How does model averaging with a categorical variable work? I have a series of models (~14) which do not include a categorical variable. One model (#15) however, does have a categorical variable with 3 levels. Normally for this model one of the categories becomes the intercept (typically based on alphabetical order). Simple enough. But what happens when I model average? The other two categories are dependent on this intercept for context, but model averaging will change the intercept! I wish to use full averaging using R package 'MuMIn' if that helps at all.
 A: I don't know MuMln. Nevertheless: If it is only about averaging predictions, it does not matter. They are all on the same scale if the models were fitted on the same response. 
If all models are linear models, you can even calculate the coefficients of the average model. To do so, you would need to average the model equations. This works even if some of the predictors were categorical since they are anyway represented by numeric (0-1) dummy variables.
Example 1: Two models with two identical numeric predictors x, z


*

*Model 1: $\beta_0^{(1)} + \beta_1^{(1)} x + \beta_2^{(1)} z$

*Model 2: $\beta_0^{(2)} + \beta_1^{(2)} x + \beta_2^{(2)} z$


To find the average model, you just sum the two formulas and divide by the number of models, 2.
$$
\frac{1}{2} \cdot [\beta_0^{(1)} + \beta_1^{(1)} x + \beta_2^{(1)} z + \beta_0^{(2)} + \beta_1^{(2)} x + \beta_2^{(2)} z]
$$ 
which can be written as a linear model as well as
$$
(\beta_0^{(1)} + \beta_0^{(2)})/2 + (\beta_1^{(1)} + \beta_1^{(2)})/2  \cdot x + (\beta_2^{(1)} + \beta_2^{(2)})/2 \cdot z
$$
Example 2: Two models with one common and one different numeric predictor


*

*Model 1: $\beta_0^{(1)} + \beta_1^{(1)} x + \beta_2^{(1)} z$

*Model 2: $\beta_0^{(2)} + \beta_1^{(2)} x + \beta_2^{(2)} t$


Again, let's add the two formulas and divide by two
$$
\frac{1}{2} \cdot [\beta_0^{(1)} + \beta_1^{(1)} x + \beta_2^{(1)} z + \beta_0^{(2)} + \beta_1^{(2)} x + \beta_2^{(2)} t]
$$ 
which can be written as a linear model as well as
$$
(\beta_0^{(1)} + \beta_0^{(2)})/2 + (\beta_1^{(1)} + \beta_1^{(2)})/2  \cdot x + \beta_2^{(1)}/2 \cdot z + \beta_2^{(2)}/2 \cdot t
$$
(same as if you would add a 0 coefficient to model one representing the effect of $z$ and vice versa).
Example 3: One model with two numeric inputs $z$ and $t$, the other with a categorical variable with three levels represented by the two dummy variables $x_1$ and $x_2$.


*

*Model 1: $\beta_0^{(1)} + \beta_1^{(1)} z + \beta_2^{(1)} t$

*Model 2: $\beta_0^{(2)} + \beta_1^{(2)} x_1 + \beta_2^{(2)} x_2$


Again, let's add the two formulas and divide by two
$$
\frac{1}{2} \cdot [\beta_0^{(1)} + \beta_1^{(1)} z + \beta_2^{(1)} t + \beta_0^{(2)} + \beta_1^{(2)} x_1 + \beta_2^{(2)} x_2]
$$ 
Here, the formula can only be simplified by averaging the two interceps.
