## SENSITIVITY OF MRQAP TESTS TO COLLINEARITY AND AUTOCORRELATION CONDITIONS (2007)

### Cached

### Download Links

Citations: | 10 - 0 self |

### BibTeX

@MISC{Dekker07sensitivityof,

author = {David Dekker and David Krackhardt and Tom A. B. Snijders},

title = { SENSITIVITY OF MRQAP TESTS TO COLLINEARITY AND AUTOCORRELATION CONDITIONS},

year = {2007}

}

### OpenURL

### Abstract

Multiple regression quadratic assignment procedures (MRQAP) tests are permutation tests for multiple linear regression model coefficients for data organized in square matrices of relatedness among n objects. Such a data structure is typical in social network studies, where variables indicate some type of relation between a given set of actors. We present a new permutation method (called “double semipartialing”, or DSP) that complements the family of extant approaches to MRQAP tests. We assess the statistical bias (type I error rate) and statistical power of the set of five methods, including DSP, across a variety of conditions of network autocorrelation, of spuriousness (size of confounder effect), and of skewness in the data. These conditions are explored across three assumed data distributions: normal, gamma, and negative binomial. We find that the Freedman–Lane method and the DSP method are the most robust against a wide array of these conditions. We also find that all five methods perform better if the test statistic is pivotal. Finally, we find limitations of usefulness for MRQAP tests: All tests degrade under simultaneous conditions of extreme skewness and high spuriousness for gamma and negative binomial distributions.

### Citations

256 | Randomization, bootstrap and Monte Carlo methods in biology. Second edition - MANLY - 1997 |

249 |
The detection of disease clustering and a generalized regression approach
- Mantel
- 1967
(Show Context)
Citation Context ...ble for all practical purposes. Within the past twenty years, two substantially different approaches to solving this problem have been proffered: linear models tested with the aid of the Mantel test (=-=Mantel, 1967-=-), which is often referred to as the quadratic assignment procedure (QAP) in social network studies, and exponential random graph models. The QAP approach (Mantel, 1967; Hubert, 1987; Krackhardt, 1987... |

202 |
Markov graphs
- Frank, Strauss
- 1986
(Show Context)
Citation Context ...ameters in an OLS model of network autocorrelated data could be tested using a multiple regression extension of the QAP test, called MRQAP. Exponential random graph models (Holland & Leinhardt, 1981; =-=Frank & Strauss, 1986-=-; Wasserman & Pattison, 1996; Snijders, Pattison, Robins, & Handcock, 2006), on the other hand, are a family of models for dichotomous or other discrete network data where the focus is on modeling the... |

191 | Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and p
- Wasserman, Pattison
- 1996
(Show Context)
Citation Context ... of network autocorrelated data could be tested using a multiple regression extension of the QAP test, called MRQAP. Exponential random graph models (Holland & Leinhardt, 1981; Frank & Strauss, 1986; =-=Wasserman & Pattison, 1996-=-; Snijders, Pattison, Robins, & Handcock, 2006), on the other hand, are a family of models for dichotomous or other discrete network data where the focus is on modeling the specific network-related de... |

169 |
UCINET for Windows: Software for Social Network Analysis
- Borgatti, Everett, et al.
- 2002
(Show Context)
Citation Context ...rk analysis it has been applied perhaps unwittingly by many researchers, because it was the only approach to MRQAP implemented in the main network analysis software package UCINET, versions 3 to 6.1 (=-=Borgatti, Everett, & Freeman, 2002-=-). Manly (1997) argued for the Y -permutation, but this approach was criticized by Kennedy and Cade (1996), especially if a non-pivotal statistic is used. Anderson and Robinson (2001) found that the a... |

