QAPLogit formulas

Dyadic logistic model

For a binary dyadic outcome $Y_{ij}$ and dyadic predictors $X_{ij1},\dots ,X_{ijp}$, the model is:

$$Pr(Y_{ij}=1)={\text{logit}}^{-1}({\beta }_0+\sum _{k=1}^p{\beta }_k{X}_{ijk})$$

Equivalently:

$$\log \left(\frac{Pr(Y_{ij}=1)}{1-Pr(Y_{ij}=1)}\right)={\beta }_0+\sum _{k=1}^p{\beta }_k{X}_{ijk}$$

Dyad extraction

For directed networks, the default dyad set is:

$$\{(i,j):i\ne j\}$$

For undirected networks, the default dyad set is the upper triangle:

$$\{(i,j):i<j\}$$

Self-dyads are excluded by default.

QAP node-label permutations

For each permutation $\pi$ of node labels, the dependent matrix is permuted as:

$$Y^{(\pi )}=P_{\pi }Y{P}_{\pi }^T$$

The dyadic predictors remain fixed. The model is then re-fit on the permuted outcome to obtain a permutation distribution of coefficients.

Empirical p-values

The initial implementation uses a plus-one empirical p-value:

$$p=\frac{1+\sum _{b=1}^B\mathbb{I}(|T_b|\ge |T_{obs}|)}{B+1}$$

where:

• $T_{obs}$ is the observed coefficient;
• $T_b$ is the coefficient from permutation $b$;
• $B$ is the number of permutations.

This plus-one correction avoids returning zero p-values from a finite permutation sample.