A note on the analysis of difference patterns - structural zeros by design

The method of finite differences is popular in the analysis of series of measures, because it allows one to analyze the shape of the curve that describes the series, and changes in the shape. In addition, it allows one to relate these changes to covariates. In the analysis of categorical variables, researchers can cross the sign patterns of first, second, and higher differences. In this note, it is shown that crossing sign patterns from different levels results in contingency tables with systematically large numbers of structural zeros. In addition, a simple algorithm is proposed that allows one to identify impossible combinations of sign patterns. Examples are presented using empirical and random number-generated data that are analyzed using log-linear models and Configural Frequency Analysis.