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I am currently interpreting some glm's and glmm's based on distributions with log link functions (gaussian - log, and negative binomial) and have started going in a bit of a loop regarding the interpretation of the parameter estimates for the fixed effects. I've found some good posts on CV regarding the interpretation of coefficients where they represent log-odds and odds ratios (1, 2, 3), but not in the case of linear predictors of continuous responses. I've been reading around in everything from primary literature to blogs (a couple of good ones; 4, 5), but the discussions I've come across always tend back toward the different approaches to model selection, or don't consider the random component of mixed models.

From what I understand, in both GLMs and GLMMs, the reported parameter estimates are on the scale of the linear predictor (ie on the opposite side of the link function to the response) and that they are both "conditional" on the chosen distribution and link functions. Furthermore, for GLMMs specifically, the fixed effect parameter estimates are also "conditional" on the grouping within the random effect(s).

If my understanding up to this point is correct, then my questions are:

  1. Can inverse-link transformed parameter estimates from glm's (e.g. exp(beta) for log-linked models) be considered analogous to marginal effects and interpreted as mean population level effects?

  2. How does one interpret the conditional parameter estimates for the fixed effects of glmm's (with the random effects), is it simply by recognising that the reported effect is conditioned on the random effects?

I'm going in loops so any clarification to set me straight or suggested targeted reading would be appreciated - cheers.

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Indeed in GLMs and because you have no random effects, the inverse-link transformed regression coefficients have an interpretation for the for the mean of the outcome.

However, in GLMMs and because there you do have random effects, the inverse-link transformed regression coefficients have an interpretation for the for the mean of the outcome conditional on the random effects. Most often you are interested in the marginal mean of the outcome averaged over the random effects distribution, but the coefficients you obtained from the GLMMs do not have this interpretation.

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  • $\begingroup$ Thanks for the response Dimitris. Can I ask for further clarification: If one is looking to draw some general inferences, would it be better to drop the random effects, re-estimate the fixed effects (does this "average over the random effects") and use the inverse-link transformed coefficients, or is it better to just interpret the fixed effects as conditional on the random effects? Alternatively, is it possible to "average over the random effects", after model fitting, in an alternative method? $\endgroup$ – Aaarrrgh's My Game Feb 14 at 10:01
  • $\begingroup$ The reason why you're using the random effects is that you want to account for the correlations in the clusters/groups you have. For example, you have students that are clustered in schools, i.e., outcomes measured on students from the same school are correlated, whereas students from different schools are assumed independent. In settings like this, ignoring these correlations (i.e., what you mentioned dropping the random effects) may have a substantial impact on inferences. $\endgroup$ – Dimitris Rizopoulos Feb 14 at 19:07
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    $\begingroup$ And yes, it is possible to average over the random effects and obtain coefficients with a marginal/population interpretation. For example, check function marginal_coefs() from the GLMMadaptive packages (drizopoulos.github.io/GLMMadaptive/articles/…) $\endgroup$ – Dimitris Rizopoulos Feb 14 at 19:09

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