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When fitting a GLMM with family=poisson(link="log") in lme4, I understand that the coefficients for the estimates of fixed effects in the summary() output must be exponentiated to be back on the scale of the original data.

  1. are the standard errors for fixed effects in the summary() output also logged and need exponentiatiating? I have seen conflicting answers on other questions, some saying the standard errors are on the original data scale and others saying they have also been logged like the estimates
  2. do the random effect standard deviations/variances need exponentiating or are they unaffected by the log link so are still on the original data scale anyway?
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Everything is on the log (expected rate) scale. However, note that standard errors don't transfer easily to the exponentiated scale, because what's a symmetric sampling distribution on the log scale isn't on the exponentiated scale. So, if you want something like confidence intervals constructed as estimate +- normal quantile * SE, you calculate those on the log scale and exponentiate the confidence interval limits. Similarly, of you want to use the random effects (either the specific estimate for a unit in the data used to fit the model, or a new previously unseen unit for which we draw from the random effects distribution), you add those on the log-scale and then exponentiate.

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