MRQAP Network Regression
Also known as: MRQAP, multiple regression QAP, Dekker double-semipartialing, QAP regression
Multiple regression quadratic assignment procedure (MRQAP) extends QAP to the regression setting: it predicts a dependent relational matrix from several independent relational matrices on the same actors — for example, modeling who collaborates with whom as a function of who is co-located, who shares a department, and who has prior friendship. Coefficients are estimated by ordinary least squares on the vectorized matrices, but significance is assessed by permutation, because dyadic dependence invalidates the standard regression standard errors.
Key highlights
- Provides valid inference for multivariate dyadic regression despite the non-independence of network ties.
- The double-semipartialing scheme remains well-calibrated under collinear predictor matrices, where naive permutation tests fail.
- Estimates familiar, interpretable regression coefficients for relational and attribute-similarity predictors.
- Handles binary, valued, directed, or undirected relations as outcomes or predictors.
Intuition
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How it works
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When to use it
Use MRQAP when the outcome is a relation between actors and you want to assess the partial effect of several relational or attribute-similarity predictors while obtaining valid significance tests under dyadic dependence. It is the standard tool for multivariate dyadic hypotheses — homophily on multiple attributes, multiplexity, proximity effects — when the full sociomatrix is observed. Prefer ERGM when endogenous structural dependence (transitivity, popularity) is the substantive interest, and prefer the network autocorrelation model when the outcome is a nodal (actor-level) variable subject to peer influence rather than a relation. MRQAP requires complete matrices and assumes a linear additive form.
Strengths & limitations
- Provides valid inference for multivariate dyadic regression despite the non-independence of network ties.
- The double-semipartialing scheme remains well-calibrated under collinear predictor matrices, where naive permutation tests fail.
- Estimates familiar, interpretable regression coefficients for relational and attribute-similarity predictors.
- Handles binary, valued, directed, or undirected relations as outcomes or predictors.
- Models dyadic association only; it does not capture endogenous structural effects such as transitivity or degree heterogeneity (use ERGM).
- Assumes a linear, additive relationship between predictor and outcome matrices.
- Requires complete relational matrices on a fixed, common actor set; missing or sampled dyads bias the estimates.
- Permutation inference is computationally intensive for large networks and many predictors.
Common pitfalls
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Applications
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Frequently asked
How does MRQAP differ from ERGM?
MRQAP regresses one relational matrix on others and tests dyadic associations with permutation inference, but treats dyads as conditionally independent given the predictors. ERGM specifies a full generative model of the network including endogenous structural terms (transitivity, reciprocity, degree). Use MRQAP for exogenous dyadic predictors and homophily; use ERGM when self-organizing structure is the hypothesis.
Why is the double-semipartialing method preferred?
When predictor matrices are correlated with each other, permuting the raw dependent or predictor matrix produces a null distribution that does not match the partial-effect being tested, inflating false positives. Double-semipartialing residualizes both the outcome and the target predictor on the other predictors before permuting, which Dekker et al. (2007) showed keeps the test valid under collinearity and autocorrelation.
Can MRQAP handle a continuous valued network?
Yes. The outcome and predictors can be valued (e.g., communication counts, similarity scores) rather than binary. The OLS estimation and permutation inference proceed identically; only the interpretation of the coefficients changes to reflect the valued scale.
Sources
- 1.Krackhardt, D. (1988). Predicting with networks: Nonparametric multiple regression analysis of dyadic data. Social Networks, 10(4), 359–381.
- 2.Dekker, D., Krackhardt, D., & Snijders, T. A. B. (2007). Sensitivity of MRQAP tests to collinearity and autocorrelation conditions. Psychometrika, 72(4), 563–581.
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ScholarGate. (2026, June 22). MRQAP Network Regression. ScholarGate. https://scholargate.app/sociology/mrqap-network-regression