Reference level and Interpretation

Asked Mar 10 '14 at 15:43

Active Jan 09 '24 at 14:50

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Suppose that we have a categorical variable $X$ that can take three values: $0$, $1$, or $2$. We use $X=0$ as the reference level. The following dummy variables are created: $$X_1 = 1 \ \text{if} \ X=1 \ \text{or} \ X_1=0 \ \text{if} \ X \neq 1$$

$$X_2 = 1 \ \text{if} \ X=2 \ \text{or} \ X_2=0 \ \text{if} \ X \neq 2$$

In a linear regression model, suppose we have $$Y = \beta_1 X_1 + \beta_2 X_2 + \epsilon$$

We know that $\beta_1$ compares $X=1$ vs $X=0$ and $\beta_2$ compares $X=2$ vs $X=0$. How would we get the coefficients that compares $X=2$ vs $X=1$? Would it be $\beta_1+\beta_2$?

asked Mar 10 '14 at 15:43

topguypoland

1

You don't mean B2-B1? – charles Mar 10 '14 at 15:49
@Charles: How did you get that? – topguypoland Mar 10 '14 at 15:50
2

This question, suitably generalized, is discussed at http://stats.stackexchange.com/questions/38337/. Searching our site turns up a few hundred closely related posts: please look some of them over. – whuber Mar 10 '14 at 15:52
@Charles: Is it $Y = \beta_{2}X_{2}+ \epsilon-\beta_{1}X_{1}+\epsilon = \beta_{2}X_{2}-\beta_{1}X_{1} = \beta_2-\beta_1$? – topguypoland Mar 10 '14 at 15:53
@topguypoland. yes – charles Mar 10 '14 at 16:11
@charles No: the two instances of "$\epsilon$" on the left hand side represent two independent random variables; they do not cancel like that (even when the second one is subtracted as intended). However, when you apply the expectation operator you obtain a correct statement. – whuber Mar 10 '14 at 20:50

Reference level and Interpretation

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