Quadratic Assignment Procedure
Also known as: QAP correlation, QAP permutation test, matrix permutation test, Hubert-Schultz QAP
The quadratic assignment procedure (QAP) is a permutation-based method for testing the association between two relational matrices measured on the same set of actors — for example, whether who advises whom is correlated with who is friends with whom. Because the dyads in a network are not independent, ordinary correlation and regression give invalid p-values; QAP fixes this by comparing the observed matrix correlation to a reference distribution generated by randomly relabeling the nodes of one matrix many times.
Key highlights
- Provides valid significance tests for matrix association despite the non-independence of network dyads.
- Distribution-free: relies on permutation rather than parametric assumptions about the data.
- Conceptually simple and applicable to binary, valued, directed, or undirected relations.
- Forms the basis for multiple-matrix regression (MRQAP) and for comparing observed networks to theoretical structures.
Intuition
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How it works
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When to use it
Use QAP correlation when you want to test whether two relations on the same actors are associated — friendship versus advice, trade versus alliance, observed network versus a hypothesized structure — and you need a significance test that respects dyadic dependence. It is the standard bivariate tool for matrix association in network analysis. Use MRQAP when you have multiple predictor matrices. QAP is not a generative model: it tests association, not endogenous structure (use ERGM for that), and it assumes the full sociomatrix is observed on a fixed node set; it cannot handle sampled dyads or value the contribution of nodal attributes beyond what is encoded as a matrix.
Strengths & limitations
- Provides valid significance tests for matrix association despite the non-independence of network dyads.
- Distribution-free: relies on permutation rather than parametric assumptions about the data.
- Conceptually simple and applicable to binary, valued, directed, or undirected relations.
- Forms the basis for multiple-matrix regression (MRQAP) and for comparing observed networks to theoretical structures.
- Tests association only; it does not model endogenous structural dependence such as transitivity or reciprocity.
- Requires complete relational matrices on a common, fixed actor set — it cannot accommodate sampled or missing dyads gracefully.
- The node-permutation null assumes exchangeability of actors, which can be questioned when actors differ systematically in ways not encoded in the matrices.
- Permutation tests become computationally heavy for very large networks when many permutations are needed for precise p-values.
Common pitfalls
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Applications
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Frequently asked
Why can't I just use an ordinary correlation test on the two matrices?
Because network dyads are not independent: every cell shares actors with the rest of its row and column, so the effective sample size is far smaller than the number of cells. Ordinary tests assume independence and therefore report standard errors that are too small and p-values that are too optimistic. QAP's node-permutation null preserves the dependence structure, yielding valid inference.
What exactly is being permuted in QAP?
The labels (identities) of the nodes are permuted, and that permutation is applied to the rows and columns of one matrix together. This keeps the matrix's internal relational pattern intact while scrambling its alignment with the other matrix, which is precisely the null hypothesis of no association between the two relations.
How is QAP related to the Mantel test?
QAP correlation is essentially the Mantel test as used in ecology and psychology: both compare two proximity or relation matrices via a permutation distribution. The network literature adopted the QAP terminology from Hubert and Schultz and extended it to directed and valued relations and to multiple regression (MRQAP).
Sources
- 1.Krackhardt, D. (1988). Predicting with networks: Nonparametric multiple regression analysis of dyadic data. Social Networks, 10(4), 359–381.
- 2.Hubert, L., & Schultz, J. (1976). Quadratic assignment as a general data analysis strategy. British Journal of Mathematical and Statistical Psychology, 29(2), 190–241.
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Cite this page
ScholarGate. (2026, June 22). Quadratic Assignment Procedure. ScholarGate. https://scholargate.app/sociology/quadratic-assignment-procedure