  This paper presents an integrated framework for estimation and inference from
generalized linear models using adjusted score equations that result in mean
and median bias reduction. The framework unifies theoretical and methodological
aspects of past research on mean bias reduction and accommodates, in a natural
way, new advances on median bias reduction. General expressions for the
adjusted score functions are derived in terms of quantities that are readily
available in standard software for fitting generalized linear models. The
resulting estimating equations are solved using a unifying quasi-Fisher scoring
algorithm that is shown to be equivalent to iteratively re-weighted least
squares with appropriately adjusted working variates. Formal links between the
iterations for mean and median bias reduction are established. Core model
invariance properties are used to develop a novel mixed adjustment strategy
when the estimation of a dispersion parameter is necessary. It is also shown
how median bias reduction in multinomial logistic regression can be done using
the equivalent Poisson log-linear model. The estimates coming out from mean and
median bias reduction are found to overcome practical issues related to
infinite estimates that can occur with positive probability in generalized
linear models with multinomial or discrete responses, and can result in valid
inferences even in the presence of a high-dimensional nuisance parameter
