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Abstract:
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Analyzing interactions is primary for multi-way models. If the number of replications per level combination is equal to one or zero in part, classic tests are not applicable. In two-way layout, the most prominent approaches are Tukey's one degree of freedom test or the test by Johnson and Graybill (1972). There exists tests of highest order interaction when only one replicate received for the highest interaction term (Boik, Marasinghe 1989). In this paper, we propose a semiparametric varying-coefficient regression, by which the main effects are specified parametrically and the interaction terms are constructed by nonlinear forms of main effects with varying coefficient functions of residuals from subgroups. Under two-way layout, our model is locally equivalent to the model by Graybill, but has flexibility to adapt functional forms of interactions. We attempt to analyze the main effects and interactions simultaneously for multi-factors under a more general unbalanced structure. We can further employ LASSO-type method to select important effects under these unbalanced designs. With a multivariate nonparametric extension, it also permits us to investigate higher order interactions.
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