121 |
Kendall’s Advanced Theory of Statistics, Volume 1 : Distribution Theory”, 6th edition
- Stuart, Ord, et al.
(Show Context)
Citation Context ... data is often skewed. We used results based on cumulants to derive the skewness of the sum of Ri, Cj, Eij in equation (19). Similar to moments, cumulants are descriptive parameters of distributions (=-=Stuart & Ord, 2005-=-, p.86). The skewness of the sum of independent random variables can be derived relatively easily, because the third cumulant equals the third central moment and the cumulant of a sum of independent r... |

83 | New specifications for exponential random graph models. Sociological Methodology 36: 99–153
- Snijders, Pattison, et al.
- 2006
(Show Context)
Citation Context ...ata could be tested using a multiple regression extension of the QAP test, called MRQAP. Exponential random graph models (Holland & Leinhardt, 1981; Frank & Strauss, 1986; Wasserman & Pattison, 1996; =-=Snijders, Pattison, Robins, & Handcock, 2006-=-), on the other hand, are a family of models for dichotomous or other discrete network data where the focus is on modeling the specific network-related dependence structure. The present paper is in th... |

83 | Syndication networks and the spatial distribution of venture capital investments
- Sorenson, Stuart
- 2001
(Show Context)
Citation Context ...kes an exciting step forward in the analysis of network relations” (Pattison, 1988). It has been frequently used to test models in network research (e.g., Borgatti & Cross, 2003; Gibbons & Olk, 2003; =-=Sorenson & Stuart, 2001-=-; Mizruchi, 1990). However, the simulations used in Krackhardt (1988) to justify this approach were confined to a very restricted set of conditions, namely that there is no correlation among the indep... |

61 | A relational view of information seeking and learning in social networks
- Borgatti, Cross
- 2003
(Show Context)
Citation Context ...y and at the time it was said that it “... takes an exciting step forward in the analysis of network relations” (Pattison, 1988). It has been frequently used to test models in network research (e.g., =-=Borgatti & Cross, 2003-=-; Gibbons & Olk, 2003; Sorenson & Stuart, 2001; Mizruchi, 1990). However, the simulations used in Krackhardt (1988) to justify this approach were confined to a very restricted set of conditions, namel... |

54 |
Predicting with networks: Nonparametric multiple regression analysis of dyadic data
- Krackhardt
- 1988
(Show Context)
Citation Context ...ervations within the rows and within the columns of network data) biases the estimates of the second moments to such an extent that it is not uncommon for type I errors of t-statistics to exceed 50% (=-=Krackhardt, 1988-=-), rendering the significance tests unusable for all practical purposes. Within the past twenty years, two substantially different approaches to solving this problem have been proffered: linear models... |

38 |
Significance Tests Which May be Applied to Samples From any Populations,” Supplement to the
- Pitman
- 1937
(Show Context)
Citation Context ...the objects O1, . . . , On. In practice, it suffices to use a relatively small random sample (1000 or 10000) 7sfrom the set of all n! permutations to approximate this reference distribution (see also =-=Pitman, 1937-=-; Mantel, 1967; Jackson & Somers, 1989). Hence, the QAP provides a permutation- or randomization-based nonparametric test of dependence between two square matrix variables of the same size, which may ... |

36 |
Multiple regression and correlation extensions of the Mantel test of matrix correspondence
- SMOUSE, LONG, et al.
- 1986
(Show Context)
Citation Context ...ely attractive approaches to MRQAP for the three matrix case as given in equation 1, are two simple extensions of the bivariate case: the X-permutation and Y -permutation procedures (e.g., Oja, 1987; =-=Smouse, Long, & Sokal, 1986-=-). In the former, we permute the X matrix associated to the tested multiple-regression coefficient and subsequently estimate the multiple regression coefficients for the permuted data sets to generate... |

32 |
Assignment Methods in Combinatorial Data Analysis
- Hubert
- 1987
(Show Context)
Citation Context ... the Mantel test (Mantel, 1967), which is often referred to as the quadratic assignment procedure (QAP) in social network studies, and exponential random graph models. The QAP approach (Mantel, 1967; =-=Hubert, 1987-=-; Krackhardt, 1987) provides a specific type of permutation test which keeps intact the dyadic data structure under the permutations. The principle of this test can be applied to many kinds of models,... |

