This tutorial aims to explain MRQAP (quadratic assignment procedure) method in a visual way.

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