Higher-order diagnosis of two-way tables, illustrated on two sets of demographic empirical distributions.

Two-way tables of all kinds often require diagnosis, usually of the residuals after some simple fit such as row-PLUS-column or row-TIMES-column. The first step in diagnosis is likely to be either one-degree-of-freedom-for-non-additivity or a diagnositic plot (for which the former is the linear regression term). Not all diagnoses can be made at the first step. Some diagnostic plots appear like a diagonal cross, or, when such an appearance is not quite clear, become converted to oppositely tilted pictures when we look only at points with high fitted values and, separately, at those with low fitted values. Such behavior diagnoses a need for a more subtle re-expression than powers and logs, in our examples a need for a re-expression like plamda -(1 - p)lamda. The appearance and treatment of such diagnoses, in two examples, lead into the use of letter-value displays and the associated plots to study the character of non-normality, instances of the effect of the method of fitting on the shape of the distribution of residuals, and convenient algorithms for the iterative, recursive fitting of a variety of additive, multiplicative, and mixed additive-multiplicative models to any kind of two-way table, APL programs for which are appended.