Probit model

In statistics, a probit model is a popular specification of a generalized linear model, 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

:

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

:

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

:

Then it is easy to show that

:

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.

*

ee also

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