In some cases, researchers hypothesize that the effects of congruence on an outcome are transmitted through a mediator variable. Mediation can be analyzed using two regression equations, one that regresses the mediator on the five quadratic terms, and another that regresses the outcome on the five quadratic terms and the mediator:
(1) M = a0 + a1X + a2Y + a3X2 + a4XY + a5Y2 + eM
(2) Z = b0 + b1M + b2X + b3Y + b4X2 + b5XY + b6Y2 + eZ
In Eqs. 1 and 2, congruence is captured by the five quadratic terms X, Y, X2, XY, and Y2, M is the mediator variable, and Z is the outcome variable. The indirect (i.e., mediated) effect of the quadratic terms on Z is specified by using the quadratic terms as predictors of M in Eq. 1 along with using M as a predictor of Z in Eq. 2. The quadratic terms are used as predictors of Z in Eq. 2, indicating that congruence influences the outcome Z directly as well as indirectly through the mediator M.
The combined effects represented by Eqs. 1 and 2 can be seen by substituting Eq. 1 into Eq. 2 to obtain the following reduced form equation:
(3) Z = b0 + b1(a0 + a1X + a2Y + a3X2 + a4XY + a5Y2 + eM) + b2X + b3Y + b4X2 + b5XY + b6Y2 + eZ
Distribution yields:
(4) Z = b0 + a0b1 + a1b1X + a2b1Y + a3b1X2 + a4b1XY + a5b1Y2 + b1eM + b2X + b3Y + b4X2 + b5XY + b6Y2 + eZ
Collecting like terms yields:
(5) Z = (b0 + a0b1) + (b2 + a1b1)X + (b3 + a2b1)Y + (b4 + a3b1)X2 + (b5 + a4b1)XY + (b6 + a5b1)Y2 + (eZ + b1eM)
The compound coefficients on X, Y, X2, XY, and Y2 represent the portion of the quadratic effect mediated by M as the products a1b1, a2b1, a3b1, a4b1, and a5b1. The portion of the quadratic effect that bypasses M is captured by b2, b3, b4, b5, and b6. Both sets of coefficients can be examined to determine whether they provide support for the hypothesized congruence effect. The coefficients representing the direct effect (b2, b3, b4, b5, and b6) can be analyzed using standard procedures for response surface methodology (Edwards, 2002; Edwards & Parry, 1993). The terms representing the mediated effect (a1b1, a2b1, a3b1, a4b1, and a5b1) can be used in a similar manner, with the caveat that these terms are products of coefficients and therefore require nonparametric techniques for conducting statistical tests. In particular, the bootstrap can be applied to Eqs. 1 and 2 to obtain a large number (e.g., 10,000) of bootstrap estimates of the coefficients in both equations (Efron & Tibshirani, 1993; Mooney & Duval, 1993; Stine, 1989). For each set of estimates, the products of the coefficients representing the mediated effect can be computed, and the distributions of these products can be examined to derive confidence intervals and draw statistical inferences. Because coefficient products such as a1b1, a2b1, a3b1, a4b1, and a5b1 are not likely to be normally distributed, confidence intervals for these products should be constructed using the percentile method with bias correction (e.g., Stine, 1989). Bias-corrected confidence intervals should also be used to analyze features of the surface represented by the mediated effect. For instance, the shape of this surface along the Y = X line can be derived by setting Y = X in Eq. 5 (cf. Edwards & Parry, 1993) and rearranging terms, which yields:
(6) Z = (b0 + a0b1) + [(b2 + b3) + (a1b1 + a2b1)]X + [(b4 + b5 + b6) + (a3b1 + a4b1 + a5b1)]X2 + (eZ + b1eM)
The coefficients in the square brackets preceding X and X2 are grouped in parentheses according to whether the coefficients correspond to the direct or indirect effect. In particular, (b2 + b3) and (b4 + b5 + b6) describe the shape along the Y = X line for the surface representing the direct effect. Analogously, (a1b1 + a2b1) and (a3b1 + a4b1 + a5b1) represent the shape along the Y = X line for the surface representing the indirect effect. For this surface, the terms (a1b1 + a2b1) and (a3b1 + a4b1 + a5b1) should be tested by computing these quantities using the bootstrap estimates and constructing bias-corrected confidence intervals.
A similar expression can be derived to examine the shapes along the Y = -X line for the surfaces representing the direct and indirect effects. This expression is obtained by substituting Y = -X in Eq. 5 and rearranging terms to obtain:
(7) Z = (b0 + a0b1) + [(b2 – b3) + (a1b1 – a2b1)]X + [(b4 – b5 + b6) + (a3b1 – a4b1 + a5b1)]X2 + (eZ + b1eM)
In Eq. 7, (b2 – b3) and (b4 – b5 + b6) indicate the shape along the Y = -X line for the surface representing the direct effect, and (a1b1 – a2b1) and (a3b1 – a4b1 + a5b1) show the shape along the Y = -X line for the surface representing the indirect effect. Again, the terms (a1b1 – a2b1) and (a3b1 – a4b1 + a5b1) can be tested using bias-corrected confidence intervals based on the bootstrap.
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., & 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.
Efron, B., & Tibshirani, R. (1993). An introduction to the bootstrap. New York: Chapman & Hall.
Mooney, C. Z., & Duval, R. D. (1993). Bootstrapping: A nonparametric approach to statistical inference. Newbury Park, CA: Sage.
Stine, R. (1989). An introduction to bootstrap methods. Sociological Methods & Research, 18, 243-291.