Interpretation of Odds in Probit Regression

Question

Logistic regression is concerned about modelling log-odds, i.e. logits. Hence, the odds of the computed probabilities can be interpreted accordingly. However, when estimating a probit model, one could also take the probabilities and compute odds. However, probit is not based on modeling odds but on the cdf of the standard normal.

So, does it make sense to compute these odds for probit models anyway? If yes, how can it be interpreted and what is exactly the difference in interpretation compared to odds in logistic regression?

Answer 1

The point of the odds ratio interpretation in logistic regression is that logistic regression is a linear model for the log odds of success. So a unit increase in an explanatory variable will result in increase or decrease of the predicted odds by a factor of $\exp(b)$, regardless of where on that explanatory variable you started or what the values of the other explanatory variables are (assuming you did not include interaction effects).

You can ofcourse compute odds ratios by converting the predicted probabilities of a probit model to odds and compute the ratio, but these odds ratios will change depending on whether you compared $x=4$ with $x=5$ or $x=40$ with $x=41$, and on the values of the other explanatory variables. So that is much less useful as a summary measure of the effect.

Answer 2

The thread has long been inactive, but I'd like to add one thing to Maarten Buis's already perfect answer.

The logit model is $\log( p(X)/(1-p(X)) ) = X\beta$, where $p(X) = P(Y=1|X)$. When $X$ increases from $x$ ($\mathit{before}$, say) to $x+\Delta x$ ($\mathit{after}$, say), the log-odds increases by $\Delta x\beta$: $$\log (\mathit{odds}_{after}) - \log (\mathit{odds}_{before}) = (x+\Delta x)\beta - x\beta = \Delta x\beta.$$ Since $\log(a)-\log(b)=\log(a/b)$, the LHS is the log odds-ratio: $$\log \left( \frac{\mathit{odds}_{after}}{\mathit{odds}_{before}} \right) = \Delta x\beta,$$ and thus the odds-ratio is $\exp(\Delta x\beta)$. When other regressors are held fixed and only $x_j$ increases by 1 unit, $\Delta x\beta = \beta_j$ and the corresponding odds-ratio is $\exp(\beta_j)$. That's why odds ratios are computed as $\exp(\hat\beta_j)$s from logit regression. Conveniently, the odds ratios do not depend on the initial $x$ but only on $\Delta x$.

The probit model is $\Phi^{-1} (p(X)) = X\beta$, i.e., $p(X) = \Phi(X\beta)$, where $\Phi(\cdot)$ is the standard normal CDF. The associated odds-ratio is $$\frac{\Phi(x\beta+\Delta x\beta)}{1-\Phi(x\beta+\Delta x\beta)} \bigg/ \frac{\Phi(x\beta)}{1-\Phi(x\beta)},$$ and just that, no further simplification. This odds-ratio depends on $x$ and $\Delta x$ given $\beta$. You can compute the odds-ratios for all observations in the sample with $x$ equal to $x_i$ for some given $\Delta x$. The $\exp(\beta_j)$ formula, or $\exp(\Delta x \beta)$ in general, makes no sense for probit.