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Selected publications of Tom A.B. Snijders
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are presented strictly for personal use only.
Abstract
Bounded confidence and negative influence are two of the most important micro-level mechanisms
employed in computational models of opinion dynamics to explain polarization of opinions.
However, empirical evidence of both mechanisms is debatable. Two common limitations
in existing empirical studies are (1) the limited external validity of laboratory experiments,
and (2) the inability in study designs to disentangle negative influence from bounded
confidence, as well as from other social influence mechanisms like assimilation.
We address both limitations, using the Stochastic Actor-Oriented Model (SAOM)
with a longitudinal field data set that tracks adolescents’ social network relations
and opinions on a set of issues. Two new SAOM effects are introduced to represent bounded
confidence and negative influence, respectively. Results show that for adolescents'
preferences on rap/ hip hop clothing style, the model containing both effects,
in addition to assimilative influence from friends, provides a good fit to the data,
but only the effects representing negative influence and assimilative influence from
friends are statistically significant. The results support that negative influence
contributes to explaining observed opinion changes, but lend little weight to bounded confidence.
Further simulation studies based on the empirically estimated model show that our model
implies only low levels of opinion polarization at the macro-level despite negative
influence at the micro-level.
We conclude that our approach not only overcomes common limitations of earlier empirical work,
but also bridges the SAOM and agent-based modeling by offering
empirically validated insights into opinion dynamics.
Abstract
We introduce a novel procedure to assess the
goodness of fit in relational event models.
Building on existing auxiliary variable approaches developed in network modelling,
the procedure involves a comparison between statistics computed
on observed relational event sequences and statistics calculated
on event sequences simulated from the fitted model.
We argue that the internal time structure of the relational mechanisms
assumed to generate the observations under the model is an
important aspect of the fit of a model to observed relational event sequences.
We establish the empirical value of the proposed goodness of fit
approach in an analysis of data that we collected on collaborative
patient-referral relations among healthcare organizations.
The illustrative case study that we develop reveals
distinctive features of relational event models that have
been ignored or overlooked in received empirical studies.
Abstract
The paper builds on the framework proposed by Pattison and Snijders (2012) for specifying exponential random
graph models (ERGMs) for social networks. We briefly review the two-dimensional hierarchy of potential
dependence structures for network tie variables that they outlined and provide proofs of the relationships among
the model forms and of the nature of their sufficient statistics, noting that models in the hierarchy have the
potential to reflect the outcome of processes of cohesion, closure, boundary and bridge formation and path
creation over short or longer network distances.
We then focus on the so-called partial inclusion dependence
assumptions among network tie variables and the pendant-triangle, or paw, statistics to which they give rise, and
illustrate their application in an empirical setting. We argue that the partial inclusion assumption leads to models
that can reflect processes of boundary and bridge formation and that the model hierarchy provides a broad and
useful framework for the statistical analysis of network data. We demonstrate in the chosen setting that pendant-triangle
(or paw) effects, in particular, lead to a marked improvement in goodness-of-fit and hence add a
potentially valuable capacity for modelling social networks.
M.A. Veldman, S.Doolaard, R.J. Bosker, and T.A.B. Snijders (2024).
Changes in Patterns of Peer Relationships in Primary Education Classroom
Networks Through Cooperative Learning.
School Psychology, in press.
DOI: https://doi.org/10.1037/spq0000617
Abstract
We studied the impact of cooperative learning on positive peer relationships,
that is, liking to work together, in classroom networks.
Cooperative learning was implemented as part of the "Success for All" program.
Longitudinal social network analysis was used to investigate the development
of structures and patterns of relationships in 16 intervention and 16 control classrooms,
including a total of 791 students.
Results showed significantly less reciprocation and transitivity in the
dynamics of the intervention networks,
while the number of nominations in the intervention classes was at least not
smaller than in the control classes, indicating less tendency toward the
formation of small clusters or cliques. We did not find that peer relationships
became more diverse with regard to gender, socioeconomic status,
or academic performance through the implementation of cooperative learning.
In intervention classrooms, children tended to nominate their teammates
with whom they work together in cooperative learning activities;
however, also in control classrooms,
children tended to nominate children with whom they are seated in the classroom.
Tom A.B. Snijders and Christian E.G. Steglich (2024).
Network Dynamics.
Chapter 34 (pp. 501-512) in John McLevey, John Scott, and Peter J. Carrington (eds.),
The SAGE Handbook of Social Network Analysis, 2nd edition. London: Sage, 2024.
Abstract
The stochastic actor oriented model (SAOM)
is a method for modelling social interactions and social behaviour over time.
It can be used to model drivers of dynamic interactions using both
exogenous covariates and endogenous network configurations,
but also the co-evolution of behaviour and social interactions.