31 |
QAP partialling as a test spuriousness
- Krackhardt
- 1987
(Show Context)
Citation Context ...st (Mantel, 1967), which is often referred to as the quadratic assignment procedure (QAP) in social network studies, and exponential random graph models. The QAP approach (Mantel, 1967; Hubert, 1987; =-=Krackhardt, 1987-=-) provides a specific type of permutation test which keeps intact the dyadic data structure under the permutations. The principle of this test can be applied to many kinds of models, but it is 2susual... |

31 | Distance-based redundancy analysis: Testing multispecies responses in multifactorial ecological experiments - Legendre, Anderson - 1999 |

31 | K.: Kendall’s advanced theory of statistics, vol 1. Distribution theory - Stuart, Ord - 1994 |

24 |
Networks of Collective Action: A Perspective on Community Influence Systems
- Laumann, Pappi
- 1976
(Show Context)
Citation Context ... frequency of linking between web pages). It has long been understood that the lack of independence of dyadic observations creates a severe barrier to reasonable interpretations of statistical tests (=-=Laumann & Pappi, 1976-=-; Laumann, Marsden, & Galaskiewicz, 1977). Indeed, using standard OLS models on dyadic data with a moderate amount of structural autocorrelation (lack of independence among observations within the row... |

22 |
The strength of strong ties: Social networks and intergroup conflict in organizations
- Nelson
- 1989
(Show Context)
Citation Context ... network data. Method This section presents a simulation study using model specifications consistent with the type of network data to which MRQAP approaches have been applied in the literature (e.g., =-=Nelson, 1989-=-; Krackhardt & Kilduff, 1999; Borgatti & Cross, 2003). Autocorrelations in the sense of within-row and within-column correlations range from nil to strong. Normally distributed data are considered as ... |

21 |
An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model
- MJ, Legendre
- 1999
(Show Context)
Citation Context ...d a variety of techniques for performing these permutation tests and concomitant advantages and disadvantages for different data situations (see Legendre, 2000, for good review on distance data) (see =-=Anderson & Legendre, 1999-=-; Anderson & Robinson, 2001, for good studies on permutation tests in ordinary regression). However, the vast majority of these studies have focused on the type of autocorrelation proposed in the orig... |

21 | A composite likelihood approach to binary spatial data - Heagerty, Lele - 1998 |

19 | A nonstochastic interpretation of reported significance levels - Freedman, Lane - 1983 |

12 | Friendship Patterns and Culture: The Control of Organizational Diversity - Kilduff - 1990 |

11 | Comparison of permutation methods for the partial correlation and partial Mantel tests - Legendre |

10 |
Permutation tests for linear models
- Anderson, Robinson
- 2001
(Show Context)
Citation Context ...the independent variables (that is, the null hypothesis was independence among all variables). More recently, literature on permutation tests has shown that collinearity 1 might cause problems (e.g., =-=Anderson & Robinson, 2001-=-). However, in this literature the problem of the typical kind of dependence of matrix data is usually not addressed and no specific 1 Here we use the term collinearity to indicate any type of correla... |

10 |
Randomization Tests in Econometrics
- Kennedy
- 1995
(Show Context)
Citation Context ...ns (not further reported below). Therefore this approach is not further elaborated here. Another approach proposed in the MRQAP literature is called partialing (Smouse et al., 1986; Krackhardt, 1988; =-=Kennedy, 1995-=-). This approach tests r(ˆɛY Z, ˆɛXZ) in the distribution of reference values r(ˆɛY Z, π(ˆɛXZ)). Anderson and Legendre (1999) found the partialing approach to be incorrect for the case of multiple reg... |

