Association between lagged exposure and state transition with msm

[CamilleMorlighem]

Hi Prof Jackson,

I would like to use the msm package to assess how lagged environmental exposure affects transition intensities between disease states. My dataset has highly irregular visit times, with a median interval of about 353 days and an average of 3.65 visits per individual, with substantial individual variability.

I would like to include the mean temperature (and similar variables) over the past 6 months prior to a transition as covariate in the model. For two consecutive visits S → R (say 2 months apart), I assume the transition to R depends on the exposure in the 6 months before R. Since msm uses covariates at the start of each interval, I attach the lagged exposure for R to observation S.
I would like to clarify whether the following points are consistent with msm model assumptions:

• This creates a misalignment in time between the covariate value and the transition period over S→R
• If S→R is short (e.g., 1 month, which sometimes happens in the dataset), then the exposure window of 6 months influencing the transition includes several months (5 months) that occurred before S. Is this acceptable within the intended covariate interpretation in msm?

A second issue is that with ~1‑year intervals, time‑varying covariates become constant over very long periods, which is unrealistic for exposures with seasonal variation. To better approximate a piecewise‑constant covariate trajectory, I am considering adding censored pseudo‑observations at fixed calendar dates (e.g., season boundaries) so that the covariate can be updated more frequently. Each real and pseudo observation would have the exposure recalculated over the past 6 months. Would using censored pseudo‑observations in this way be an appropriate strategy ?

Thank you in advance for your answer !

Kind regards
Camille Morlighem

[chjackson]

It's not so clear what you're asking, so it's hard to answer. I can clarify what the msm assumption is (covariates are assumed to be constant on intervals between observations, and the transition intensity within that interval is taken from the covariate value at the start).

But I can't advise what would be the most appropriate covariate for you to use for your application - you are best placed to answer that, given your knowledge of the msm assumptions. The covariate can be anything you like, as long as you are willing to make that assumption.

If you think the variation in covariates within observation intervals is important, I think it would be reasonable to include censored pseudo-observations, as you say.

[CamilleMorlighem]

Thank you for your answer!

Sorry if my earlier questions weren’t clear. If I understand correctly, the effects of covariates on transition intensities do not depend on the interval length, so in practice, it doesn’t really matter if covariates aren’t measured at exactly the same time as the observation (as long as the conceptual/biological assumptions make sense) ?

Regarding the censored pseudo‑observations, can these be included as often as needed? For example, if the interval between observations is about one year but the covariate changes daily, could pseudo‑observations theoretically be added every day, or would this affect the model in practice (beyond computational time)?

[chjackson]

In the underlying msm model, the intensity at any time $t$ is modified by the covariate at the same time. We don't always know the value of the covariate at that time, so when you input the data, you are assuming that the covariate value at any time $t$ is equal to the value at the most recent observation before $t$. Does that resolve the query in the first paragraph - not quite sure what you mean by "effects ... depend on the interval length".

Yes you can include as many censored observations as computation will allow, if you believe this gives a more accurate picture of how the covariate changes over time. This changes the data you are inputting to the model, but it doesn't change the model structure.

[CamilleMorlighem]

I think I originally mixed up model assumptions and conceptual assumptions, thank you very much for the clarifications! I’ll implement this using censored observations and I may get back to you later at that point. Many thanks again!!