In its standard implementations, it assumes that all individual
have the same interaction evaluation function.
This lack of heterogeneity is one of its limitations.
The aim of this paper is to extend the inference framework for the SAOM
to include random effects, so that the heterogeneity of individuals can be modelled more accurately.
We decompose the linear evaluation function that models the
probability of forming or removing a tie from the network,
in a homogeneous fixed part and a random, individual-specific part.
We extend the algorithm so that the variance of the random parameters
can be estimated with method of moments.
Our method is applicable for the general random effect formulations.
We illustrate the method with a random out-degree model and
show the parameter estimation of the random components,
significance tests and model evaluation.
We apply the method to the Kapferer’s Tailor shop study.
It is shown that a random out-degree
constitutes a serious alternative to including transitivity and higher-order dependency effects.
Editorial for the special issue of Network Science om Relational Event Modeling.
Abstract
Relational event models (REMs) for the analysis of social interaction were first introduced 15 years ago.
Since then, a number of important substantive and methodological contributions have produced their progressive
refinement and hence facilitated their increased adoption in studies of social and other networks.
Today REMs represent a well-established class of statistical models for relational processes. This special
issue of Network Science demonstrates the standing and recognition that REMs have achieved within the
network analysis and networks science communities.We wrote this brief introductory editorial essay with
four main objectives in mind: (i) positioning relational event data and models in the larger context of
contemporary network science and social network research; (ii) reviewing some of the most important
recent developments; (iii) presenting the innovative studies collected in this special issue as evidence of
the empirical value of REMs, and (iv) identifying open questions and future research directions.
Abstract
Stochastic actor-oriented models (SAOMs)
are a modelling framework for analysing network dynamics using network panel data.
This paper extends the SAOM to the analysis of multilevel network panels through
a random coefficient model, estimated with a Bayesian approach.
The proposed model allows testing theories about network dynamics,
social influence, and interdependence of multiple networks.
It is illustrated by a study of the dynamic interdependence of friendship
networks and minor delinquency. Data were available for 126 classrooms in
the first year of secondary school, of which 82 were used,
containing relatively few missing data points and having not too much network turnover.
Abstract
Despite the central role of self-assembled groups in animal and human societies, statistical tools to explain
their composition are limited. The authors introduce a statistical framework for cross-sectional observations
of groups with exclusive membership to illuminate the social and organizational mechanisms that
bring people together. Drawing from stochastic models for networks and partitions, the proposed framework
introduces an exponential family of distributions for partitions. The authors derive its main mathematical
properties and suggest strategies to specify and estimate such models. A case study on hackathon
events applies the developed framework to the study of mechanisms underlying the formation of selfassembled
project teams.
Software available from https://github.com/marion-hoffman/ERPM/
Abstract
The study investigates to what degree Indonesias large-scale decentralization and
democratization changed corruption networks. A role structure approach is developed
to move current analysis of dyad-level structures to the network level. This approach
is empirically tested by comparing the relational content and third-party structures of
96 corruption networks operating in the first phase of decentralization (2001–2004),
characterized by a powerful local council, with 94 corruption networks detected in
the second phase (2005–2013), when direct local elections were introduced, and the
power of the local council declined. Building on Fiske’s relational model theory, it is
argued and found that the local executive’s reduced dependence on the local council
provided the opportunity to initiate corrupt exchanges with a broader set of players
both inside and outside the bureaucracy. Whereas deep dependence power relations
(i.e. formal authority) remained important, an increasing proportion of corruption
networks involved compound role structures characterized by both shallow (nonembedded
profit and work relations) and deep interdependence (kin- and friendship).
Emmanuel Lazega, Tom A.B. Snijders, and Rafael P.M. Wittek (eds.) (2022).
A Research Agenda for Social Networks and Social Resilience.
Cheltenham: Edward Elgar.
ISBN: 978 1 80392 577 6
Contents
- Emmanuel Lazega, Rafael P.M. Wittek and Tom A.B. Snijders
Introduction to social networks and social resilience
- Miranda J. Lubbers
Social networks and the resilience of marginalized
communities
- G. Robin Gauthier and Kelly L. Markowski
Gender, social networks and resilience during the
COVID-19 pandemic 17
- Birgit Pauksztat
Social networks and resilience in work teams
- Paola Tubaro
Social networks and resilience in emerging labor
markets
- Tobias H. Stark and Verena Seibel
Inter-ethnic relationships in social networks and
their effect on the resilience of ethnically diverse
societies
- Clemens Kroneberg
Ethnic diversity, social networks, and the social
resilience of schools
- Paolo Campana
Criminal networks and social resilience
- James Hollway
A framework for resilience of and in international
networks
- Karin Ingold, Dimitris Christopoulos and Manuel Fischer
Resilience in political networks
- Isabelle Langrock and Sandra González-Bailón
Protest networks, mobilization, and resilience
- Camille Roth
Resilience of socio-semantic bubbles
- Laura Prota
Food systems between resilience and change:
a social network analysis perspective
- Kayla de la Haye
Social networks to support food and nutrition
security: a case study in the United States
- Mathieu Thomas, Christian Leclerc, Isabelle
Goldringer, Baptiste Rouger, Vanesse Labeyrie
and Sélim Louafi
The importance of seed circulation networks in
the resilience of seed systems
- Petr Matous
Community resilience under rural development
projects and technocratic interventions: agenda
for social network research
- Lea Ellwardt
The social safety net: implications for resilience in
old age
- Kerstin Sailer and Xiaoming Li
Spatial opportunity structures for resilient social
networks: the role of architectural and urban form
- Michele L. Barnes
An integrative network approach for
understanding resilience to environmental change
Christian E.G. Steglich and Tom A.B. Snijders (2022).
