Multiple Regression – Interpreting Betas When There Are Multiple Categorical Variables

categorical datacontrastsinterpretationmultiple regressionregression coefficients

I understand the concept that $\hat\beta_0$ is the mean for when the categorical variable is equal to 0 (or is the reference group), giving the end interpretation that the regression coefficient is the difference in mean of the two categories. Even with >2 categories I would assume each $\hat\beta$ explains the difference between that category's mean and the reference.

But, what if more variables are brought into the multivariable model? Now what does the intercept mean given that it doesn't make sense for it to be the mean for the reference of two categorical variables? An example would be if gender (M(ref)/F) and race (white(ref)/black) were both in a model. Is the $\hat\beta_0$ the mean for only white males? How does one interpret any other possibilities?

As a separate note: do contrast statements serve as a way to method for investigating effect modification? Or just to see the effect ($\hat\beta$) at different levels?

Best Answer

You are right about the interpretation of the betas when there is a single categorical variable with $k$ levels. If there were multiple categorical variables (and there were no interaction term), the intercept ($\hat\beta_0$) is the mean of the group that constitutes the reference level for both (all) categorical variables. Using your example scenario, consider the case where there is no interaction, then the betas are:

  • $\hat\beta_0$: the mean of white males
  • $\hat\beta_{\rm Female}$: the difference between the mean of females and the mean of males
  • $\hat\beta_{\rm Black}$: the difference between the mean of blacks and the mean of whites

We can also think of this in terms of how to calculate the various group means:
\begin{align} &\bar x_{\rm White\ Males}& &= \hat\beta_0 \\ &\bar x_{\rm White\ Females}& &= \hat\beta_0 + \hat\beta_{\rm Female} \\ &\bar x_{\rm Black\ Males}& &= \hat\beta_0 + \hat\beta_{\rm Black} \\ &\bar x_{\rm Black\ Females}& &= \hat\beta_0 + \hat\beta_{\rm Female} + \hat\beta_{\rm Black} \end{align}

If you had an interaction term, it would be added at the end of the equation for black females. (The interpretation of such an interaction term is quite convoluted, but I walk through it here: Interpretation of interaction term.)

Update: To clarify my points, let's consider a canned example, coded in R. 

d = data.frame(Sex  =factor(rep(c("Male","Female"),times=2), levels=c("Male","Female")),
               Race =factor(rep(c("White","Black"),each=2),  levels=c("White","Black")),
               y    =c(1, 3, 5, 7))
d
#      Sex  Race y
# 1   Male White 1
# 2 Female White 3
# 3   Male Black 5
# 4 Female Black 7

enter image description here

The means of y for these categorical variables are: 

aggregate(y~Sex,  d, mean)
#      Sex y
# 1   Male 3
# 2 Female 5
## i.e., the difference is 2
aggregate(y~Race, d, mean)
#    Race y
# 1 White 2
# 2 Black 6
## i.e., the difference is 4

We can compare the differences between these means to the coefficients from a fitted model: 

summary(lm(y~Sex+Race, d))
# ...
# Coefficients:
#             Estimate Std. Error  t value Pr(>|t|)    
# (Intercept)        1   3.85e-16 2.60e+15  2.4e-16 ***
# SexFemale          2   4.44e-16 4.50e+15  < 2e-16 ***
# RaceBlack          4   4.44e-16 9.01e+15  < 2e-16 ***
# ...
# Warning message:
#   In summary.lm(lm(y ~ Sex + Race, d)) :
#   essentially perfect fit: summary may be unreliable

The thing to recognize about this situation is that, without an interaction term, we are assuming parallel lines. Thus, the Estimate for the (Intercept) is the mean of white males. The Estimate for SexFemale is the difference between the mean of females and the mean of males. The Estimate for RaceBlack is the difference between the mean of blacks and the mean of whites. Again, because a model without an interaction term assumes that the effects are strictly additive (the lines are strictly parallel), the mean of black females is then the mean of white males plus the difference between the mean of females and the mean of males plus the difference between the mean of blacks and the mean of whites.

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