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The Analysis of Nonadditivity in Two-Way Analysis of Variance
Leo A. Goodman and Shelby J. Haberman
Journal of the American Statistical Association
Vol. 85, No. 409 (Mar., 1990), pp. 139-145
Stable URL: http://www.jstor.org/stable/2289536
Page Count: 7
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Singular value decompositions are used to study interactions in two-way layouts. In contrast to traditional treatments, emphasis is on description of the nature of the interactions rather than on tests of whether they do or do not exist. The first model considered assumes that the matrix of interactions has rank 1, so in a singular value decomposition of the interactions, only one singular value is not 0. Given that this model holds and interactions are present, normal approximations are obtained under conditions described for the estimates of this singular value and for the corresponding scores for row categories and column categories. Thus approximate confidence intervals can be obtained for the parameters in this singular value decomposition of interactions. In addition, under this model, approximate t and F tests can be constructed for hypotheses that assign specified values to the row and column scores in the singular value decomposition. Results are illustrated in a reanalysis of data previously analyzed by Johnson and Graybill (1972). The model that assumes that the interactions matrix is of rank 1 is also used to test the appropriateness of the scoring system that is associated with Tukey's one degree of freedom for nonadditivity. In addition, models are considered in which the singular value decomposition of the interactions has rank greater than 1, and normal approximations for parameter estimates and approximate t and F tests for hypotheses concerning the scores in the singular value decomposition are derived for these models as well.
Journal of the American Statistical Association © 1990 American Statistical Association