Stochastic Network Modeling as Generative Social Science.
Chapter in
Handbook of Sociological Science; Contributions to Rigorous Sociology, edited by
Klarita Gërxhani, Nan Dirk de Graaf and Werner Raub (Edward Elgar, 2022).
ISBN: 978 1 78990 942 5
Abstract
Stochastic models of sociocentric networks were originally developed for testing
hypotheses about micro-level dependencies on the basis of empirical network
data. Such dependencies can lead to phenomena like clustering, hub
formation, and network autocorrelation. Due to the complex nature of sociocentric
networks, parameter estimates of these models are typically obtained
by simulation-based inference. This opens the possibility of re-purposing
these models as simulation tools for the study of emergent macro-level phenomena.
The combination of fitting micro-level models to empirical data sets
and explanation of macro-level outcomes renders these models powerful tools
for sociological inquiry into interdependent social systems. In this chapter,
the use of stochastic actor-oriented models as generative models for such networked
social systems is discussed. This is illustrated with an investigation
of the emergence of subgroups in adolescents' friendship networks.
Supplementary materials are in
www.stats.ox.ac.uk/~snijders/HoRS-Chapter.pdf
Abstract
We review the empirical comparison of Stochastic Actor-oriented Models (SAOMs)
and Temporal Exponential Random Graph Models (TERGMs) by Leifeld & Cranmer
in this journal [Network Science 7(1):20–51, 2019].
When specifying their TERGM, they use exogenous nodal attributes calculated
from the outcome networks’ observed degrees instead of endogenous ERGM
equivalents of structural effects as used in the SAOM. This turns the modeled endogeneity
into circularity and obtained results are tautological.
In consequence, their out-of-sample predictions using TERGMs are based
on out-of-sample information and thereby predict the future using observations from the future.
Thus, their analysis rests on erroneous model specifications that invalidate
the article’s conclusions.
Finally, beyond these specific points, we argue that their evaluation
metric—tie-level predictive accuracy—is unsuited for the task of comparing model performance.
Tomáš Diviák, Casper S. van Nassau, Jan Kornelis Dijkstra, and Tom A.B. Snijders (2022).
Dynamics and disruption: Structural and individual changes in two Dutch Jihadi networks
after police interventions.
Social Networks, 70, 364-374.
DOI:
https://doi.org/10.1016/j.socnet.2022.04.001
Abstract
Longitudinal studies of criminal networks are rare yet necessary to
assess the adaptation and recovery of criminal networks after disruptions.
We study the structural and individual effects of law enforcement disruption
on criminal networks using longitudinal data on two Dutch jihadi networks
(n1 = 57; n2 = 26). We expect that actors in criminal networks under disruption
seek security and to remain concealed as much as possible.
We argue how the structural properties of networks as well as mechanisms
operating at the actor level might change after a disruption.
We used descriptive measures of network cohesion together with core-periphery model
fit and modularity to analyse the structural changes in the networks.
To obtain the actor-level tendencies, we used undirected stochastic actor
oriented models. At the structural level, both networks display opposite
tendencies – the larger one is becoming less cohesive after the disruption,
whereas the smaller network counter-intuitively becomes structurally more cohesive.
Despite the structural differences, the actor-level mechanisms are similar
in both studied networks with actors being inclined towards triadic
closure and reliance on pre-existing ties.
Our study demonstrates the importance of considering the analytical
levels of networks and actors by showing that a disruption may disrupt
the network at the structural level, while triggering contradictory
unintended consequences by increasing individual-level connectivity
as in the case of our second network.
Abstract
Our aim in this study was to determine the
effect of social network characteristics on condom use among FSWs,
considering individual characteristics of the
FSWs and of their sexual partners, characteristics of their relationship,
and the FSW's personal network.
A cross-sectional ego-centric network survey of 170 FSWs was carried out
in Tehran between January and June 2017.