9 |
Randomization Tests, 3rd Edition
- Edgington
- 1995
(Show Context)
Citation Context ...α = .10 (similar to Krackhardt, 1988). To asses the consistency of the different tests based on the Monte Carlo simulations we use the lower and upper boundaries of the 99% expected outcome interval (=-=Edgington, 1995-=-). For the power study we generated data for which the null hypothesis was false, i.e. β �= 0. Data were generated in a similar fashion as in the “type I error”- study described above. The difference ... |

9 |
Individual and structural origins of friendship and social position among professionals
- Gibbons, Olk
- 2003
(Show Context)
Citation Context ... said that it “... takes an exciting step forward in the analysis of network relations” (Pattison, 1988). It has been frequently used to test models in network research (e.g., Borgatti & Cross, 2003; =-=Gibbons & Olk, 2003-=-; Sorenson & Stuart, 2001; Mizruchi, 1990). However, the simulations used in Krackhardt (1988) to justify this approach were confined to a very restricted set of conditions, namely that there is no co... |

9 | Randomization tests for multiple regression - Kennedy, Cade - 1996 |

9 |
Community-Elite Influence Structures: Extension of a Network Approach
- Laumann, Marsden, et al.
- 1977
(Show Context)
Citation Context ...etween web pages). It has long been understood that the lack of independence of dyadic observations creates a severe barrier to reasonable interpretations of statistical tests (Laumann & Pappi, 1976; =-=Laumann, Marsden, & Galaskiewicz, 1977-=-). Indeed, using standard OLS models on dyadic data with a moderate amount of structural autocorrelation (lack of independence among observations within the rows and within the columns of network data... |

9 | Permutation versus bootstrap significance tests in multiple regression and ANOVA - Braak, C - 1992 |

8 | Modeling brain evolution from behavior: a permutational regression approach. Evolution - Legendre, Lapointe, et al. - 1994 |

7 | Permutation tests for multi-factorial analysis of variance - Anderson, Braak, et al. - 2003 |

7 |
On permutation tests in multiple regression and analysis of covariance problems
- Oja
- 1987
(Show Context)
Citation Context ...ns Intuitively attractive approaches to MRQAP for the three matrix case as given in equation 1, are two simple extensions of the bivariate case: the X-permutation and Y -permutation procedures (e.g., =-=Oja, 1987-=-; Smouse, Long, & Sokal, 1986). In the former, we permute the X matrix associated to the tested multiple-regression coefficient and subsequently estimate the multiple regression coefficients for the p... |

6 |
The analysis of social interaction data
- Baker, Hubert
- 1981
(Show Context)
Citation Context ...zero. Application of the QAP method subsequently was proposed for other structurally autocorrelated data such as social network data where square matrices reflect relations between individuals (e.g., =-=Baker & Hubert, 1981-=-; Krackhardt, 1987). Matrices that reflect social networks are specific in that dyadic relations are often asymmetric. This is usually not the case with the distance matrices analyzed in other fields ... |

6 |
Are probability estimates from the permutation model of Mantel’s test stable? Canadian Journal of Zoology 67:766
- Jackson, Somers
- 1989
(Show Context)
Citation Context ...In practice, it suffices to use a relatively small random sample (1000 or 10000) 7sfrom the set of all n! permutations to approximate this reference distribution (see also Pitman, 1937; Mantel, 1967; =-=Jackson & Somers, 1989-=-). Hence, the QAP provides a permutation- or randomization-based nonparametric test of dependence between two square matrix variables of the same size, which may represent, e.g., distances or similari... |

5 | Friendship Among the French Financial Elite - Kadushin - 1995 |

5 |
Network models: Some comments on papers in this special issue
- Pattison
- 1988
(Show Context)
Citation Context ...irst approach. Krackhardt’s MRQAP approach had an appeal of simplicity and accessibility and at the time it was said that it “... takes an exciting step forward in the analysis of network relations” (=-=Pattison, 1988-=-). It has been frequently used to test models in network research (e.g., Borgatti & Cross, 2003; Gibbons & Olk, 2003; Sorenson & Stuart, 2001; Mizruchi, 1990). However, the simulations used in Krackha... |

