Dummy variables in multiple regression, why use an intercept?

Question

When performing a multiple regression with dummy variables, is it really necessary to include an intercept term in the design matrix?

By dummy variables, I mean indicator variables; a one in the design matrix if some effect is present, and a zero if not. It seems to me that without the intercept it is simpler to interpret the OLS solution. Instead of

$\beta_{0}$ = $\mu_{A}$ (where $\beta_{0}$ is the intercept)

$\beta_{1}$ = $\mu_{B} - \mu_{A}$

$\beta_{2}$ = $\mu_{C} - \mu_{A}$

etc.

We have

$\beta_{1}$ = $\mu_{A}$

$\beta_{2}$ = $\mu_{B}$

$\beta_{3}$ = $\mu_{C}$

etc.

Do the computations of $R^{2}$, the F-statistic and t-statistics change?

What if a continuous independent variable is then included?

Answer 1

Things like the predictions, residuals, full-reduced model tests, etc. will not be affected by the change that you propose, but what does change is the interpretation and tests on the individual terms.

Most regression routines will provide an automatic test of whether a term is 0 or not. This is meaningful when a term represents the difference between two group means (what we get when we include an intercept), but is testing whether each of the group means equals 0 meaningful? The same goes for the confidence intervals and we usually want to know if groups differ from each other. If every term just represents a mean then we compute confidence intervals for the means, then people try to interpret the amount of a difference by seeing if the intervals overlap, but this is very inferior to looking at a confidence interval on a difference.

Answer 2

@GregSnow is right that this change doesn't really matter. Let me add a few details to extend that. What you are talking about is sometimes called level means coding, whereas the default coding scheme is called reference level coding. Note that there are many possible valid coding schemes. If you have a categorical variable with only two levels, then the t-test of the beta that comes by default in the regression output with any statistical software is more meaningful when you had used reference level coding.

On the other hand, when you have a categorical variable with multiple categories, which coding scheme you use is just a matter of taste. To get a predicted value, you will have to solve the regression equation for $\hat y$ at the relevant spot in the covariate space either way. To test if the categorical variable is related to the response, you will need to use an F change test (discussed here) either way, etc.

To address your specific questions more concretely: if you use level means coding instead of reference level coding, the F-statistic will be the same; the t-statistics will change in that they will now test whether your level means are zero, as @GregSnow explains; and the inclusion of continuous covariates will be the same.