A multilevel ordered logistic regression analysis was conducted to examine
the effects of individual and relational characteristics simultaneously.
The main determinants at the level of the individual FSW were FSWs' age and HIV knowledge.
At the level of the sexual partner,
age and education of sexual partners, as well as intimacy, duration of tie,
frequency of contacts with a given partner,
frequency of contact, perceived social support,
and perceived safe sex norms were significantly associated with condom use.
Abstract
This study aimed to determine the association between social
support and frequency of condom use among FSWs in Tehran, Iran.
Using mixed sampling methods, we recruited 170 FSWs in Tehran in 2017.
We measured self-reported social
support by face-to-face interviews using a standardized questionnaire.
Linear regression was used to assess the association between socio-demographic
characteristics (age, education level, marital status, and place of living),
transactional sex characteristics (age at first transactional sex and
frequency of transactional sex in the last month), HIV knowledge, social
support network characteristics (social network size, duration of tie,
intimacy, social support), and condom use behavior.
The participants perceived moderate social support from sexual partners, low
from peer sex workers, and very low from family members.
Family and peer sex worker social support are associated with condom use,
but less strongly than HIV knowledge or place of living.
However, very few FSWs are socially connected with families.
Interventions to promote condom
use among this vulnerable population should also consider social and familial support.
Abstract
This research focuses on ambivalence in family networks
and presents a framework for investigating how triadic configurations,
beyond specific dyads, may affect individual outcomes such as well-being.
First, the paper introduces the ambivalent triad census,
counting the frequencies of 18 non-isomorphic triads in which ties can be
positive, negative, or ambivalent, in non-directed networks,
and proposes the linear combinations of three theoretical mechanisms
(ambivalent balance, diffusion of stress, divide and conquer) predicting
how embeddedness in an ambivalent triad may affect individual well-being.
Second, the ambivalent triad census is applied to 300 non-directed family networks,
150 stepfamilies and 150 first-time families, in which mothers reported
about the relationships with and between family members.
Results show that mothers embedded in triads exhibiting more ambivalent
balance and less diffusion of stress score higher on social self-esteem.
Abstract
Success for All (SfA) is a comprehensive school reform program with a
strong emphasis on cooperative learning that aims to improve students'
social emotional learning alongside students' cognitive learning.
In the present study it was examined whether SfA led to improved students'
social behavior in Grade 1-3 of primary education. Peer sociometric data
was collected for 974 students aged 6-9. Using multivariate multilevel
analysis we found no significant effect of SfA on students' pro-
and antisocial behavior over time. However, a significant interaction
effect was found showing that antisocial behavior of students from
disadvantaged backgrounds decreased in the intervention condition in Grade 2.
This is a promising finding given that the SfA program was especially
developed for schools serving large numbers of disadvantaged students.
Abstract
This study examines ethnic integration in secondary school.
Social identity theory suggeststhat perception of relevant individual attributes
plays a crucial role in defining ingroupsand outgroups,
contributing to befriending, and disliking others.
Therefore, we analyze therole of peer-perceived ethnicity in social ties.
Networks of friendship, dislike, and perceived ethnicity were modeled together
using dynamic stochastic actor-oriented models, separating the effect of perceived
ethnicity on social ties from that of social ties on perceived ethnicity.
Data came from a Hungarian sample of 12 school classes with one minority group: the Roma.
Treating friendship and dislike as mutually exclusive and comparing them to neutral
relations, we found evidence for the role of perceived ethnicity in dislike
— majority students disliked those they perceived as minority peers.
However, we saw no direct effect of ethnicity on the friendship network.
Implications of the joint modeling of mutually exclusive relationship aspects are discussed.
Robert W. Krause, Mark Huisman, Christian Steglich, Tom A.B. Snijders (2020).
Missing data in cross-sectional networks –
An extensive comparison of missing data treatment methods.
Social Networks, 62, 99-112.
DOI: https://doi.org/10.1016/j.socnet.2020.02.004
Abstract
This paper compares several missing data treatment methods
for missing network data on a diverse set of simulated networks under
several missing data mechanisms. We focus the comparison on three different outcomes:
descriptive statistics, link reconstruction, and model parameters.
The results indicate that the often used methods (analysis of available cases
and null-tie imputation) lead to considerable bias on descriptive statistics
with moderate or large proportions of missing data.
Multiple imputation using sophisticated imputation models based on exponential
random graph models (ERGMs) lead to acceptable biases in descriptive statistics
and model parameters even under large amounts of missing data.
For link reconstruction multiple imputation by simple ERGM performed well on
small data sets, while missing data was more accurately imputed in larger
data sets with multiple imputation by complex Bayesian ERGMs (BERGMs).
