# reporting regression results with categorical variables and interactions

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?

• 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. Jan 15, 2019 at 9:02

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.
• 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? Jan 16, 2019 at 0:55
• 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. Jan 17, 2019 at 1:56