Generalized Additive Models: Some Applications
本文介绍了广义可加模型，它扩展了广义线性模型，允许响应变量与预测变量之间通过非参数平滑函数建立关系。作者使用局部评分算法估计这些函数，并通过协方差分析和逻辑回归两个实例展示其应用，同时提供了推断工具来评估估计函数的相关性和显著性。
Abstract Generalized additive models have the form η(x) = α + σ fj (x j ), where η might be the regression function in a multiple regression or the logistic transformation of the posterior probability Pr(y = 1 | x) in a logistic regression. In fact, these models generalize the whole family of generalized linear models η(x) = β′x, where η(x) = g(μ(x)) is some transformation of the regression function. We use the local scoring algorithm to estimate the functions fj (xj ) nonparametrically, using a scatterplot smoother as a building block. We demonstrate the models in two different analyses: a nonparametric analysis of covariance and a logistic regression. The procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis. A variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions: these include confidence curves, degrees of freedom estimates, and approximate hypothesis tests. The local scoring algorithm is analogous to the iterative reweighted least squares algorithm for solving likelihood and nonlinear regression equations. At each iteration, an adjusted dependent variable is formed and an additive regression model is fit using the backfitting algorithm. The backfitting algorithm cycles through the variables and estimates each coordinated function by smoothing the partial residuals.