Probit model

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In statistics, a probit model is a popular specification of a generalized linear model, using the probit link function. 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

\Pr(Y=1|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 \ \stackrel{\mathrm{def}}{=}\ 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

\Pr(Y=1 | X=x) = \Phi(x'\beta).

[edit] 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.

[edit] See also

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