logbinomial

Logbinomial refers to statistical models that use a log link for binomial outcomes, commonly known as log-binomial regression. It is used to model the probability of a binary event as p = exp(Xβ), rather than the logit transformation p = exp(Xβ)/(1+exp(Xβ)) used in logistic regression. The parameters β represent the log risk associated with covariates, and exp(βj) gives a relative risk for a one-unit increase in the covariate, holding others constant.

Because p must lie in the interval (0,1), the linear predictor Xβ must be non-positive for all observations, which imposes a set of bound constraints on β. This can make maximum likelihood estimation challenging and cause convergence problems, especially with small samples or covariates with strong effects.

In practice, log-binomial models are sometimes estimated using constrained optimization techniques, or via a common workaround: fitting a Poisson regression model with a log link to the binary outcome and obtaining robust standard errors to produce valid inferences about relative risks. This approach provides a convenient approximation that often works well in epidemiological studies, though it does not strictly estimate the binomial likelihood.

Compared with logistic regression, the log-binomial approach yields risk ratios directly, which can be more interpretable when outcomes are not rare. Limitations include instability and convergence issues, and the need for careful model specification to respect the probabilistic bounds.

Related topics include generalized linear models, link functions, and methods for estimating relative risk, such as Poisson regression with robust standard errors.