QAP correlation and QAP regression

QAP Correlation calculates measures of nominal, ordinal, and interval association between the relations in two matrices, and uses quadratic assignment procedures to develop standard errors to test for the significance of association.

网络相关、网络回归 网络议程设置 network agenda setting

二次指派过程（Quadratic Assignment Procedure）

QAP (Quadratic Assignment Procedure， 二次指派过程)是一种对两个方阵中各个格值的相似性进行比较的方法，即它对方阵的各个格值进行比较，给出两个矩阵之间的相关性系数，同时对系数进行非参数检验，它以对矩阵数据的置换为基础。

原理：

（1）首先，计算已知的两个矩阵之间的相关系数。具体地说，为了比较两个矩阵之间的相关性，首先把每个矩阵中的所有取值看成是一个长向量，每个向量包含n(n- 1)个数字(对角线.上的数字忽略不计)。然后像比较任何两个变量之间的相关性那样计算这两个向量之间的相关性系数

（2）其次，对其中的一个矩阵的行和相应的列同时进行随机的置换(而不是仅仅置换行或者列，否则破环原始数据)， 然后计算置换后的矩阵与另一个矩阵之间的相关系数（1），保存计算的结果;重复这种计算过程几百次甚至几千次，将得到一个相关系数的分布，从中可以看到这种随机置换后计算出来的几百或几千个相关系数大于或等于在第一步中计算 出来的观察到的相关系数的比例。

（3）最后，比较在第一步中计算 出来的观察到的相关系数与根据随机重排计算出来的相关系数的分布，看观察到的相关系数是落入拒绝域还是接受域，进而做出判断。也就说，如果上述比例低于0.05，就在统计意义.上表明所研究的两个矩阵之间存在强关系，或者说二者之间出现在相关系数不太可能是随机带来的。

Ucinet

Network Correlation

Correlation between two networks with the same actors

If there is a tie between two particular actors in one relation, is there likely to be a tie between them in another relation? If two actors have a strong tie of one type, are they also likely to have a strong tie of another?

When we have information about multiple relations among the same sets of actors, it is often of considerable interest whether the probability (or strength) of a tie of one type is related to the probability (or strength) of another. Consider the Knoke information and money ties. If organizations exchange information, this may create a sense of trust, making monetary exchange relations more likely; or, if they exchange money, this may facilitate more open communications. That is, we might hypothesize that the matrix of information relations would be positively correlated with the matrix of monetary relations - pairs that engage in one type of exchange are more likely to engage in the other. Alternatively, it might be that the relations are complementary: money flows in one direction, information in the other (a negative correlation). Or, it may be that the two relations have nothing to do with one another (no correlation).

Tools>Testing Hypotheses>Dyadic (QAP)>QAP Correlationcalculates measures of nominal, ordinal, and interval association between the relations in two matrices, and uses quadratic assignment procedures to develop standard errors to test for the significance of association. Figure 18.8 shows the results for the correlation between the Knoke information and monetary exchange networks.

Figure 18.8. Association between Knoke information and Knoke monetary networks by QAP correlation

The first column shows the values of five alternative measures of association. The Pearson correlation is a standard measure when both matrices have valued relations measured at the interval level. Gamma would be a reasonable choice if one or both relations were measured on an ordinal scale. Simple matching and the Jaccard coefficient are reasonable measures when both relations are binary; the Hamming distance is a measure of dissimilarity or distance between the scores in one matrix and the scores in the other (it is the number of values that differ, element-wise, from one matrix to the other).

The third column (Avg) shows the average value of the measure of association across a large number of trials in which the rows and columns of the two matrices have been randomly permuted. That is, what would the correlation (or other measure) be, on the average, if we matched random actors?

The idea of the "Quadratic Assignment Procedure" is to identify the value of the measure of association when their really isn't any systematic connection between the two relations.

