Reiss Philip T, Huang Lei, Mennes Maarten
New York University and Nathan S. Kline Institute for Psychiatric Research, NY, USA.
Int J Biostat. 2010;6(1):Article 28. doi: 10.2202/1557-4679.1246.
Regression models for functional responses and scalar predictors are often fitted by means of basis functions, with quadratic roughness penalties applied to avoid overfitting. The fitting approach described by Ramsay and Silverman in the 1990 s amounts to a penalized ordinary least squares (P-OLS) estimator of the coefficient functions. We recast this estimator as a generalized ridge regression estimator, and present a penalized generalized least squares (P-GLS) alternative. We describe algorithms by which both estimators can be implemented, with automatic selection of optimal smoothing parameters, in a more computationally efficient manner than has heretofore been available. We discuss pointwise confidence intervals for the coefficient functions, simultaneous inference by permutation tests, and model selection, including a novel notion of pointwise model selection. P-OLS and P-GLS are compared in a simulation study. Our methods are illustrated with an analysis of age effects in a functional magnetic resonance imaging data set, as well as a reanalysis of a now-classic Canadian weather data set. An R package implementing the methods is publicly available.
用于函数响应和标量预测变量的回归模型通常通过基函数进行拟合，并应用二次粗糙度惩罚来避免过度拟合。拉姆齐（Ramsay）和西尔弗曼（Silverman）在20世纪90年代描述的拟合方法相当于系数函数的惩罚普通最小二乘（P-OLS）估计器。我们将此估计器重铸为广义岭回归估计器，并提出一种惩罚广义最小二乘（P-GLS）替代方法。我们描述了两种估计器都可以实现的算法，能够以比以往更高效的计算方式自动选择最优平滑参数。我们讨论了系数函数的逐点置信区间、通过置换检验进行的同时推断以及模型选择，包括一种新的逐点模型选择概念。在模拟研究中对P-OLS和P-GLS进行了比较。我们通过对功能磁共振成像数据集中年龄效应的分析以及对一个现已成为经典的加拿大天气数据集的重新分析来说明我们的方法。一个实现这些方法的R包已公开发布。
