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MRQAP tutorial for newbies

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.

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MRQAP tutorial Fariba Karimi Fariba.karimi@gesis.org 24.11.2015
Mul=ple Regression Quadra=c Assignment Procedure
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?
OLS (Ordinary Least Square) Y = β0 + β1X1 + β2 X2 +...+ε Dependent variable coefficients Explanatory/independent variables residual
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
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
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.
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.
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.
In other words … •  We are preserving the dependence within rows / columns—but removing the rela=onship between the dependent and independent variables.
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
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
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
QAP process – graph representa=on before reshuffling ajer
Available func=ons •  UCINET: tools -> tes=ng hypothesis -> dyadic - > regression (QAP) •  R: library(statnet) -> netlm •  c/python ?
Example 1 – there is no correla=on
Example 1 – there is no correla=on
Example 2 – there is a correla=on
Example 2 – there is a correla=on
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
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