A note on adjusted responses, fitted values and residuals in Generalized Linear Models

Adjusted responses, adjusted fitted values and adjusted residuals are known to play in Generalized Linear Models the role played in Linear Models by observations, fitted values and ordinary residuals. We think this parallelism, which was widely recognized and used in the early literature on Generalized Linear Models, has been somewhat overlooked in more recent presentations. We revise this parallelism, systematizing and proving some results that are either scattered or not satisfactorily spelled out in the literature. In particular, we formally derive the asymptotic dispersion matrix of the (scaled) adjusted residuals, by proving that in Generalized Linear Models the fitted values are asymptotically uncorrelated with the raw residuals and hence deriving the asymptotic dispersion matrix of these latter residuals. Also, we show that an orthogonal decomposition of the error vector between adjusted response and true linear predictor, parallel to the familiar decomposition in Linear Models, holds approximately. Finally, we provide some new perspective, both in Linear and Generalized Linear Models, on adjusted residuals for model comparison, and their relationships with test-statistics used to compare the fit of nested models.