Quasilikelihoods

Quasi-likelihoods, or quasi-likelihood methods, are inference tools that provide likelihood-like procedures without requiring a full probability model for the data. Introduced by Wedderburn in 1974, a quasi-likelihood function Q(y; mu) is defined so that its derivative with respect to the mean mu equals (y − mu) / V(mu), where mu = E[Y] and V(mu) is a specified variance function. The function Q is determined up to an additive constant by choosing V(mu); it need not correspond to the log-likelihood of any distribution.

In regression settings, one specifies a mean-variance relationship Var(Y) = V(mu) phi, where phi is a dispersion parameter, and links mu to a linear predictor via mu = g^{-1}(X beta). Parameter estimates are obtained by solving quasi-likelihood estimating equations, often through iteratively reweighted least squares or generalized estimating equations with a working variance function. Quasi-likelihood methods yield consistent estimates of the mean parameters under the correct mean-variance specification and can be robust to certain distributional misspecifications.

Relation to true likelihoods and inference is nuanced: quasi-likelihoods are not generally true likelihoods, so standard likelihood-based tests and information criteria may not be valid unless the variance function aligns with an exponential-family form or robust adjustments are used. Despite this, quasi-likelihood techniques underpin many generalized linear model approaches, especially for overdispersed or heteroscedastic data, and form the basis of the GEE framework with working variance functions to obtain efficient estimates and robust standard errors.

Examples include Poisson and binomial cases with familiar variance forms V(mu) = mu and V(mu) = mu(1 − mu); in the normal case with constant variance, the quasi-likelihood reduces to the Gaussian log-likelihood up to a constant.