Betaregressio

Betaregressio, sometimes referred to as beta regression, is a regression framework for modeling a response variable that lies strictly between 0 and 1. It is well suited for proportions, rates, or fractions that are not bounded by discrete counts. The response is assumed to follow a Beta distribution with a mean parameter mu_i in (0,1) and a dispersion (or precision) parameter phi > 0.

In the standard parameterization, the model specifies y_i ~ Beta(mu_i * phi, (1 - mu_i) * phi). The mean of y_i is mu_i, and the variance is mu_i(1 − mu_i) / (1 + phi). The mean mu_i is linked to a linear predictor via a link function: g(mu_i) = eta_i = X_i^T beta, where X_i is the vector of covariates for observation i and beta is the coefficient vector. The link function is typically chosen from common options such as logit, probit, or cloglog.

Estimation and inference are usually performed by maximum likelihood, using GLM-like software. Standard errors are obtained from the observed information matrix. In many applications, phi is treated as a constant across observations, but extensions allow phi to vary with covariates or to be modeled with random effects.

Extensions and variations include zero- or one-inflated beta regression to handle observations that are exactly 0 or 1, beta regression with covariate-dependent dispersion, and beta regression for hierarchical or mixed-effect data. Beta regression is used in fields such as ecology, economics, psychology, and biomedical research to model proportions and rates with realistic heteroscedastic behavior.

The Hungarian term betaregressio is associated with the same modeling approach, emphasizing its role in analyzing outcomes restricted to the unit interval.