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 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

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 ).

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. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: publisher location (link)

References

See also