 Quasibinomial regression

The logit mean model $p_i=(1+e^{-x_i^T\beta }{)}^{-1}$ with working variance function $\varphi p_i(1-p_i)$ has the same coefficient estimates as ordinary binomial logistic regression, but estimates dispersion from residuals. For an actual individual binary random variable, $Y^2=Y$ forces variance $p(1-p)$, so a nonunit dispersion is a working specification rather than a new independent binary distribution.