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I am working on a measurement error model for presence/absence data using a variant of a Monte Carlo EM (MCEM) algorithm for model fitting. As part of this approach, I simulate random effects by taking random draws from a known distribution [1]. I then need to pass these simulated random effects as an offset to a logistic GLM.

Most comments I have found on specifying offsets in a GLM address Poisson regression with a log-link or Binomial regression when adjusting the number of trials (Using offset in binomial model to account for increased numbers of patients, How to formulate the offset of a GLM).

I am wondering how I might write my model down in statistical notation to see how these offsets should be correctly specified (similar to what is shown here, but for my purposes).

My instinct is that they should be specified linearly, as in

glm(y ~ W1 + W2, 
    offset = V_sim,
    family = binomial(link = "logit"))

where y is Bernoulli, W1 and W2 are covariates, and V_sim is my vector of simulated random effects. However, from the comments on some of these posts, it seems that passing offsets in the logistic setting might be different from other settings.


[1] The variance of the random effects is estimated in a previous GLMM fit with a similar model form to the GLM with offsets, except that group membership for the random effects was specified as a random intercept (instead of being specified as an offset).

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