Екі мәнді регрессия және пробит модельдері

Statistical regression where the dependent variable can take only two values

In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from probability + unit. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model. A probit model is a popular specification for a binary response model. As such it treats the same set of problems as does logistic regression using similar techniques. When viewed in the generalized linear model framework, the probit model employs a probit link function. It is most often estimated using the maximum likelihood procedure, such an estimation being called a probit regression.

The suitability of an estimated binary model can be evaluated by counting the number of true observations equaling 1, and the number equaling zero, for which the model assigns a correct predicted classification by treating any estimated probability above 1/2 (or, below 1/2), as an assignment of a prediction of 1 (or, of 0). See for details.

Consider the latent variable model formulation of the probit model. When the variance of conditional on is not constant but dependent on , then the heteroscedasticity issue arises. For example, suppose and where is a continuous positive explanatory variable. Under heteroskedasticity, the probit estimator for is usually inconsistent, and most of the tests about the coefficients are invalid. More importantly, the estimator for becomes inconsistent, too. To deal with this problem, the original model needs to be transformed to be homoskedastic. For instance, in the same example, can be rewritten as , where Therefore, and running probit on generates a consistent estimator for the conditional probability

When the assumption that is normally distributed fails to hold, then a functional form misspecification issue arises: if the model is still estimated as a probit model, the estimators of the coefficients are inconsistent. For instance, if follows a logistic distribution in the true model, but the model is estimated by probit, the estimates will be generally smaller than the true value. However, the inconsistency of the coefficient estimates is practically irrelevant because the estimates for the partial effects, , will be close to the estimates given by the true logit model. To avoid the issue of distribution misspecification, one may adopt a general distribution assumption for the error term, such that many different types of distribution can be included in the model. The cost is heavier computation and lower accuracy for the increase of the number of parameter. In most of the cases in practice where the distribution form is misspecified, the estimators for the coefficients are inconsistent, but estimators for the conditional probability and the partial effects are still very good. One can also take semi parametric or non parametric approaches, e. g., via local likelihood or nonparametric quasi likelihood methods, which avoid assumptions on a parametric form for the index function and is robust to the choice of the link function (e. g., probit or logit).

The probit model is usually credited to Chester Bliss, who coined the term "probit" in 1934, and to John Gaddum (1933), who systematized earlier work. However, the basic model dates to the Weber–Fechner law by Gustav Fechner, published in , and was repeatedly rediscovered until the 1930s; see and
A fast method for computing maximum likelihood estimates for the probit model was proposed by Ronald Fisher as an appendix to Bliss' work in 1935.