Open Access: PDF ISBN 978-3-11-064749-5, EPUB ISBN 978-3-11-064761-7.
DOI:
https://doi.org/10.1515/9783110647495
Abstract
This chapter proposes an integrated statistical approach to studying
the micro-macro transition by combining a random coefficient multilevel
approach with the Stochastic Actor-Oriented Model.
This is elaborated for the substantively interesting and topical
question whether the growing ethnic and religious
diversity in our societies, along with the well-known tendency
for homophily, necessarily lead to a decline in social cohesion.
Abstract
We propose two complementary ways to deal with a
nesting structure in the node set of a network; such
a structure may be called a multilevel network, with a
node set consisting of several groups. First, within-group
ties are distinguished from between-group ties by considering
them as two distinct but interrelated networks.
Second, effects of nodal variables are differentiated
according to the levels of the nesting structure, to prevent
ecological fallacies. This is elaborated in a study
of two repeated observations of a sociability network in
seven villages in Senegal, analyzed using the Stochastic
Actor-oriented Model.
Abstract
Using data on 32 actors and ties among them drawn from available court files,
we combine analytical sociology with statistical models for networks in order to
analyse a case of a counterfeit alcohol distribution network from the Czech Republic.
We formulate a theory of action and identify relational mechanisms which
could explain how the structure of the network emerged. We use the exponential random
graph model to test these mechanisms. The analysis reveals that the two actors
capable of manufacturing the poisonous mixture were considerably,
though not optimally, proximate to others enabling fast distribution of the mixture.
Our model results that the structure was formed by mechanisms of triadic closure,
negative tendency to concentrate ties, and tie translation of pre-existing ties into operational ties.
Abstract
We propose a Mahalanobis distance–based Monte Carlo goodness of
fit testing procedure for the family of stochastic actor-oriented
models for social network evolution.
A modified model distance estimator is proposed to help the researcher
identify model extensions that will remediate poor fit.
Nynke M. D. Niezink, Tom A. B. Snijders, and Marijtje van Duijn (2019).
No longer discrete:
Modeling the dynamics of social networks and continuous behavior.
Sociological Methodology, 49, 295-340.
DOI:
http://dx.doi.org/10.1177/0081175019842263
Abstract
The dynamics of individual behavior are related to the dynamics of the social structures
in which individuals are embedded. This implies that to study social mechanisms
such as social selection or peer influence, we need to model the evolution of
social networks and the attributes of network actors as interdependent processes. The
stochastic actor-oriented model is a statistical approach to study network-attribute coevolution
based on longitudinal data. In its standard specification, the co-evolving
actor attributes are assumed to be measured on an ordinal categorical scale. Continuous
variables first need to be discretized to fit into such a modeling framework. This
article presents an extension of the stochastic actor-oriented model that does away
with this restriction by using a stochastic differential equation to model the evolution
of a continuous attribute. A measure for explained variance is proposed, and an interpretation
of parameter sizes is given. The proposed method is illustrated by a study
of the relationship between friendship, alcohol consumption and self-esteem among
adolescents.
Kim Titlestad, T.A.B. Snijders, Kevin Durrheim, Michael Quayle, and Tom Postmes (2019).
The dynamic emergence of cooperative norms in a social dilemma.
Journal of Experimental Social Psychology, 84, 103799.
DOI: https://doi.org/10.1016/j.jesp.2019.03.010
Abstract
This paper addresses the formation of social norms of cooperation through interaction
in repeated Public Goods Games, using novel multilevel techniques.
Cooperation has traditionally been understood as the interplay of
static factors such as shared social identity and pre-existing norms.
This study investigates the dynamic emergence of cooperative norms in
the presence or absence of social categorization.
A small effect of categorization was found: Categorization helps initiate and maintain
higher levels of cooperation. However, the differences in emergent cooperation
between small groups were much stronger than the differences between the
Categorization and Non-Categorization conditions.
Using explorative analyses, three distinct classes of groups were found.
Within groups, group members follow nearly identical rules for
their choice of cooperative behavior.
We argue that individual behavior converged because of the social
interactions within these groups. Overall, the development of cooperation
is best predicted by the process of norm formation that occurs when social identities emerge.
Abstract
This paper proposes that common measures for network transitivity,
based on the enumeration of transitive triples, do not reflect the
theoretical statements about transitivity they aim to describe.
These statements are often formulated as comparative conditional probabilities,
but these are not directly reflected by simple functions of enumerations.
We think that a better approach is obtained by considering the
probability of a tie between two randomly drawn nodes,
conditional on selected features of the network.
Two measures of transitivity based on correlation coefficients between
the existence of a tie and the existence, or the number,
of two-paths between the nodes are developed, and called “Transitivity Phi”
and “Transitivity Correlation.”
