# Hypothesis testing with quotient of regression coefficients

Suppose we have the following multiple logistic regression model $\beta_0 + \beta_1 X_1 + \beta_2 X_2$, where $X_1$ and $X_2$ are binary variables, and $\theta = \beta_1 / \beta_2$. Then I have two questions:

1. Since $\theta = \beta_1 / \beta_2$, is it true that $$\frac{\hat \theta - \theta_0}{\hat{\mathrm{se}}(\hat \theta)} \sim \mathcal{N}(0, 1) \enspace ?$$ I mean, can I use the Wald statistic to test whether $\theta = \theta_0$?

2. On the other hand, if we wanted to use a t-test for the same purpose, how many degrees of freedom should we use? I mean, as we are really using 2 coefficients, would it be a t-test with $n - 1$ or $n - 2$ degrees of freedom?

• You should look into the delta method, which can handle exactly this situation. This is by far the easiest way to trace out the standard error. Commented Jun 21, 2016 at 10:59

Consider the hypothesis

$H_0: \frac{\beta_1}{\beta_2}=\theta_0$ against
$H_1: \frac{\beta_1}{\beta_2}\neq\theta_0$

And assume that $\beta_2\neq 0$ (because otherwise the hypothesis isn't really meaningful).

Then we can simply multiply through by $\beta_2$:

$H_0: {\beta_1}=\theta_0{\beta_2}$ against
$H_1: {\beta_1}\neq\theta_0{\beta_2}$

Now recall that $\theta_0$ is a specified constant.

This is now easily seen to be a particular form of general linear hypothesis. It's perfectly amenable to an ordinary $F$ or $t$-test from information that can be extracted from a regression, following standard theory. Indeed, some packages will allow you to test such a hypothesis directly.