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January, 1976 Spanning Sets for Estimable Contrasts in Classification Models
David Birkes, Yadolah Dodge, Justus Seely
Ann. Statist. 4(1): 86-107 (January, 1976). DOI: 10.1214/aos/1176343349

Abstract

Two algorithms, the $R$-process and the $Q$-process, are presented which can be effective tools for determining the estimable contrasts in a classification model. Both algorithms operate on the incidence matrix of the model as opposed to the design matrix. If the model is partitioned as $E(Y_u) = h_u\cdot \xi + t_u \cdot \theta, u \in U$, the $R$-process produces a spanning set for the estimable $\theta$-contrasts (i.e., contrasts involving only $\theta$ parameters) whenever the set of distinct $h_u$ vectors is linearly independent. If the distinct $h_u$ vectors are dependent, the $R$-process is still useful and often simplifies the problem of obtaining a spanning set for the estimable $\theta$-contrasts. After the $R$-process has been applied in a case when the distinct $h_u$ vectors are dependent, the $Q$-process produces a spanning set for the estimable $\theta$-contrasts provided a partition $h_u\cdot \xi = f_u\cdot \phi + g_u\cdot\omega, u \in U$, can be made such that the sets of distinct $f_u$ vectors and distinct $g_u$ vectors are both linearly independent. As examples, the $R$-process is used to investigate the additive two-way model; and the $R$-process and $Q$-process together are used to investigate an additive three-way model, a two-way model with interaction, and a Graeco-Latin square model.

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David Birkes. Yadolah Dodge. Justus Seely. "Spanning Sets for Estimable Contrasts in Classification Models." Ann. Statist. 4 (1) 86 - 107, January, 1976. https://doi.org/10.1214/aos/1176343349

Information

Published: January, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0321.62081
MathSciNet: MR395063
Digital Object Identifier: 10.1214/aos/1176343349

Subjects:
Primary: 62J99
Secondary: 62K99

Keywords: Classification models , Degrees of freedom , estimable contrasts , maximal rank design matrices , missing observations

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 1 • January, 1976
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