Section 13.4 Alternative Specification with Different Reference Category
One final thing I want to show you is that our results will be in a slightly different format but will be in one sense equivalent if we decide to use a different category as our omitted category when using a qualitative independent variable. Let’s say we want to make Black (race_2) our reference category. Compare the following results to the previous regression:
. reg vote_3 race_1 race_3 race_4 race_5 race_6
Source | SS df MS Number of obs = 3,036
-------------+---------------------------------- F(5, 3030) = 2.23
Model | .20833556 5 .041667112 Prob > F = 0.0490
Residual | 56.6836275 3,030 .018707468 R-squared = 0.0037
-------------+---------------------------------- Adj R-squared = 0.0020
Total | 56.8919631 3,035 .018745293 Root MSE = .13678
------------------------------------------------------------------------------
vote_3 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
race_1 | .0142558 .0082213 1.73 0.083 -.0018642 .0303757
race_3 | .0066378 .017388 0.38 0.703 -.0274557 .0407313
race_4 | -.0063492 .0402287 -0.16 0.875 -.0852274 .072529
race_5 | .0022345 .0118186 0.19 0.850 -.0209387 .0254077
race_6 | .0444983 .0147623 3.01 0.003 .015553 .0734435
_cons | .0063492 .0077064 0.82 0.410 -.0087611 .0214595
Now, our constant tells us that a Black voter has a .6% chance of voting third-party. This is the same prediction we would get from our prior model where race_1 was the omitted category: to find our prediction for Black voters from the prior results, we would have added the coefficient for race_2 (-.014) to the constant (.021), yielding .006 or .6% (or .007 if we use the rounded numbers shown in parentheses).
The coefficient for race_1 tells us about how White voters differ from Black voters. Notice that the p-value is exactly the same as what we saw in the prior table for race_2, and the coefficient for race_1 in this table is the same as the coefficient for race_2 in the prior table, except the sign has changed. That’s because comparing White to Black is the same as comparing Black to White, except that we’re going in the opposite direction.
If you download the data yourself and have access to statistical software, you can go on to play around with these two sets of results more on your own if you’d like. Both regression equations will yield the same prediction for a voter of any given race. The difference lies only in the starting point, as represented by the constant. However, the p-values will usually differ because they are describing a different comparison (e.g., comparing Asian to Black in this table versus comparing Asian to White in the prior table). Thus, it doesn’t really matter which category you pick as your omitted category, except that you may care more about some comparisons than others. You can also run the same regression multiple times but with different omitted categories so that you can get the p-values for a full set of comparisons across groups.
