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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?

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  • $\begingroup$ 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. $\endgroup$ – Repmat Jun 21 '16 at 10:59
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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.

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