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Based on the emmeans vignette on transformations and link functions, it seems like bias adjustment is recommended when back-transforming estimated marginal means from linear models or linear mixed effects models where the response is transformed [i.e., lm(log(y)...) or lmer(log(y)...)] or generalized linear models with random effects [i.e., glmmTMB(y~ ... (1|RE)].

However, when using random effects in linear mixed-effects models without transformations (i.e., lmer(y ~ ... + (1|RE))), is bias adjustment still necessary to ensure that the estimated marginal means reflect the expected means of the response?

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No. The bias adjustment is a correction on the transformation. If the inverse transformation is the function $h$, then without bias adjustment you are estimating $h(E(Z))$ where $Z$ is the transformed response, and with bias adjustment you are estimating $E(h(Z)) = E(Y)$ where $Y$ is the untransformed response. With no transformation, $h$ is the identity and these are not different.

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