For a GLM *η*_{i} = *g*(μ_{i}) = ∑_{j}β_{j}*x*_{ij}, the likelihood equations

depend on the assumed probability distribution for *y*_{i} only through *μ*_{i} and the variance function, *v*(μ_{i}) = var(*y*_{i}). The choice of distribution for *y*_{i} determines the relation *v*(μ_{i}) between the variance and the mean. Higher moments such as the skewness can affect properties of the model, such as how fast converges to normality, but they have no impact on the value of and its large-sample covariance matrix.

An alternative approach, *quasi-likelihood estimation*, specifies a link function and linear predictor *g*(μ_{i}) = ∑_{j}β_{j}*x*_{ij} like a generalized linear model (GLM), but it does not assume a particular probability distribution for *y*_{i}. This approach estimates {β_{j}} by solving equations that resemble the likelihood equations (8.1) for GLMs, but it assumes only a mean–variance relation for the distribution of *y*_{i}. The estimates are the solution of Equation (8.1) with *v*(μ_{i}) replaced by whatever variance function seems appropriate in a particular situation, with a corresponding adjustment for standard errors. To illustrate, a standard modeling approach for counts assumes that {*y*_{i}} are independent Poisson ...