's a negative sign. It's completely counterintuitive sign. So this is one of the telltale signs that you may have some collider bias lurking. 

Kendra Plourde, PhD: Exactly. So for instance, in my example of smoking, we might see that smoking is protective [12:00] of cognitive decline. That's something that we might see and scratch our heads as to why that is.

But really, that just means maybe that the smokers are dropping out of your study before you could see their cognitive decline. And that's the bias leaking in. 

Terrence E. Murphy, PhD, MS: So Kendra's paper in Epidemiology in 2022 is a very nice example where someone's more interested in the longitudinal model, and they want to jointly model to prevent the survival event from injecting bias into the associations of interest with their longitudinal model.

In just some email dialogue we had, I asked her what was her favorite joint modeling paper, and she pointed me to one by Van Oudenhoven from 2019 in BMC Medical Research Methodology. And in that [13:00] paper, they model a longitudinal outcome called the clinical dementia rating. And they also model time to diagnosis of dementia, and they model these jointly.

And there's an exposure called Fortasyn Connect. And what's very nice about this paper is they show how, I think their primary interest is looking at the association of this exposure called Fortasyn Connect on the time to dementia, but they want to model this clinical dementia rating and, and they do it for a separate one called the neuropsychological test battery.

And they want to show that the effect of the exposure on the outcome is [14:00] affected by the longitudinal model, and they, as you mentioned in your slides, Kendra, they actually show that you can calculate how much of the effect of the exposure on the outcome is transmitted through the longitudinal model, meaning that the longitudinal model can be serving as a mediator as well as the direct effect of the exposure on on that outcome.

So this, in my view, this paper, it has more interest, I think, in the time to event model, but it paints very clearly how critically important it is to jointly model the longitudinal outcomes with this added benefit of being able to see some mediation effects. Why don't you give us your take on that paper?

I'll just note too that in that paper, the, the senior author, the last author is Dimitris Rizopoulos, who also [15:00] turns out to be the author of the JMR package that you mentioned in your slide. So he's a world famous joint modeler. So it's perhaps not surprising that this paper does a beautiful job laying out the interrelationships between the longitudinal model and the time to event model.

Kendra Plourde, PhD: Yes, exactly. It's his specification of a joint model that I focused on in my module because there are different specifications for a joint model. There's lots of different types of joint models, even other types of joint longitudinal and survival models. So yes, it's not surprising that one of my favorite papers is his, and yes, he does focus on, or they, I should say, focus on the time to event model, which actually the joint model that I focus on in the module, that was the purpose.

They were focused on how do we include [16:00] repeated measures more elegantly or with better assumptions, I should say. In our Cox model, because as you mentioned at the beginning of our conversation, incorporating repeated measures in a Cox model is great. But you still have that issue of assuming that measures are staying constant until the next measurement time in a Cox model.

And so smoothing out the repeated measures allows for a more, for better assumptions in our Cox model, because we know they're changing over time, and it just smooths that all out for getting a better predictions or better estimation of our association between that time varying covariate. And the Cox model, this paper in particular, besides the mediation component that you mentioned, which I also really like I'm not many papers that I've seen, I'm sure they're out there, though, [17:00] talk about not only the mediating aspects of a joint model.

So we have this exposure of interest. What is the direct effect of that exposure on my time to event of interest here dementia diagnosis and what's the indirect so I know that this exposure has an effect on say, I think, in the paper. It was some cognitive measure, and I know that my exposure has an effect on this cognitive measure, which then also has an effect on the time to event.

And so being able to split up all of those different, the indirect and direct effects, I thought they did that very well in this paper. And on top of that I also really enjoy this paper because they talk so much about how do we best model the longitudinal outcome. It's association with that time to event outcome.

Because [18:00] in the module, you know, there's only so much space and time, so I include resources to, you know, finding out about different specifications of the, how do we specify the relationship of that longitudinal outcome with the time to event outcome? Is it just the current measure of our longitudinal outcome that's affecting our time to event of interest? Or is it really the slope, the change of the longitudinal outcome that's affecting our time to event? Or is it a cumulative effect? So is there a building, an accumulation of this exposure over time that then has an effect on our time to event outcome? 

And they explore all of these different components of the joint model. So I found it very nice. Because, again, the joint model was created with the time to event [19:00] model as its focus, as the interest, and a how do we incorporate [indistinct] covariate in a Cox model.

My paper was a little bit going off on, well, I want to see how the longitudinal model is benefited from this simultaneous estimation. That's why I thought that this paper by Rizopoulos and Floor van Oudenhoven, and I apologize for, I'm sure I just butchered her name. I thought that they did a really wonderful job of going through all of those aspects that I think are very prevalent in aging research.

Terrence E. Murphy, PhD, MS: Now, Kendra, we're coming to the end of this conversation and the purpose of these modules is to help other folks who want to get into aging research. We're trying to introduce to them some useful analytic techniques. In my view, there are one, lots of advantages to joint modeling that you've pointed out, [20:00] but it is more complex.

It has these two submodels, and it involves some, you know, use certain R packages of things. So how would you help aging researchers who are maybe a little less statistically savvy than yourself? What words of encouragement or advice would you give them about Not shying away from embracing joint modeling in their own work.

Kendra Plourde, PhD: Well, first, I would say work with a statistician. I'm always, I'm a biased statistician, so I always enjoy working with investigators and if they don't have the funding, you know, I work with investigators and researchers in the program of aging, on aging and they can come to me and ask, you know, I have this issue.

Can you help me out? Can we just have a conversation about it and how best to go about modeling this? So I would say first talk to your local [21:00] bio statistician. But also, I think just being aware of the assumptions and of the model of the joint model. I think you mentioned already, Rizopoulos has a book on joint modeling, and I know he also has a website with a lot of resources and just understanding what are the assumptions that you're making if you're interested in using a joint model and also thinking about what that association really is, maybe creating a a DAG plot of I think the this is the association of my longitudinal.

This is my time to event or. And just seeing if collider bias, for instance, is going to be an issue in your study, or maybe it's not. Just because you have dropout in a longitudinal study doesn't mean you should use a joint model. It really needs to be informative dropout. Does your longitudinal outcome in particular have, [22:00] an association with the dropout mechanism?

If the dropout mechanism is completely random then you don't need a joint model. So I want to make that distinction in particular that just because you see a longitudinal study and oh no, there's an issue of dropout, joint models may not be helpful. I, I would highly suggest though, just if you have a biostatistician, everyone should have one on call.

Terrence E. Murphy, PhD, MS: Okay. So we hear we hear a shout out for our friendly neighborhood biostatisticians. I think to me that the point is joint modeling is a wonderful option to pursue, but it is of a higher level of a complexity. And so it is best practiced in league with a biostatistician because the higher complexity means there are possibly more pitfalls or ways to, to get stuck, but don't let that scare you.

Well, Kendra, [23:00] any closing remarks? 

Kendra Plourde, PhD: I hope that people find the module helpful. I hope that the resources I provided for them to find more information are also helpful if they decide to investigate joint models further. And thank you for having me. 

Terrence E. Murphy, PhD, MS: Thank you so much, Dr. Plourde, for sharing your expertise on joint modeling. Thank you.