Difference Score Regression Macro

I have created a fairly comprehensive macro that undertakes model comparison, analysis and graphing of difference score regression (situations where there are two related predictors of a dependent variable, say expected versus actual levels of something - another example is the classic Person-Organisation Fit literature, in which the difference between the individual and organisation on various measures is of interest as a predictor of work-related outcomes). Outputs compare algebraic, absolute and squared comparisons up to third order models, and graphs response surfaces and the like. The output helps the user choose a best model for assessment of the differences, assess the important shapes / coefficients of this model, and graph the relationships. I attach scoping document. Interested users are welcome to the macro, just contact me.

1. Statistical Background

This macro creates output based on Jeffrey R. Edwards’ development of theory behind the analysis of differences between two variables as independent variables. The next section briefly describes the core concepts and models of this development.

1.1 Core Difference Analysis Models

Let C1 and C2 be the predictors being compared, Z be the dependent variable, and let w=1 if C1<C2 and w=0 if C1>=C2. For instance, C1 might be expected pay, C2 actual pay, and Z satisfaction. The core question is how differences between expected and actual pay affect satisfaction. There may also be a string of control/covariate variables.

Table 1 shows various models assessing differences between C1 and C2 as predictors of Z:

Table 1: Models analyzing difference scores

1) Base model for comparison	Z = controls
2) Constrained algebraic differences	Z = Controls + (C1-C2)
3) Unconstrained algebraic differences (i.e. C1 and C2 separated)	Z = Controls + C1 + C2
4) Higher order algebraic differences (i.e. curvilinear effects, same as model 9)	Z = Controls + C1 + C2 + C1² + C1xC2 + C2²
5) Constrained absolute differences	Z = Controls + abs(C1-C2)
6) Unconstrained absolute differences	Z = Controls + C1 + C2 + w + wC1 + wC2
7) Higher order absolute differences	Z = Controls + C1 + C2 + w + wC1 + wC2 + C1Squared C1xC2 C2Squared wxC1Squared wxC1xC2 wxC2Squared
😎 Constrained squared differences	Z = Controls + (C1-C2)²
9) Unconstrained squared differences i.e. curvilinear effects (same model as 3)	Z = Controls + C1 + C2 + C1² + C1C2 + C2²
10) Higher order squared differences (i.e. curvilinear + cubic effects)	Z = Controls + C1 + C2 + C1² + C1C2 + C2² + C1³ + C1²C2 + C1xC2² + C2³*

The key tasks in the analysis are to identify which of these 10 models seems to fit best and then to assess key parameters and shapes of the chosen model, including regression coefficients, graphical representation of the relationship (especially important when the relationship is unconstrained and therefore forms a 3D response surface), and shapes along or on key parts of the relationship (e.g. what the relationship is long the line of congruence, i.e. that line where C1 = C2).

1.2 Statistical References

Some references include:

Edwards, J. R., Cable, D. M., Williamson, I. O., Lambert, L. S., & Shipp, A. J. (2006). The phenomenology of fit: Linking the person and environment to the subjective experience of person-environment fit. Journal of Applied Psychology, 91, 802-827.

Edwards, J. R. (2002). Alternatives to difference scores: Polynomial regression analysis and response surface methodology. In F. Drasgow & N. W. Schmitt (Eds.), Advances in measurement and data analysis (pp. 350-400). San Francisco: Jossey-Bass.

Edwards, J. R. (2001). Ten difference score myths. Organizational Research Methods, 4, 264-286.

Edwards, J. R. (1994a). Regression analysis as an alternative to difference scores. Journal of Management, 20, 683-689.

Edwards, J. R. (1994b). The study of congruence in organizational behavior research: Critique and a proposed alternative. Organizational Behavior and Human Decision Processes, 58, 51-100 (erratum, 58, 323-325).

Edwards, J. R., & Parry, M. E. (1993). On the use of polynomial regression equations as an alternative to difference scores in organizational research. Academy of Management Journal, 36, 1577-1613.