Tag Info

This refers generally to statistical procedures that utilize the probit function. The primary example of which is probit regression where the probit transformation of the parameter p of a binary response distribution is used as a link.

The probit function is the inverse standard normal cumulative distribution function. That is, it takes in a probability and outputs a z-score. One important use of the probit is as a link function for the binomial distribution in the Generalized Linear Model.

For GLMs, we motivate the probit link function as follows:

Instead of modeling the response $Y$ directly in terms of $X$, we can model it through a latent variable, $Z$.

$$Z = X'\beta - \epsilon$$

Where $\epsilon_i \sim F( \cdot)$, i.i.d from some distribution with cdf $F$. In binary regression, we could classify observations, $Y_i$ as class 1 if $Z \ge 0$ or class 0 otherwise. So

$$P(Y=1|x) = P(Z \ge 0 ) = P(X'\beta - \epsilon \ge 0)$$ $$=P(\epsilon \le X'\beta) = F(X'\beta)$$

When $F$ is a Normal distribution, then the expression above is $\Phi(X'\beta)$, corresponding to a probit regression.

'Probit' is short for probability unit. The idea was originally proposed in the 1930s and predates the (now more common) logistic function.