Statistics and Its Interface

Volume 3 (2010)

Number 2

Quasi-least squares with mixed linear correlation structures

Pages: 223 – 233

DOI: http://dx.doi.org/10.4310/SII.2010.v3.n2.a9

Authors

Mary Frances Cotch (Division of Epidemiology and Clinical Applications, National Eye Institute, National Institutes of Health, Bethesda, Maryland, U.S.A.)

Jon Peet (Department of Ophthalmology, University of Pennsylvania School of Medicine, Philadelphia, Penn., U.S.A.)

Justine Shults (Department of Biostatistics and Epidemiology, University of Pennsylvania School of Medicine, Philadelphia, Penn., U.S.A.)

Dwight Stambolian (Department of Ophthalmology, University of Pennsylvania School of Medicine, Philadelphia, Penn., U.S.A.)

Jichun Xie (Department of Biostatistics and Epidemiology, University of Pennsylvania School of Medicine, Philadelphia, Penn., U.S.A.)

Abstract

Quasi-least squares (QLS) is a two-stage computational approach for estimation of the correlation parameters in the framework of generalized estimating equations. We prove two general results for the class of mixed linear correlation structures: namely, that the stage one QLS estimate of the correlation parameter always exists and is feasible (yields a positive definite estimated correlation matrix) for any correlation structure, while the stage two estimator exists and is unique (and therefore consistent) with probability one, for the class of mixed linear correlation structures. Our general results justify the implementation of QLS for particular members of the class of mixed linear correlation structures that are appropriate for analysis of data from families that may vary in size and composition. We describe the familial structures and implement them in an analysis of optical spherical values in the Old Order Amish (OOA). For the OOA analysis, we show that we would suffer a substantial loss in efficiency, if the familial structures were the true structures, but were misspecified as simpler approximate structures. To help bridge the interface between Statistics and Medicine, we also provide R software so that medical researchers can implement the familial structures in a QLS analysis of their own data.

Keywords

quasi-least squares, linear correlation structure, mixed correlation structure, familial data

2010 Mathematics Subject Classification

60K35

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