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Biometrika 1998 85(3):689-700; doi:10.1093/biomet/85.3.689
© 1998 by Biometrika Trust
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Generalised bilinear regression

K. RUBEN GABRIEL

Department of Statistics, University of Rochester Rochester, New York 14627, U.S.A.krgl{at}dbl.cc.rochester.edu

This paper discusses the application of generalised linear methods to bilinear models by criss-cross regression. It proposes an extension to segmented bilinear models in which the expectation matrix is linked to a sum in which each segment has specified row and column covariance matrices as well as a coefficient parameter matrix that is specified only by its rank. This extension includes a variety of biadditive models including the generalised Tukey degree of freedom for non-additivity model that consists of two bilinear segments, one of which is constant. The extension also covers a variety of other models for which least squares fits had not hitherto been available, such as higher-way layouts combined into the rows and columns of a matrix, and a harmonic model which can be reparameterised so a lower rank fit is equivalent to a constant phase parameter. A number of practical applications are provided, including displaying fits by biplots and using them to diagnose models.

Key Words: Bilinear model • Biplot • Criss-cross regression • Exploratory data analysis • Generalised linear model • Iteratively reweighted least squares • Reduced rank regression


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