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What is the relationship between regression coefficients for categorical variables and their estimated marginal means?

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The regression coefficients may be used to make predictions. Marginal averages of such predictions over a regular grid produce estimated marginal means.

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  • $\begingroup$ In the case of a categorical variable, shouldn't it be possible to calculate the estimated means from each treatment using the coefficients. If you get predicted means from coefficients, would they aggree with the estimated marginal means? If not, why? $\endgroup$ Commented Mar 29, 2020 at 15:07
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    $\begingroup$ Yes. It is possible. That's what is stated in my answer. Follow those two steps: (1) make predictions for each factor combination; these are of the form A*b for some matrix A, where b is the vector of regression coefficients. (2) Obtain marginal averages of those predictions. That is equivalent to averaging together the corresponding rows of A. The resulting averaged rows of A give you the linear functions of the regression coefficients corresponding to the EMMs. $\endgroup$
    – Russ Lenth
    Commented Mar 29, 2020 at 17:54
  • $\begingroup$ See the vignette on basics of EMMs: cran.r-project.org/web/packages/emmeans/vignettes/basics.html $\endgroup$
    – Russ Lenth
    Commented Mar 29, 2020 at 17:56
  • $\begingroup$ Thanks Russ, that helps. I had read through the vignette but I didn't find as succinct and clear an explanation. I'm trying to decide if I should plot EMMs in response scale, or coefficients in response scale. $\endgroup$ Commented Mar 29, 2020 at 17:57
  • $\begingroup$ It makes no sense to put the coefficients on the response scale. They are only coefficients for the linear predictor -- not the predictions themselves. $\endgroup$
    – Russ Lenth
    Commented Mar 29, 2020 at 18:29

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