MULTIPLICATIVE MODELS FOR INTERACTION IN UNBALANCED TWO-WAY ANOVA (REDUCED RANK REGRESSION, PRINCIPAL COMPONENT ANALYSIS, SINGULAR VALUE DECOMPOSITION).

摘要

Interactions often present interpretive difficulties in two-way ANOVA. Multiplicative models for interaction can be of assistance. For balanced designs, a variety of multiplicative models have been proposed. These models include Tukey's one degree of freedom for non-additivity, Mandel's bundle of lines and Gollob's FANOVA. The purpose of this thesis is to construct a generalized multiplicative model useful in unbalanced two-way ANOVA. The problems addressed are: (a) parameter estimation, (b) hypothesis testing, and (c) interpretation.;Hypothesis Testing. For balanced designs and asymptotically for unbalanced designs, the test statistic for a rank 1 model is distributed as the maximum root of a central Wishart matrix divided by an independent chi-square random variable. For small designs {min(a,b) (LESSTHEQ) 4}, tables of exact critical values of this test statistic are constructed. For larger designs and/or models of rank 2 or more, a procedure for obtaining approximate critical values is described. Simulation results suggest that for small sample sizes, lack of balance does not greatly influence the null distribution of the rank 1 test statistic.;Interpretation. Presence of an interaction of rank 1 or more indicates that some row (column) contrast is heterogeneous among columns (rows). Presence of interaction does not rule out a significant row (column) contrast which is homogeneous among columns (rows). A modified Newton algorithm is constructed for finding maximum row (column) contrasts subject to minimum heterogeneity (i.e., interaction) among columns (rows). A new type of plot is constructed which displays, for a set of contrasts defined by a continuous function, (a) the contrast coefficient vector; (b) the contrast sum of squares, and (c) the associated interaction. The plots and algorithms are illustrated using a variety of real data sets.;Parameter Estimation. For balanced designs, parameter estimation is accomplished by a singular value decomposition of the interaction matrix. For unbalanced designs, parameters are estimated by means of a generalized singular value decomposition (GSVD). In GSVD, generalized least squares rather than ordinary least squares is the estimation criterion. The covariance matrix of the interaction estimators need not have any particular structure. Three algorithms are constructed for performing GSVD: a modified Newton algorithm for rank 1 estimation, a modified Gauss algorithm for rank 1 estimation, and a version of Wold's criss-cross algorithm for estimation of any rank.