# Marginal model versus random-effects model – how to choose between them? An advice for a layman

In searching for any info about marginal model and random-effects model, and how to choose between them, I have found some info but it was more-or-less mathematical abstract explanation (like for example here: https://stats.stackexchange.com/a/68753/38080). Somewhere I have found that there were observed substantial differences between a parameter estimates between these two methods/models (http://www.biomedcentral.com/1471-2288/2/15/), however the opposite was wrote by Zuur et al. (2009, p. 116; http://link.springer.com/book/10.1007%2F978-0-387-87458-6). Marginal model (generalized estimating equation approach) brings population-averaged parameters, while outputs from random-effects model (generalized linear mixed model) take into account random effect – subject (Verbeke et al. 2010, pp. 49–52; http://link.springer.com/chapter/10.1007/0-387-28980-1_16).

I would like to see some layman-like explanation of these models illustrated on some model (real-life) examples in language familiar to non-statistician and non-mathematician.

In detail, I would like to know:

When should be used marginal model and when should be used random-effects model? For which scientific questions are these models suitable?

How should be outputs from these models interpreted?

Thank you for linking my answer! I will try to give an explicit explanation. This question has been discussed many times at this site (see the related questions on the right side), but it is really confusing and important for a "layman".

First of all, for linear models (continuous response), the estimates of marginal and conditional (random-effects) models coincide. So I will focus on nonlinear models, especially logistic regression for binary data.

## Scientific questions

The mostly used example to distinguish marginal and conditional models is:

If you are a doctor and you want an estimate of how much a statin drug will lower your patient’s odds of getting a heart attack, the subject-specific coefficient is the clear choice. On the other hand, if you are a state health official and you want to know how the number of people who die of heart attacks would change if everyone in the at-risk population took the stain drug, you would probably want to use the population–averaged coefficients. (Allison, 2009)

The two kinds of scientific questions correspond to these two models.

## Illustration

The best illustration I have seen so far is the following figure in Applied Longitudinal Analysis (Fitzmaurice, Laird and Ware, 2011, Page 479), if we change the covariate from "statin drug" to "time". It is clear that the two models differ in the scale of coefficients, which can be essentially explained by the fact that the mean of a nonlinear function of a random variable does not equal the nonlinear function of the mean. ## Interpretation

In the above figure, the dotted lines are from a random intercept model. It shows that we need to control the random effects constant when interpreting the fixed effects, i.e. only go along a line when interpreting the slope. This is why we call the estimates from random effects models "subject specific". Specifically,

• For conditional models, the interpretation is that, how would the log odds change with one unit change of time for a given subject? (See Page 403 of Fitzmaurice, Laird and Ware (2011) about the discussion about why the interpretation of time-invariant covariates in conditional models is potentially misleading.)
• For marginal models, the interpretation is exactly the same as the interpretation of linear regressions, i.e., how would the log odds change with one unit change of time, or the log odds ratio of drug vs. placebo.

There is another example on this site.

• Thank you very much for nice answer! I have yet one question: you have wrote that the estimates of marginal and random-effects models coincide for linear models - does this hold also for random-effects model with random intercepts and slopes, if there are differences in the random slopes? Feb 13 '14 at 6:40
• Yes, the estimates for the fixed effects in random-effects model and the estimates for the mean model in marginal models coincide, regardless of the random effects structure. Feb 13 '14 at 14:24
• Was just wondering if anybody would perhaps happen to have a worked example of both modelling approaches in R? Maybe for this specific example, as it seems quite didactical? Nov 25 '14 at 18:00