I want to perform logistic regression in R, where one of my predictors, $x_i$ is categorical (takes on the values A, B, and C). A is the reference category. The model is set up as follows:
$logit(P(Y_i = 1)) = \beta_0 + \beta_1I(x_i=B) + \beta_2I(x_i=C)$
I want do a hypothesis test for whether the percentage chance of success is the same given $x_i = A$, $x_i = B$, or $x_i = C$. For example, to test if there is a significant difference in probability of success between $x_i = A$ and $x_i = B$, I can test the null hypothesis, $H_0: \beta_1 = 0$. To test if there is a significant difference in probability of success between $x_i = A$ and $x_i = C$, I can test the null hypothesis, $H_0: \beta_2 = 0$.
But how do I test whether there is a significant difference between $x_i = B$ and $x_i = C$? My confusion is that $A$ is the reference category, so I don't know how to directly test the hypothesis.
UPDATE: For the difference between B and C, would I be testing $H_0: \beta_2 - \beta_3 = 0$? And if so, how could this be done from the logistic regression output in R?