Some desirable properties for these measures are studied and compared
to existing clustering coefficients, in both random (Erdös–Renyi)
and in stylized networks (windmills). Furthermore, it is shown that in a directed graph,
under the condition of zero Transitivity Correlation,
the total number of transitive triples is determined by four underlying features:
size, density, reciprocity, and the covariance between in- and outdegrees.
Also, it is demonstrated that plotting conditional probability of ties,
given the number of two-paths, provides valuable insights into empirical
regularities and irregularities of transitivity patterns.
Abstract
Stochastic actor-oriented models (SAOMs) can be used to analyse dynamic network data,
collected by observing a network and a behaviour in a panel design.
The parameters of SAOMs are usually estimated by the method of moments (MoM)
implemented by a stochastic approximation algorithm, where statistics defining
the moment conditions correspond in a natural way to the parameters.
Here, we propose to apply the generalized method of moments (GMoM),
using more statistics than parameters. We concentrate on statistics
depending jointly on the network and the behaviour, because of the
importance of their interdependence, and propose to add contemporaneous
statistics to the usual cross-lagged statistics. We describe the stochastic
algorithm developed to approximate the GMoM solution. A small simulation study
supports the greater statistical efficiency of the GMoM estimator compared to the MoM.
Abstract
Next to homophily, three other mechanisms are discussed that may play
a role in how numerical actor attributes affect directed social networks:
aspiration, conformity, and sociability. A four-parameter specification
is proposed for the effects of numerical actor attributes on directed
social networks.
Abstract
In some panel data studies, respondents are nested in social contexts,
like classrooms or organizations. The social relations, such as friendship
or collaboration, between the individuals in such contexts can be represented
by social networks. Like individual outcomes (e.g., behavior or performance),
the social relations between individuals are not static.
Social networks and individual outcomes change over time and can mutually affect each other.
In this chapter, we present a statistical model to study the interdependent
dynamics (or coevolution) of network structure and individual outcomes.
We assume panel observations of networks and individual attributes to be
the discrete-time realizations of an underlying continuous-time process,
which is modeled by a stochastic differential equation for the attribute
dynamics and a Markov chain model for the network dynamics.
We illustrate the proposed method by a study of the coevolution of
friendship ties and mathematics grades among 1160 students in 39 classrooms.
Tomáš Diviák, Jan Kornelis Dijkstra, and Tom A. B. Snijders (2018).
Structure, multiplexity, and centrality in a corruption
network: the Czech Rath affair.
Trends in Organized Crime , 22, 274–297.
DOI: https://doi.org/10.1007/s12117-018-9334-y
Abstract
The present study is an analysis of a Czech political corruption network known as
the Rath affair reconstructed with publicly available data. We argue that for the study of
criminal networks it is fruitful to follow a multiplex approach, i.e., to distinguish several
interdependent network dimensions and study how they are interrelated. Relational elements
in corruption are identified, and we propose three dimensions that are essential for understanding
the Rath network: pre-existing ties (e.g., marriage or co-membership of the same
party), resource transfer (e.g., bribing), and collaboration (e.g., communication).
The aim of the study was threefold. We aimed to examine if the network exhibits
a core/periphery structure,
to investigate the multiplex structure of the network by assessing
the overlap of the main dimensions of the network,
and to determine the central and multiplex actors while
considering the differentiation of centrality according to the three network dimensions. The
core/periphery model appears to have a perfect fit to the aggregated network, leading to a
four-block adjacency matrix. Studying the frequency of ties in these blocks shows that
collaboration ties are present in all the blocks, while resource transfer ties are mainly
locatedbetween the core and periphery, and pre-existing ties are rare generally. We also
identify central actors,none of which are strategically positioned, occupying more visible
positions instead. The majority of actors display strong multiplexity in the composition of
their own ties. In the conclusion the potential usefulness of multiplex descriptive measures
and of mixed methods approaches, implications of our results for trust in criminal networks,
and potential merits of analytical sociology approach are discussed.
Abstract
The aim of the present study was to develop and validate a safe sex norm
questionnaire as an appropriate instrument which would be adaptable to the female sex
worker (FSW) population.
A list of 34 items was finalized, with subscales for actual behavioral norms and for
perceived norms.
Non-parametric Mokken scale analysis was used for assessing
scalability and unidimensionality of the questionnaire in a sample of 170 FSWs in Tehran,
and showed that the two subscales were distinct constructs, and all
items are good indicators for the constructs.
Our findings support that the safe sex norm questionnaire is a valid and reliable
measure that would be useful to harm reduction programs and help effective HIV prevention
among female sex workers.
Robert W. Krause, Mark Huisman, Christian E.G. Steglich, and Tom A.B. Snijders (2018).
Missing network data: A comparison of different imputation methods.
Proceedings of the 2018 IEEE/ACM International Conference on
Advances in Social Networks Analysis and Mining.
