Probit

In probability theory and statistics the probit function is the inverse cumulative distribution function, or quantile function of the normal distribution.

The probit function is often denoted as $\Phi ^{-1}$ and is of type:

$\Phi ^{-1}:[0;1]\to (-\infty ;+\infty )\!$

Plot of probit function

Like the logit (log odds) function, it may be used to transform a variable $p$ ranging over the interval $[0,1]$ into a derived quantity $\Phi ^{-1}(p)$ ranging over the real numbers. This has applications in probit models, which are generalized linear models.

The probit function may be expressed in terms of the inverse of the error function (this can almost a definition of the error function):

$\Phi ^{-1}(p)={\sqrt {2}}\,\operatorname {erf} ^{-1}(2p-1)$

The probit model was developed by C. I. Bliss in 1934. A popoular, but almost identical, alternative to probit is the logit model.

Bliss, C. I. (1934). The method of probits. Science 79:38-39.