Probitform

Probitform refers to the probit form of a statistical model used for binary outcome data. It denotes the representation in which the probability of a positive outcome is linked to a linear predictor through the standard normal cumulative distribution function (CDF). In this form, the latent propensity z_i = x_i^Tβ + ε_i with ε_i ~ N(0,1) yields Y_i = 1 if z_i > 0. Consequently, P(Y_i = 1 | x_i) = Φ(x_i^Tβ), where Φ denotes the standard normal CDF. The probit transform, η_i = Φ^-1(P(Y_i = 1 | x_i)), is linear in the covariates: η_i = x_i^Tβ. This relationship underpins the probit model and defines the probitform.

Estimation of the probitform is typically done via maximum likelihood methods, leveraging the cumulative normal distribution in the likelihood function. Software implementations include glm with a probit link in various environments (for example, R’s glm(..., family = binomial(link = "probit"))) and corresponding probit-based discrete choice models in Python’s statsmodels. Bayesian approaches also employ data augmentation with latent variables to fit probit models.

Probitform is commonly contrasted with the logit form, which uses the logistic CDF as the link function. While the two forms yield similar fitted probabilities in many applications, they imply different underlying error distributions and have distinct interpretive characteristics for coefficients. The choice between probitform and logit form often rests on theoretical considerations, historical precedent, or software convenience.

Limitations of the probitform include sensitivity to sample size and potential interpretability challenges for direct effect sizes, particularly when translating coefficients into probabilities. See also probit model, logistic regression, and link functions.