This value, as you can see, is not necessarily zero -- because different measures of association will have limited ranges of values based on the distributions of scores in the two matrices. We note, for example, that there is an observed simple matching of .456 (i.e. if there is a 1 in a cell in matrix one, there is a 45.6% chance that there will be a 1 in the corresponding cell of matrix two). This would seem to indicate association. But, because of the density of the two matrices, matching randomly re-arranged matrices will display an average matching of .475. So the observed measure differs hardly at all from a random result.

To test the hypothesis that there is association, we look at the proportion of random trials that would generate a coefficient as large as (or as small as, depending on the measure) the statistic actually observed. These figures are reported (from the random permutation trials) in the columns labeled "P(large)" and "P(small)." The appropriate one of these values to test the null hypothesis of no association is shown in the column "Signif."

Network regression

Rather than correlating one relation with another, we may wish to predict one relation knowing the other. That is, rather than symmetric association between the relations, we may wish to examine asymmetric association. The standard tool for this question is linear regression, and the approach may be extended to using more than one independent variable.

Suppose, for example, that we wanted to see if we could predict which of the Knoke bureaucracies sent information to which others. We can treat the information exchange network as our "dependent" network (with N = 90).

We might hypothesize that the presence of a money tie from one organization to another would increase the likelihood of an information tie (of course, from the previous section, we know this isn't empirically supported!).

Furthermore, we might hypothesize that institutionally similar organizations would be more likely to exchange information. So, we have created another 10 by 10 matrix, coding each element to be a "1" if both organizations in the dyad are governmental bodies, or both are non-governmental bodies, and "0" if they are of mixed types.

We can now perform a standard multiple regression analysis by regressing each element in the information network on its corresponding elements in the monetary network and the government institution network. To estimate standard errors for R-squared and for the regression coefficients, we can use quadratic assignment. We will run many trials with the rows and columns in the dependent matrix randomly shuffled, and recover the R-square and regression coefficients from these runs. These are then used to assemble empirical sampling distributions to estimate standard errors under the hypothesis of no association.

Version 6.81 of UCINET offers four alternative methods for Tools>Testing Hypotheses>Dyadic (QAP)>QAP Regression. Figure 18.9 shows the results of the "full partialling" method.

Figure 18.9. QAP regression of information ties on money ties and governmental status by full partialling method

The descriptive statistics and measure of goodness of fit are standard multiple regression results -- except, of course, that we are looking at predicting relations between actors, not the attributes of actors.

The model R-square (.018) indicates that knowing whether one organization sends money to another, and whether the two organizations are institutionally similar reduces uncertainty in predicting an information tie by only about 2%. The significance level (by the QAP method) is .120. Usually, we would conclude that we cannot be sure the observed result is non-random.

Since the dependent matrix in this example is binary, the regression equation is interpretable as a linear probability model (one might want to consider logit or probit models -- but UCINET does not provide these). The intercept indicates that, if two organizations are not of the same institutional type, and one does not send money to the other, the probability that one sends information to the other is .61. If one organization does send money to the other, this reduces the probability of an information link by .046. If the two organizations are of the same institutional type, the probability of information sending is reduced by .124.

Using the QAP method, however, none of these effects are different from zero at conventional (e.g. p < .05) levels. The results are interesting - they suggest that monetary and informational linkages are, if anything, alternative rather than re-enforcing ties, and that institutionally similar organizations are less likely to communicate. But, we shouldn't take these apparent patterns seriously, because they could appear quite frequently simply by random permutation of the cases.

The tools in the this section are very useful for examining how multi-plex relations among a set of actors "go together." These tools can often be helpful additions to some of the tools for working with multi-plex data that we examined in chapter 16.

https://faculty.ucr.edu/~hanneman/nettext/C18_Statistics.html#twor

http://www.umasocialmedia.com/socialnetworks/networks-lecture-13-qap-correlation/

Gender, Voting, and Cosponsorship in the Maine State Legislature JAMES M. COOK

Python

https://github.com/SocratesAcademy/methods_seminar_2020_network_science

https://github.com/socrateslab/mrqap-python