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I'm interested in using a Poisson approximation for modeling a common (~40% of the time) binary outcome. But my data has some clustering in it, having come from three different sites, which suggests the use of a random effects model.

Getting the actual estimates are easy enough, using something like lme4 in R:

model <- glmer(Outcome ~ Sex + Age + (1|Site), data=study, family=poisson)

The problem is that Zou G. A modified Poisson regression approach to prospective studies with binary data. Am J Epidemiol. 2004;159(7):702–706 suggests that a robust variance estimator is needed using this method, but the standard sandwich library doesn't work (by, as I understand it, design) with lme4.

Which suggests two questions:

  1. Is a robust variance estimator still needed in this circumstance?
  2. How does one obtain one?
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I would model the sites as fixed effects: you should have the power to do so and it theoretically achieves the same effect as including a random effect. Adjusting for site as either a fixed or random effect makes participants within the same site conditionally independent of each other. The random effects variance estimates becomes a little unwieldy when nearly 50% of the off-diagonal covariance matrix entries are non-zero.

If the model has entirely fixed effects it is amenable to estimation with a GEE and that solves the problem easily.

If you had more sites, another approach would be using the robust sandwich- variance estimate to the generalized least squares model using an exchangeable correlation structure for site-based clusters. Random intercepts and exchangeable correlation lead to equivalent estimates when the intracluster correlation is positive and that's usually the case. Sandwich-variance calculation can be done by-hand as shown here.

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