To Linearize Generalized Linear Models: A Dual Role of the IRLS Algorithm

對於廣義線性模式（generalized linear models，GLMs）中迴歸係數之最大概似估計量（maximum likelihood estimates，MLEs）通常可藉由一疊代重新加權最小平方法（iterative reweighted least squares，IRLS）算則獲得。在本研究中，我們發現在IRLS算則收斂時最後一次疊代後的線性迴歸模式可視為等同於原廣義線性模式之一個線性表示式。這樣的發現不僅含有其原本等式之統計意義，更可以進一步提供我們一個十分有用的工具，即將廣義線性模式的線性化用於模式配適之目的。因此，藉由利用IRLS算則的雙重角色，我們對於廣義線性模式提出一個新的判定係數（coefficient of determination，R2）以評估模式配適好壞，及各種常用的迴歸診斷（regression diagnostics）統計量。

The maximum likelihood estimates (MLEs) of the regression coefficients in generalized linear models (GLMs) are usually obtained by an iterative reweighted least squares (IRLS) algorithm (Charnes, Frome, and Yu 1976). In this study, we find that at the end of the last iteration of the IRLS algorithm, an equivalent linear regression representation of a GLM can also be obtained. This finding is not only very interesting in its own rights, but it provides us with a useful tool to linearize GLMs for model fitting purposes. By making use of the dual role of the IRLS algorithm, we propose a new coefficient of determination and various regression diagnostic statistics for GLMs and their quasi-likelihood extensions as an illustration of this tool.