Bias Correction in Generalized Linear Models

SUMMARY

In this paper we derive general formulae for first-order biases of maximum likelihood estimates of the linear parameters, linear predictors, the dispersion parameter and fitted values in generalized linear models. These formulae may be implemented in the GLIM program to compute bias-corrected maximum likelihood estimates to order n  ⁻¹, where n is the sample size, with minimal effort by means of a supplementary weighted regression. For linear logistic models it is shown that the asymptotic bias vector of $\widehat{\beta }$ is almost collinear with β. The approximate formula βp/m₊ for the bias of $\widehat{\beta }$ in logistic models, where p = dim(β) and m₊ = ∑ m_i is the sum of the binomial indices, is derived and checked numerically.

bias correction, canonical link, deviance, dispersion parameter, generalized linear model, logistic model, maximum likelihood estimate