Abstract
This paper compares several imputation methods for
missing data in network analysis on a diverse set of simulated
networks under several missing data mechanisms. Previous work
has highlighted the biases in descriptive statistics of networks
introduced by missing data. The results of the current study
indicate that the default methods (analysis of available cases
and null-tie imputation) do not perform well with moderate
or large amounts of missing data. The results further indicate
that multiple imputation using sophisticated imputation models
based on exponential random graph models (ERGMs) lead to
acceptable biases even under large amounts of missing data.
Robert W. Krause, Mark Huisman, and Tom A.B. Snijders (2018).
Multiple imputation for longitudinal network data.
Italian Journal of Applied Statistics, 30, 33-57.
DOI: https://doi.org/10.26398/IJAS.0030-002.
Abstract
Missing data on network ties are a fundamental problem for network analysis.
In this paper, we present a new method with two variants to handle missing data
due to actor non-response in the framework of stochastic actor-oriented models (SAOMs).
The proposed method imputes missing tie variables in the first wave either by
using a Bayesian exponential random graph model (BERGMs) or a stationary SAOM
and imputes missing tie variables in later waves utilizing a SAOM.
The proposed method is compared to the standard SAOM missing data treatment
as well as recently proposed methods.
The multiple imputation procedure provided more reliable point estimates than the default treatment.
Abstract
Longitudinal social network studies can easily suffer from insufficient statistical
power. Studies that simultaneously investigate change of network ties
and change of nodal attributes (selection and influence studies) are particularly
at risk because the number of nodal observations is typically much
lower than the number of observed tie variables. This article presents a
simulation-based procedure to evaluate statistical power of longitudinal
social network studies in which stochastic actor-oriented models are to be
applied. Two detailed case studies illustrate how statistical power is strongly
affected by network size, number of data collection waves, effect sizes,
missing data, and participant turnover. These issues should thus be explored
in the design phase of longitudinal social network studies.
Abstract
We examined social mechanisms that account for similarity
in the social context of school-sponsored extracurricular sports activities
among friends. We distinguish two social mechanisms: "shared sports activities that
lead to friendship," whereby friendship formation and maintenance are conditioned
by joint sports participation, and "friendship that leads to shared sports activities,"
a form of social influence whereby adolescents join or maintain certain
sports activities based on their friends' choices.
Using a longitudinal sample of 1,776 10th graders at five high schools in
Southern California, we employed a stochastic actor-oriented multivariate
dynamic model to model the dynamic interplay between the two-mode affiliation
network of adolescents' participation in sports activities and the one-mode
friendship network. As a corresponding descriptive method,
we propose a quantitative measure for the relative strength of the
two-mentioned mechanisms as explanations of the association between
the one-mode and the two-mode network.
Further, we introduce two specifications that represent homophily effects
in the two-mode network and apply them to test gender homophily
in sports participation. The results provide strong evidence for both mechanisms,
with friendship leading to shared sport activities as stronger than shared
sports activities leading to friendships in explaining adolescents'
friendships with co-participants.
Nynke M. D. Niezink and Tom A.B. Snijders (2017).
Co-evolution of Social Networks and Continuous Actor Attributes.
The Annals of Applied Statistics, 11, 1948-1973.
DOI:
http://dx.doi.org/10.1214/17-AOAS1037.
Abstract
Social networks and the attributes of the actors in these networks are not
static; they may develop interdependently over time. The stochastic
actor-oriented model allows for statistical inference on the mechanisms driving
this co-evolution process. In earlier versions of this model, dynamic actor
attributes are assumed to be measured on an ordinal categorical scale. We
present an extension of the stochastic actor-oriented model that does away
with this restriction using a stochastic differential equation to model the
evolution of continuous actor attributes. We estimate the parameters by a
procedure based on the method of moments. The proposed method is applied to
study the dynamics of a friendship network among the students at an
Australian high school. In particular, we model the relationship between
friendship and obesity, focusing on body mass index as a continuous co-evolving
attribute.
Tom A.B. Snijders (2017).
Comment: Modeling of Coordination, Rate Functions, and Missing Ordering Information.
Sociological Methodology, 47, 41-47.
DOI: https://doi.org/10.1177%2F0081175017723346
Abstract
This is a comment on the paper "Dynamic Network Actor Models:
Investigating Coordination Ties through Time"
by Christoph Stadtfeld, James Hollway, and Per Block, published in this volume of
Sociological Methodology.
First the position of their proposed model in relation to other publications for statistical dynamic
network models is described. This is followed by a discussion of the model for
coordination between the two actors at either side of the tie.
Next, it is argued that the rate function may be modeled more in detail in event models,
compared to models for panel data with a small number of waves.
Finally, a remark is made about how to deal with unknown time orderings in event data.
