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

<math> P(Y=1 \mid X=x) = \Phi(x'\beta), </math>

where Φ is the cumulative distribution function of the standard normal distribution. The parameters β 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

<math> Y^* = x'\beta + \varepsilon, </math>

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

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

0&\text{otherwise}\end{array}\right. </math>

Then it is easy to show that

<math> P(Y=1 \mid X=x) = \Phi(x'\beta). </math>

References

• Bliss, C.I. (1935). The calculation of the dosage-mortality curve. Annals of Applied Biology (22)134-167.

• Bliss, C.I. (1938). The determination of the dosage-mortality curve from small numbers. Quarterly Journal of Pharmacology (11)192-216.

• McCullagh, Peter (1989). Generalized Linear Models. London: Chapman and Hall. ISBN 0-412-31760-5.

See also

• Generalized linear model
• Logit Model
• Multivariate probit models
• Ordered probit and Ordered logit model