Polynomial Regression with Response Surface Analysis
Also known as: Response Surface Methodology, RSA, Polynomial Regression for Congruence, Edwards Polynomial Regression, Surface Analysis for Fit
Polynomial regression with response surface analysis is the methodological gold standard for testing congruence, fit, and agreement hypotheses in organizational behavior, introduced by Jeffrey Edwards and Mark Parry in 1993. It replaces the once-common practice of subtracting two scores and regressing the outcome on that difference, a practice that conflates several distinct effects and discards information. Instead, the two component variables are entered together with their squares and cross-product, and the resulting equation is interpreted as a three-dimensional surface relating the two predictors to the outcome. Edwards and Parry showed that difference scores impose untestable and usually false constraints, and that the polynomial approach recovers the constrained model as a special case while exposing far richer patterns. Shanock and colleagues' 2010 tutorial made the method accessible by providing surface coefficients, tests, and plotting tools. The technique is now standard wherever person-environment fit and rater agreement are studied.
Key highlights
- Avoids the well-documented confounds and information loss of difference scores while nesting them as a testable special case.
- Separates the effect of agreement level (the congruence line) from the effect of the direction and degree of mismatch (the incongruence line).
- Detects curvilinear and asymmetric fit patterns, such as over-qualification differing from under-qualification, that linear gap models cannot represent.
- Yields an interpretable three-dimensional surface and principal axes that communicate complex joint effects visually.
Intuition
This section is available to Pro members. Upgrade to Pro
How it works
This section is available to Pro members. Upgrade to Pro
When to use it
Use polynomial regression with response surface analysis whenever a hypothesis concerns the congruence, fit, agreement, or discrepancy between two commensurate variables and their joint effect on an outcome. It is the appropriate tool for person-organization and person-job fit, self-other rating agreement, expectation-experience discrepancies, and any setting where a researcher might otherwise be tempted to use a difference score. It requires that the two predictors be measured on the same scale and that the sample be large enough to estimate five terms stably and to detect curvature. It is unnecessary when only the simple level of a single variable matters, when the two variables are not genuinely commensurate, or when theory clearly specifies a purely additive relationship; in those cases the added complexity buys nothing.
Strengths & limitations
- Avoids the well-documented confounds and information loss of difference scores while nesting them as a testable special case.
- Separates the effect of agreement level (the congruence line) from the effect of the direction and degree of mismatch (the incongruence line).
- Detects curvilinear and asymmetric fit patterns, such as over-qualification differing from under-qualification, that linear gap models cannot represent.
- Yields an interpretable three-dimensional surface and principal axes that communicate complex joint effects visually.
- Requires larger samples and adequate variance in both predictors to estimate and detect the higher-order terms reliably.
- Demands genuinely commensurate measurement of the two predictors, which is not always achievable.
- Surface and principal-axis interpretation is technically demanding and easy to misread without careful plotting.
- Like all regression, it is observational and does not by itself establish that congruence causes the outcome.
Common pitfalls
This section is available to Pro members. Upgrade to Pro
Applications
This section is available to Pro members. Upgrade to Pro
Frequently asked
Why not just use a difference score?
Edwards and Parry showed that a difference score forces a more general quadratic model into a rigid form: it assumes the two component variables have effects of equal magnitude and opposite sign, that only their gap matters and not their level, and that the relationship is linear. These constraints are rarely true and are usually rejected when tested. The difference score also discards the information in the individual scores, so a low-low match and a high-high match look identical. Polynomial regression keeps both scores and tests the constraints rather than imposing them, recovering the difference-score model only if the data actually support it.
What do the four surface coefficients mean?
They translate the regression into fit language along two diagonals of the predictor plane. Along the line of congruence, where the two predictors are equal, a1 is the slope (does agreement at higher levels help or hurt) and a2 is the curvature (does the outcome peak or trough at agreement). Along the line of incongruence, where the predictors are opposite, a3 captures whether the direction of mismatch matters and a4 captures the curvature of the mismatch effect. Shanock and colleagues provide the formulas and significance tests, and the coefficients are best understood alongside a plot of the surface.
How large a sample do I need?
Larger than for a simple regression, because the model estimates five coefficients and must detect curvature, which is harder than detecting linear effects. Adequate, non-restricted variance in both predictors is essential, and there must be enough cases off the congruence diagonal to estimate the incongruence effects. There is no single magic number, but underpowered samples will fail to find real curvature and may produce unstable surfaces. Shanock and colleagues encourage checking the joint significance of the quadratic block and inspecting the surface before drawing strong conclusions about fit.
Sources
- 1.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(6), 1577-1613.
- 2.Shanock, L. R., Baran, B. E., Gentry, W. A., Pattison, S. C., & Heggestad, E. D. (2010). Polynomial regression with response surface analysis: A powerful approach for examining moderation and overcoming limitations of difference scores. Journal of Business and Psychology, 25(4), 543-554.
You have read it. What now?
Cite this page
ScholarGate. (2026, June 23). Polynomial Regression with Response Surface Analysis. ScholarGate. https://scholargate.app/organizational-behavior/polynomial-regression-response-surface