Two-way plots: I. The case of straight line segments
The two-way plot is a visualization tool for tables by means of intersections of geometrical objects. This paper studies such plots in the case, where the geometrical objects are linear segments. The original two-way plot for additive fits has been introduced in Tokey (1977, Chapter 10E). It consists of two sets of parallel segments, one segment for each row and one segment for each column, chosen in such a manner that all row segments intersect all column segments and such that the ordinates of the intersections are equal to the fitted values. A two-way plot shows immediately and precisely the important features of a fit, not only locally within a single row or column, but also globally. As such it is a very useful tool to go along with the numerical descriptions we generally use. The additive model, however, is too restrictive and this paper generalizes the two-way plot to models with interactive terms. The most general graph we will consider, retains from Tukey's original only the fact that each row and each column is represented by a line segment and that they all mutually intersect each other. The fits that correspond to such a figure turn out to be sums of row and column effects and multiplicative terms involving additional row and column effects. It turns out in practice, that requiring at least one of the row or the column line segments to be parallel or in a fan shape often leads to more visually pleasing charts, without much restricting the quality of the fit.
interaction analysis of variance non-additivity two-way tables bilinear models