1. Social Network Analysis
QAP Procedure

2. OLS and Network Analysis
A very simple approach to Network Analysis would be to use OLS (or, if the data were binary, to use logistic regression—or if the data were a count, to use a negative binomial, etc.)

3. OLS and Network Analysis
The basic idea would be that you would create a data set that had the “dyad” or pair (in the Arizona case, pair of legislators) as the unit of analysis.
The independent variables would be either attributes of each of one or both members of the pairs, or of similarities and / or matches between the pairs.

4. OLS and Network Analysis
For example, in the case of the Arizona legislature, we would have a data set that might include each pair of legislators, a dummy variable for whether they were of the same party, a variable that indicated how close they were in terms of their ideology, a variable for the seniority of the first cosponsor (the “column” cosponsor), and a variable for the seniority of the second cosponsor (the “row” cosponsor).

5. OLS and Network Analysis
All could reasonably be expected to influence the likelihood of cosponsorship. More senior legislators may be relatively active in cosponsorship (since they’ve had the time to build up relationships), and legislators who are ideologically similar and who are affiliated with the same political party may be relatively likely to cosponsor with each other.

6. OLS and Network Analysis
This approach is sometimes used in International Relations, in order to analyze the behavior of dyads (or pairs) of nations.
However, there is a problem.

7. OLS and Network Analysis
The problem is that the observations are not independent on each other. If A cosponsors with B, and B cosponsors with C, it may be relatively likely that A cosponsors with C.
Moreover, the fact that there are repeating observations means that the errors are correlated with each other. Observations in individual rows or in individual columns tend to be highly correlated, which inflates or deflates standard errors.

8. OLS and Network Analysis
We could employ a random effects model, which requires modeling and estimating the covariance matrix. But the validity of this depends on whether we have estimated the model correctly—a difficult challenge to meet.

9. OLS and Network Analysis
In IR, it is common to address some of these issues by clustering on one variable. But we have two variables—rows and columns—that need clustering.
In the QAP procedure for network analysis, the standard errors are estimated using permutations of the data set.

10. QAP (Quadratic Assignment Procedure)
Essentially, what the QAP does is to “scramble” the dependent variable data through several permutations. By taking the data, and “scrambling” it repeatedly, resulting in multiple random datasets with the dependent variable—and then multiple analyses can be performed.
Those datasets and analyses form an empirical samping distribution, and we can compare our coefficient with this sampling distribution of coefficients from all the permuted datasets.

11. QAP (Quadratic Assignment Procedure)
(Note that the QAP permutes the rows and columns—but for a single node, the row and column remain the same, and are permuted in the same way, so that the rows and columns for a single node are not separated.)
Essentially, you are preserving the dependence within rows / columns—but removing the relationship between the dependent and independent variables.

12. UCINET & QAP UCINET will allow you to run a QAP.
Go to “Tools”, to “Testing Hypotheses” to “Dyadic” and to “Regression (QAP)”. Choose “Double Dekker Semi-Partialling MRQAP”.

13. UCINET & QAP
Enter in the name of the file with the dependent (cosponsorship variable)
Enter in the name of the two files, one with the “sameparty” information, and one with the “diffideol” information.

14. UCINET & QAP
And click ok.
The results?

15. UCINET & QAP Unstandardized Coefficient Signif. Intercept 8.920
Same Party
-.483
.275
Difference, Ideology
-2.823
.000

16. UCINET & QAP
Once we control for the difference in ideology – which is substantially, significantly, and negatively related to cosponsorship – we find that being of the same party is not a significant influence on cosponsorship.

17. For more information...
“QAP – the Quadratic Assignment Procedure” (William Simpson, Harvard Business School, 2001).
Decker, Krackhardt, Snijders, “Sensitivity of MRQAP Tests to Collinearity and Autocorrelation Conditions”