How to interpret regression intercept with one dummy coded categorical predictor?

This is nearly the same headline as my previous question, but it turned out I was asking about multiple issues. I still have not understood the more basic issues, which have to do with the interpretation of the intercept and the role of the reference category.

Let's say I do a regression (in this case, a logistic regression) with a single predictor, "color", a dummy coded categorical variable having three categories ("red", "blue", and "green"). The regression and results have two predictor variables, let's say "red" and "blue" with "green" as the omitted reference category.

In the results, the intercept is the log odds when "red" and "blue" are zero. But in that case, it's also the log odds when "green"=1.

The results have an intercept and a test of its significance. If the intercept is significant but the coefficients for "red" and "blue" are not, what does this say about:

• the value "green" as a predictor of the outcome?

• the categorical variable "color" as a predictor of the outcome?

(I have an inkling that the answers may depend on how perfectly "color" predicts the outcome, but I don't know how to talk about this.)

Thanks for your help with these elementary questions.

First, you need to be careful about how your program is coding the categorical variable. Your interpretation is correct if the program uses dummy coding (reference cell coding) but other schemes are possible. What program are you using?

Next, what can we say about color or individual colors predicting the outcome? You are correct that the intercept is the log odds when red = 0 and blue = 0. But the parameter estimates for red and blue are compared to green. If red is (say) higher than green then, perforce, green is lower than red.

Here, as usually, the intercept is probably not of much interest. It doesn't compare green to another color; it will simply be the proportion of outcomes that are "true" when the color is green.

• Thanks for your reply. In my case, I did the coding by hand. If my data were this example, a value is 1 if the case is a member of that category (blue=1) and 0 for the other two.
– Rico
Commented Aug 21, 2013 at 15:14
• What still don't quite get is how a specified category is compared to green alone just because it is omitted. In principle, if red=1, then green must equal zero. That makes sense to me. But if red=1, blue must also = 0. Therefore, how does the coefficient for red not reflect a comparison with all other cases? I realize that the reference category's omission gives it special status, but I don't understand how.
– Rico
Commented Aug 21, 2013 at 15:21
• So, one other thought. You said that the intercept 'will simply be the proportion of outcomes that are "true" when the color is green.' Is the statistical significance of the intercept related to the relationship between the outcome (Y/N) and Green (Y/N), as in a Chi-Square test of independence? (I realize in any case this would not affect interpretation of the logistic regression overall, since the point is to detect differences in the probability of the outcome when comparing colors.
– Rico
Commented Aug 21, 2013 at 15:38
• No, it's related to whether more than half of the outcomes are Y when color = green. Commented Aug 21, 2013 at 15:59