This document provides an overview of multiple regression quadratic assignment procedure (MRQAP) for network analysis. It explains that MRQAP is useful when observations are not independent due to network dependencies. The procedure works by permuting the dependent variable many times to build a sampling distribution of coefficients to test significance while preserving network dependencies. Examples show how MRQAP can detect correlations between network variables when they exist versus random permutations when there is no correlation. Functions for performing MRQAP in software packages like UCINET and R are also outlined.

1. MRQAP
tutorial
Fariba Karimi
Fariba.karimi@gesis.org
24.11.2015

2. Mul=ple Regression Quadra=c
Assignment Procedure

3. Why regression in network analysis?
• Inferen=al sta=s=cs have proven to have very
useful applica=ons to social network analysis.
At a most general level, the ques=on of
"inference" is: how much confidence can I
have that the pa6ern I see in the data I've
collected is actually typical of some larger
popula?on, or that the apparent pa6ern is
not really just a random occurrence?

4. OLS (Ordinary Least Square)
Y = β0 + β1X1 + β2 X2 +...+ε
Dependent
variable
coefficients
Explanatory/independent
variables
residual

5. OLS (Ordinary Least Square) - test
null-hypothesis à
Small p-value suggests that coefficients are significant.
E.g. p-value 0.01 means that coefficients are significant
with 99% confidence interval.
Y = β0 + β1X1 + β2 X2 +...+ε
β = 0

6. OLS (Ordinary Least Square) - test
• P-value:
null-hypothesis à
Small p-value suggests that coefficients are
significant. E.g. p-value 0.01 means that coefficients
are significant with 99% confidence interval.
• R-squared: quan=fying model performance.
E.g. R-squared = 0.4 means that the model explains
40% of the varia=ons in the dependent variables.
Y = β0 + β1X1 + β2 X2 +...+ε
β = 0

7. Problem
• Observa=ons are not independent of each
other. If A are connected to B and B is
connected C, it maybe likely that A is
connected to C.
• Repea=ng observa=ons à error correlated
with each other. Observa=ons in rows and
columns tend to be highly correlated which
influence the standard error.

8. Problem
• Repea=ng observa=ons à error correlated
with each other. Observa=ons in rows and
columns tend to be highly correlated which
influence the standard error.

9. What does QAP do?
• Essen=ally, what the QAP does is to “scramble”
the dependent variable data through several
permuta?ons. By taking the data, and
“scrambling” it repeatedly, resul=ng in mul=ple
random datasets with the dependent variable—
and then mul=ple analyses can be performed.
• Those datasets and analyses form an empirical
sampling distribu=on, and we can compare our
coefficient with this sampling distribu?on of
coefficients from all the permuted datasets.

10. In other words …
• We are preserving the dependence within
rows / columns—but removing the
rela=onship between the dependent and
independent variables.

11. Friendship, age , class
A B C D E F G
A 0 1 0 0 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
A B C D E F G
A 0 1 0 2 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
A B C D E F G
A 0 1 0 2 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
≈ +
Friendship =e Age difference educa=on

12. Friendship, age , class
A B C D E F G
A 0 1 0 0 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
A B C D E F G
A 0 1 0 2 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
A B C D E F G
A 0 1 0 2 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
≈ +
Friendship =e Age difference educa=on

13. A B C D E F G
A 0 1 0 0 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
A B C D E F G
A 0 1 0 2 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
A B C D E F G
A 0 1 0 2 1 0 0
B 1 0 3 5 1 4 2
C 0 3 0 4 5 8 10
D 2 5 4 0 0 3 2
E 1 1 3 0 0 2 2
F 0 4 2 3 3 0 1
G 0 2 1 2 2 1 0
≈ +
Friendship =e Age difference educa=on
• Permutes dependent variables lots of =me. Measure
the sampling distribu=on of the coefficients.
• P-value is a propor=on of =mes that the observa=on is
Falling outside the sampling distribu=on.
QAP procedure

14. QAP process – graph representa=on
before reshuffling ajer

15. Available func=ons
• UCINET: tools -> tes=ng hypothesis -> dyadic -
> regression (QAP)
• R: library(statnet) -> netlm
• c/python ?

16. Example 1 – there is no correla=on

17. Example 1 – there is no correla=on

18. Example 2 – there is a correla=on

19. Example 2 – there is a correla=on

20. Recap
• QAP is useful when we have dyadic
rela=onship in the data.
• Use netlm func=on in R for the regression
analysis.
• Disadvantage: it is slow for large network size

21. References
• Predic=ng with networks: nonparametric
mul=ple regression analysis of dyadic data, D.
Krackhardt (1981)
• The SNA package, CT Buos (2014)
• hop://svitsrv25.epfl.ch/R-doc/library/sna/html/
qaptest.html
• hop://www.stata.com/mee=ng/1nasug/
simpson.pdf
• hop://www.erikgjesqeld.net/uploads/
3/7/6/8/37685481/
sna_code_(gjesqeld_and_phillips_2013).pdf