# Common approaches to illustrating “central” responsiveness in linear regression predictions when including categorical variables?

Consider the simple linear regression framework: $y=\beta_0+\beta_T T+X\beta_X+\varepsilon$

Where $T$ is an indicator for treatment in an RCT and $X$ is a vector of controls.

One common approach I know of to illustrate the impact of treatment for "typical" individuals (or whatever the observation level is) on $y$ in such a framework is to simply get $\hat{y}$ by setting all the $X$ to their sample averages (or perhaps some other measure of central tendency, like median) like so:

$\hat{y}_{control}=\hat{\beta}_0+\bar{X}\hat{\beta}_X$

$\hat{y}_{treatment}=\hat{\beta}_0+\hat\beta_T+\bar{X}\hat{\beta}_X$

There are of course drawbacks to this approach, but that's not my primary concern here.

I'm wondering what are some options available to illustrate this sort of "central responsiveness" when $X$ contains categorical variables--e.g. geographic region or martial status--that can't be ordered or assigned numeric values (so plain averaging is meaningless/impossible).

That is, let's suppose the framework is more specifically:

$y=\beta_0+\beta_T T+\beta_{married}D_{married}+\beta_{divorced}D_{divorced}+X\beta_X+\varepsilon$

What are some options/commonly used approaches to repeating the above exercise in this particular setup?

Note that I have no interest in providing group-by-group breakdowns; I strongly prefer to have two simple summary numbers, $\hat y_{control}$ and $\hat y_{treatment}$.

Note that this was inspired by my related post at SO; I've tried to translate things into more general terms than that specific problem, and I hope I've done so clearly.

• Use the sample proportions of each categorical variable. – Jason Sanchez Jun 21 '15 at 0:25
• that was exactly my approach. it was, however, rebuffed (see the comments on my SO post); could you provide some references/more justification for why this is your suggestion? – MichaelChirico Jun 21 '15 at 0:32

Use the sample proportions of each category.

For simplicity, imagine a linear regression includes a three class categorical variable (Single, Married, and Divorced) dummied so Single is the reference case. The response variable is household income and the estimated parameter for Married, $\beta_{Married}$, is +40,000 while that for Divorced, $\beta_{Divorced}$, is -5,000.

Remember that these $\beta$ shift the intercept above or below the base case, Single. So in this example someone who is married is expected to make \$40,000 more than someone who is single (given the other$X$variables are the same.) If someone is married, we could just add 40,000 to the intercept term and the rest of the equation is identical. Any good method of finding the "average" of a categorical variable should be able to clearly handle the obvious cases. First, if the vast majority of the category belongs to one class, the "average" class effect should look very similar to the predominant class effect. By using class proportions, if the vast majority of the data set is married, the "average" person would have income very near that of a married person. Similarly, if the vast majority is single, no major adjustment (intercept shift) would be made from the base case. Second, if half of the categorical variable is one class and nearly half of it is another class, then the "average" effect should be somewhere between the two classes. If half of the observations are married and the vast majority of the other observations are single, it makes sense that the "average" class has a \$20,000 higher $y$ value than someone who is just single.

This second point really gets to the crux of the argument. If half of the observations are married and the other half are single then the expected intercept shift due to marital status is 20,000, which is what is exactly accomplished using the class proportions method.

By using the class proportions, you are weighting the $\beta$ terms so the net effect is the expected intercept shift based on the distribution of classes for the category. To me this seems ideal.

Although this example has focused on linear regression, I hope you could see how it would easily apply to logistic regression. instead of influencing real values, you're finding the expected intercept shift of the odds ratio.

• This jibes exactly with my initial intuition; thanks for spelling out the reasoning so carefully as I'd neglected to do so – MichaelChirico Jun 21 '15 at 13:27
• I'd still love to see a reference for confirmation--either a text or a well-published/cited paper that uses this approach – MichaelChirico Jun 21 '15 at 13:28
• I think this can be a reasonable approach if the rationale for doing so is provided. Even if you find a paper that uses this method, you'll still need to provide the rationale for your application. It might be that the proportions used would be found from a US Census or some known industry standards rather than the proportions you happened to have in your sample. In addition, this somewhat falls into the category of "ignoring interaction terms" so, again, providing a justifiable rationale is critical and application specific. – JimB Jun 21 '15 at 15:36