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I have always wondered about this point :

  • Longitudinal Regression models for repeated measures essentially allow for previous response measurements for the same patient to influence future response measurements for the same patient.
  • In a very naive sense, one could make the argument that Longitudinal Regression models the next observation based on some function of past measurements (correlation structure), thus giving it the illusion of a N-th order Markov Chain (i.e. most Markov Chains simply assume that the next state is only decided by the current state ... but Markov Chains can be constructed in which the next state is decided by the previous "n" states, i.e. N-th order).
  • Therefore, could someone make the argument that a Longitudinal Regression Model is a Infinite State Markov Process?
  • For some reason, I think the answer is "NO" because I have never heard the term "Memoryless Property" used in the same sentence as "Longitudinal Regression".

In general, are there any statistical modelling frameworks which allow to use a Markov Regression (e.g. https://www.jstatsoft.org/article/view/v038i08) but for a continuous variable instead of a discrete variable?

stats_noob
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1 Answers1

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I hope that someone can find an article where this is fully worked out, but I ran some simple simulations to get at this interesting question. Related to random effects being non-directional and inducing unreasonable correlation structures (compound symmetry with a random intercepts model) with serial data, a random intercepts model is very close to a $k$-order Markov model where instead of conditioning on the previous $k$ values of Y one conditions on the average of all previous Y for the subject. You can get a sense of how this aligns with compound symmetry, and note that the earliest values of Y get the same weight as the most recent Y, which is seldom what we see in real data.

Much is written about the continuous case in the time series literature. A general semiparametric Markov model that handles binary, ordinal, and continuous Y is described here.

Frank Harrell
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    Wow! a reply from Dr. Harrell himself! I also hope that someone finds an article about this. In the future, would you be able to show us your simulations? – stats_noob Dec 02 '23 at 15:47
  • Re time series: Time series is the only thing which comes to mind when thinking about continuous markov models. If I wanted to forcefully "shoehorn" all this together ... I was imagining that is it possible to take a Mixed Effects/Hierarchical Time Series model (e.g. https://otexts.com/fpp2/hts.html) and add exogenous variables (e.g. https://robjhyndman.com/hyndsight/arimax/)? Although I have a feeling this will end in a disaster ... – stats_noob Dec 02 '23 at 15:51
  • @wulaganof I just posted this example as a blog article: https://www.fharrell.com/post/corstruct/ – Frank Harrell Dec 03 '23 at 14:26
  • Super cool! After having written the blog post, do you have a general conclusion on this topic? Markov models are similar to Longitudinal? Does one have advantages in certain situations over the other? – stats_noob Dec 04 '23 at 01:14
  • And do you have an opinion on the topic I raised about the validity of the mixed effects/hierarchical time series model? – stats_noob Dec 04 '23 at 01:14
  • Yes a Markov model is one structure for modeling longitudinal data. It is the most general structure if one does not need nested clusters, e.g., don’t have data like city within state within country. Its generality is due to flexibility in how many previous values to condition on, how to summarize them (the last one vs the average vs the slope, etc.), and the types of Y that Markov can handle (binary, ordinal, nominal, continuous, censored). Markov models are more likely to fit correlation patterns in serial data than are random effects models. But you can mix both model types. – Frank Harrell Dec 04 '23 at 13:15