4 |
Whether close or far: Perceptions of balance in friendship networks in organizations
- Krackhardt, Kilduff
- 1999
(Show Context)
Citation Context ... Method This section presents a simulation study using model specifications consistent with the type of network data to which MRQAP approaches have been applied in the literature (e.g., Nelson, 1989; =-=Krackhardt & Kilduff, 1999-=-; Borgatti & Cross, 2003). Autocorrelations in the sense of within-row and within-column correlations range from nil to strong. Normally distributed data are considered as well as skewed and discrete ... |

4 | Exploring the dynamics of international trade by combining the comparative advantages of multivariate statistics and network visualizations - Krempel, Plumper - 2003 |

4 | Randomization Tests. 3rd edn - Edgington - 1995 |

4 | Predicting with networks—Nonparametric multiple-regression analysis of dyadic data - Krackhardt - 1988 |

3 |
An exponential family of probability-distributions for directed-graphs
- Holland, Leinhardt
- 1981
(Show Context)
Citation Context ...ardt (1988) showed that parameters in an OLS model of network autocorrelated data could be tested using a multiple regression extension of the QAP test, called MRQAP. Exponential random graph models (=-=Holland & Leinhardt, 1981-=-; Frank & Strauss, 1986; Wasserman & Pattison, 1996; Snijders, Pattison, Robins, & Handcock, 2006), on the other hand, are a family of models for dichotomous or other discrete network data where the f... |

3 | Construction of permutation tests - WELCH - 1990 |

2 |
Analyzing pair data and point data on social relationships, attitudes and background characteristics of Costa Rican Census Bureau employees
- Proctor
- 1969
(Show Context)
Citation Context ...s: MRQAP, Mantel tests, Permutation tests, Social Networks, Network Autocorrelation, Collinearity, Dyadic data 1sIntroduction The problem of statistical analysis of social network data is an old one (=-=Proctor, 1969-=-). Such data are dyadic in the sense that for a basic set of objects O1, . . . , On, the observed variables refer to pairs of objects and therefore are doubly indexed, e.g., the variable Yij refers to... |

1 |
Comparison of permutation methods fot the partial correlation and partial mantel tests
- Legendre
- 2000
(Show Context)
Citation Context ... no specific 1 Here we use the term collinearity to indicate any type of correlation among the independent variables used in an analysis. 3sattention is given to QAP-like tests (for an exception, see =-=Legendre, 2000-=-). In this paper we study what the literature on permutation tests in linear models has to offer for MRQAP tests for social network data. We assess the problems that might occur due to collinearity in... |

1 |
Capitalist collective action: Competition, cooperation, and conflict in the coal industry, by J.R
- Mizruchi
- 1990
(Show Context)
Citation Context ...ard in the analysis of network relations” (Pattison, 1988). It has been frequently used to test models in network research (e.g., Borgatti & Cross, 2003; Gibbons & Olk, 2003; Sorenson & Stuart, 2001; =-=Mizruchi, 1990-=-). However, the simulations used in Krackhardt (1988) to justify this approach were confined to a very restricted set of conditions, namely that there is no correlation among the independent variables... |

1 |
An investigation of 3-matrix permutation tests
- Oden, Sokal
- 1992
(Show Context)
Citation Context ...AP Permutation Test for Regression Coefficients Background The QAP approach to studying dyadic data in general has been widely used by researchers in a broad array of disciplines, such as statistics (=-=Oden & Sokal, 1992-=-), biology (Legendre, 2000), and psychology (Hubert, 1987). The approach was first suggested by the statistician Mantel (1967) to address the epidemiological question of whether the distribution of di... |

1 | A composite likelihood approach to binary spatial data - DEKKER, KRACKHARDT, et al. - 1998 |