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

Abstract 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. Key words: series of measures; method of differences; signs of differences; structural zeros; log-linear modeling; Configural Frequency Analysis In behavioral research, measurement and analysis of change are of importance in many contexts. For example, intervention and therapy are concerned with changes in behavior over time. This applies accordingly to prevention and education. The number of methods of analysis of change is large. Among these, the method of finite differences plays a prominent role. It can be used for a number of important purposes. The following five applications are most prominent. First, the method allows one to determine a polynomial function that either fits or sufficiently approximates a series of values (for a description of this technique, see Zurmuhl, 1965; for applications, see von Eye, 2002). Second, it allows one to identify and correct errors in a series of measures (Hermite interpolation; see, e.g., Zurmuhl, 1965). Third, the method can be used to devise numerical solutions of partial differential equations (Smith, 1986). Fourth, the method has been used in the context of modeling. McArdle and collaborators (McArdle, 2001; McArdle & Hamagami, 2001) have devised structural models of developmental change that are based on latent first difference scores. Fifth, the method has been used to describe the shape of trajectories over time (Krauth,1973; lienert & Krauth, 1973 a, b). The present note is concerned with characteristics of the patterns that result when the signs of the differences are used. That is, this note examines the method of differences in the context of categorical data analysis. Specifically, (1) it will be shown that, when differences of various orders are categorized, the resulting cross-classifications contain structural zeros by design (that is, cells that, by definition, cannot contain any cases), and (2) an algorithm will be presented that allows one to identify those patterns that are structural zeros. Effects of these structural zeros include constraints when it comes to modeling the resulting cross-classifications or performing configurai analyses. 1. The method of differences There exist many variants of the method of finite differences (see Zurmuhl, 1965). Here, we use the method affinite ascending differences (MFAD) as an example. The following arguments apply accordingly to any of the methods of differences. Consider a series of I measures, X, with values x, and i=1,..., I. Then, the MFAD creates first differences as Δ^sub 1,j^ = x^sub j + 1^ - x^sub j^, for j=1,..., I - 1, second differences as Δ^sub 2, k^ = Δ^sub 1,k+1^ - Δ^sub 1, k^, for k= 1,...,1- 2, and so forth. If a series of measures can be described using a polynomial of lth order, with l ≤ I 1, the lth differences are constant, and all higher-order differences are zero. Consider the data example in Table 1. The table shows, in the second line, the values of the 3rd-order polynomial y = 25 + 3x + 2x^sup 2^ +0.6x^sup 3^, for values of X from O to 6, in steps of 1. In the following rows, it shows the first, second, and higher differences. Obviously, the third differences are constant, and the fourth differences are zero. …