Foundations of Linear and Generalized Linear Models

CHAPTER 8 Quasi-Likelihood Methods

For a GLM η_i = g(μ_i) = ∑_jβ_jx_ij, the likelihood equations

(8.1)

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) = ∑_jβ_jx_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 ...