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Several sources recommend reporting regression coefficients in a table for every mixed-effects model. For continuous predictors that's fine because I only get one coefficient for that predictor. But what if I have categorical predictors with more than two levels? For example

mod <- lmer(angle~temp*recipe + (1|replicate), data=cake)
summary(mod)

Fixed effects:
               Estimate Std. Error         df t value   
(Intercept)    2.379365   6.199942 262.454162   0.384  
temp           0.153714   0.029819 250.000012   5.155
recipeB       -3.649206   8.464773 250.000006  -0.431  
recipeC       -1.941270   8.464773 250.000006  -0.229  
temp:recipeB   0.010857   0.042170 250.000006   0.257   
temp:recipeC   0.002095   0.042170 250.000006   0.050

How should I report this information? Should I report the coefficients for temp, recipeB and recipeC and their corresponding std. errors? Recipe has 3 levels (A,B,C) and mod used A as reference. Should I also change the reference category in recipe (eg to B) and re-run the model? What about the interactions? Does it make sense to include all of the different combinations in the table?

EDIT: Model with two categorical variables, three levels each

cake2 <- cake[which(cake$temp < 200),]
mod <- lmer(angle~temperature*recipe + (1|replicate), data=cake2)
summary(mod)

Fixed effects:
                       Estimate Std. Error        df t value    
(Intercept)            29.33333    1.67210  17.27956  17.543 
temperature.L           3.44125    1.12498 112.00000   3.059 
temperature.Q          -0.08165    1.12498 112.00000  -0.073  
recipeA                 1.15556    0.91854 112.00000   1.258     
recipeC                 0.20000    0.91854 112.00000   0.218  
temperature.L:recipeA  -2.26274    1.59096 112.00000  -1.422   
temperature.Q:recipeA  -1.19753    1.59096 112.00000  -0.753     
temperature.L:recipeC  -0.75425    1.59096 112.00000  -0.474    
temperature.Q:recipeC   0.81650    1.59096 112.00000   0.513

Here I'm not sure what is irrelevant and can be be left out of the table. For example the way the model is set up, the reference is always temperature=175 and recipe=B. I think I should also report the interaction effects using other references right? Or will readers still be able to calculate the values of the other effects only using the values from the table above?

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  • $\begingroup$ You should include everything, so that you won't be accused of obfuscation. If you habe too many levels you could show the output of an ANOVA. $\endgroup$ – user2974951 Jan 15 '19 at 9:02
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Regardless of where, why, & to whom you're reporting results, some general considerations are likely to apply.

In general tabulating some coefficient estimates but not others may well cause confusion about what model you've in fact fitted; & in particular reporting coefficient estimates for "main effects" but not for the interactions in which they participate is not very informative. In this case if you excluded interactions from the table we'd learn about the effect of temperature only for Recipe A, the reference level (from the temp coefficient). The interactions (temp:recipeB, temp:recipeC) show the effect of temperature for the other two recipes, so there's no point in not showing them.

However, reporting on various models that differ only in how they're parametrized is likely over-kill. (Just two three-level categorical predictors, & you've got nine different ways to specify the reference levels.) Readers can easily enough calculate estimates & standard errors for any contrasts they might be interested in that aren't explicitly tabulated. For example, the recipeB coefficient represents the effect of changing from Recipe A to Recipe B at 0°F, while recipeC represents the effect of changing from Recipe A to Recipe C at 0°F. If anyone's wondering about the effect of changing from Recipe B to Recipe C at 0°F, its the difference between recipeC & recipeB; there's no need to re-fit the model using Recipe B as the reference level.

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  • $\begingroup$ thanks for your answer, but I'm a bit confused. Your second paragraph suggests to me I should include interactions with other references, otherwise we only learn about the effect of temperature for recipeA. But in the third paragraph you seem to suggest that reporting all those coefficients for the different interaction effects (using different references) is less than ideal. $\endgroup$ – locus Jan 16 '19 at 0:54
  • $\begingroup$ I edited my question to depict exactly your example of two categorical predictors with three levels. For model mod, what information would you present in the table that avoids the over-kill, but includes enough information that allows readers to do calculate any contrasts they like? $\endgroup$ – locus Jan 16 '19 at 0:55
  • $\begingroup$ I've rolled back that edit, as part of my answer is concerned with the model you first presented. I'll expand that answer to be clearer on the points you've raised in your comment, & then you can decide if you still want to ask something about the model with two categorical variables - by appending it to the question or by asking a new question, whichever seems best. $\endgroup$ – Scortchi - Reinstate Monica♦ Jan 16 '19 at 9:34
  • $\begingroup$ Thanks @Scortchi that was very helpful. I think the problem is that I still struggle with the interpretation of the output from the summary() function, but your answer helped. I understood the part with the main effects, but deciding which interaction effects with which reference level I should report still bothers me. I appended the example with two categorical variables you mention, in case you have some time to respond. $\endgroup$ – locus Jan 17 '19 at 1:56

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