Quasi-likelihood specifies a mean and a variance function without necessarily specifying a full response distribution. For independent responses, the estimating equation is . A common scalar dispersion parameter rescales coefficient covariance while leaving the roots of the quasi-score equation unchanged.
The logit mean model with working variance function has the same coefficient estimates as ordinary binomial logistic regression, but estimates dispersion from residuals. For an actual individual binary random variable, forces variance , so a nonunit dispersion is a working specification rather than a new independent binary distribution.
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Quasi-likelihood is a statistical framework used to estimate parameters in models where the likelihood function may not be fully specified or is difficult to derive. It extends the concept of likelihood by using a quasi-likelihood function that approximates the true likelihood of the observed data. The quasi-likelihood approach is particularly useful in situations where the distribution of the response variable is unknown or when the underlying data-generating process is complex.