Suppose a logistic regression model has three binary explanatory variables $x_1$, $x_2$ and $x_3$ used to estimate the probability of success. This model includes all three main effects, the three $2$- way interactions, and the one $3$-way interaction.
Derive the odds ratio to examine the odds of a success across two levels of one of the explanatory variables, say $x_1$, and derive its variance.
I assume you start by:
$\log(\frac{\pi}{1-\pi}) = \beta_1x_1 + \beta_2x_2 + \beta_3x_3 +\beta_4x_1x_2 + \beta_5x_1x_3 + \beta_6x_2x_3 + \beta_7x_1x_2x_3$
and then taking exponetials on both sides:
$\frac{\pi}{1-\pi} = e^{\beta_1x_1 + \beta_2x_2 + \beta_3x_3 +\beta_4x_1x_2 + \beta_5x_1x_3 + \beta_6x_2x_3 + \beta_7x_1x_2x_3}$
However I am not sure how to proceed.