Probit model

In statistics, a probit model is a popular specification of a generalized linear model. In particular, it is used for Binomial regression using the probit link function. A probit regression is the application of this model to a given dataset. Probit models were introduced by Chester Ittner Bliss in 1935, and a fast method of solving the models was introduced by Ronald Fisher in an appendix to the same article. Because the response is a series of binomial results, the likelihood is often assumed to follow the binomial distribution. Let Y be a binary outcome variable, and let X be a vector of regressors. The probit model assumes that


 * $$ P(Y=1 \mid X=x) = \Phi(x'\beta), $$

where &Phi; is the cumulative distribution function of the standard normal distribution. The parameters &beta; are typically estimated by maximum likelihood.

While easily motivated without it, the probit model can be generated by a simple latent variable model. Suppose that


 * $$ Y^* = x'\beta + \varepsilon, $$

where $$ \varepsilon | x \sim \mathcal{N}(0,1) $$, and suppose that $$ Y $$ is an indicator for whether the latent variable $$ Y^* $$ is positive:


 * $$ Y \ := \  1_{(Y^* >0)}=\left\{\begin{array}{ll}1&\text{if}\ \ Y^* >0\\

0&\text{otherwise}\end{array}\right. $$

Then it is easy to show that


 * $$ P(Y=1 \mid X=x) = \Phi(x'\beta). $$