Model robust regression: combining parametric, nonparametric, and semiparametric methods

Abstract

In obtaining a regression tit to a set of data, ordinary least squares regression depends directly on the parametric model formulated by the researcher. Ifthis model is incorrect, a least squares analysis may be misleading. Altematively, nonparametric regression (kemel or local polynomial regression, for example) has no dependence on an underlying parametric model, but instead depends entirely on the distances between regressor coordinates and the prediction point of interest. This procedure avoids the necessity of a reliable model, but in using no information from the researcher, may Ht to irregular pattems in the data. The proper combination of these two regression procedures can overcome their respective problems. Considered is the situation where the researcher has an idea of which model should explain the behavior of the data, but this model is not adequate throughout the entire range of the data. An extension of partial linear regression and two methods of model robust regression are developed and compared in this context. These methods involve parametric tits to the data and nonparametric tits to either the data or residuals. The two tits are then combined in the most efficient proportions via a mixing parameter. Performance is based on bias and variance considerations.

A semi-parametric approach to dual modeling when no replication exists

“…when cost of data collection is prohibitive), nonparametric fitting techniques may fit irregularities in the data too closely, thereby creating estimated mean and variance functions which are highly variable. Mays et al (2001) (henceforth referred to as MBS) introduce semi-parametric methods for estimating the mean in constant variance situations, which are essentially hybrids of parametric and nonparametric methods. Pickle et al (2008) extend the method of MBS to the dual model problem for the replicated case.…”

Several nonparametric and semiparametric approaches to linear mixed model regression

“…We adopt here to the mixed model scenario a penalized version of PRESS, proposed in the context of fixed effects models for uncorrelated data by Mays et al [31] For the LM model, PRESS * * is defined as…”

Modeling Nonmonotonic Dose-Response Curves

“…Since tr [H k ] reflects the kernel fits' " model degrees of freedom" (Cleveland, 1978), it is seen that the denominator of PRESS*(6) penalizes the PRESS statistic for choosing b too small. Empirical studies by the authors and others (Einsporn and Birch, 1993;Mays, 1995) have demonstrated that using PRESS*(b) is often superior to using PRESS as a bandwidth selector.…”