The standard approach to model fitting for designed experiments
begins with an ANOVA decomposition for the factor effects and their
interactions.  Then it uses significance tests and empirical
principles (such as hierarchical ordering, effect sparsity, and effect
heredity) for model selection. This can produce a fitted model that is
hard to interpret, especially if it contains more than a very few
interaction terms.  We propose an alternative approach that yields a
tree-structured piecewise additive model, with interaction effects
carried by the tree structure.  The method relies on data segmentation
and it selects a model from a class of hierarchical models by
cross-validation estimation of prediction error.  Examples from
least-squares, Poisson, and logistic regression for replicated and
unreplicated factorial experiments are used to compare the two
approaches.