Per Block, Christoph Stadtfeld, and Tom A. B. Snijders (2019).
Forms of dependence: Comparing SAOMs and ERGMs from basic principles.
Sociological Methods and Research,48, 202-239.
DOI: http://dx.doi.org/10.1177/0049124116672680
Abstract
Two approaches for the statistical analysis of social network generation
are widely used; the tie-oriented exponential random graph model
(ERGM) and the stochastic actor-oriented model (SAOM) or Siena
model. While the choice for either model by empirical researchers often
seems arbitrary, there are important differences between these models
that current literature tends to miss. First, the ERGM is defined on the
graph level, while the SAOM is defined on the transition level. This allows
the SAOM to model asymmetric or one-sided tie transition dependence.
Second, network statistics in the ERGM are defined globally but are
nested in actors in the SAOM. Consequently, dependence assumptions in
the SAOM are generally stronger than in the ERGM. Resulting from both,
meso- and macro-level properties of networks that can be represented
by either model differ substantively and analyzing the same network
employing ERGMs and SAOMs can lead to distinct results. Guidelines for
theoretically founded model choice are suggested.
András Vörös and Tom A.B. Snijders (2017).
Cluster analysis of multiplex networks: Defining composite network measures.
Social Networks , 49, 93-112.
DOI: http://dx.doi.org/10.1016/j.socnet.2017.01.002
Abstract
Social relations are multiplex by nature: actors in a group are tied together
by various types of relationships. To understand and explain group processes
it is, therefore, important to study multiple social networks simultaneously
in a given group. However, with multiplexity the complexity of data also
increases. Although some multivariate network methods (e.g. Exponential Random
Graph Models, Stochastic Actor-oriented Models) allow to jointly analyze
multiple networks, modeling becomes complicated when it focuses on more than
a few (2–4) network dimensions. In such cases, dimension reduction methods
are called for to obtain a manageable set of variables.
Drawing on existing statistical methods and measures, we propose a procedure
to reduce the dimensions of multiplex network data measured in multiple groups.
We achieve this by clustering the networks using their pairwise similarities,
and constructing composite network measures as combinations of the networks
in each resulting cluster. The procedure is demonstrated on a dataset of 21
interpersonal network dimensions in 18 Hungarian high-school classrooms.
The results indicate that the network items organize into three
well-interpretable clusters: positive, negative, and social role
attributions. We show that the composite networks defined on these three
relationship groups overlap but do not fully coincide with the network measures
most often used in adolescent research, such as friendship and dislike.
Kayo Fujimoto, Tom A.B. Snijders, and Thomas W. Valente (2017).
Popularity breeds contempt: The evolution of reputational dislike relations
and friendships in high school.
Social Networks, 48, 100-109.
DOI: http://dx.doi.org/10.1016/j.socnet.2016.07.006
Abstract
We examined the dynamics of the perception of 'dislike' ties (reputational dislike)
among adolescents within the contexts of friendship, perceived popularity,
substance use, and Facebook use. Survey data were collected from a
longitudinal sample of 238 adolescents from the 11th and 12th grades
in one California high school. We estimated stochastic actor-based network
dynamic models, using reports of reputational dislike, friendships,
and perceived popularity, to identify factors associated with the maintenance
and generation reputational dislike ties. The results showed that high-status
adolescents and more frequent Facebook users tended to become perceived as or
stay disliked by their peers over time. There was a tendency for friendships
to promote the creation and maintenance of reputational disliking but not
vice versa. Adolescents tended to perceive others as disliked when their friends
also perceived them as disliked. There was no evidence that either cigarette
smoking or drinking alcohol affected reputational dislike dynamics.
This study highlights the important role that the hierarchical peer system,
online peer context, and friendships play in driving information diffusion
of negative peer relations among adolescents.
Abstract
This chapter gives a general introduction to Stochastic Actor-Oriented
Models for Network Dynamics from a statistical point of view.
An e-print link to this article is here at the Siena website.
Snijders, Tom A.B. and Pickup, Mark (2017).
Stochastic Actor-Oriented Models for Network Dynamics.
Pp. 221-247 in
Oxford Handbook of Political Networks, edited by
Jennifer Nicoll Victor, Alexander H. Montgomery, and Mark Lubell.
Oxford: Oxford University Press.
Also available in
OUP Handbooks-online.
DOI: http://dx.doi.org/10.1093/oxfordhb/9780190228217.013.10.
Abstract
Stochastic Actor Oriented Models for Network Dynamics are used for the statistical
analysis of longitudinal network data collected as a panel. The probability model defines
an unobserved stochastic process of tie changes, where social actors add new ties or drop
existing ties in response to the current network structure; the panel observations are
snapshots of the resulting changing network. The statistical analysis is based on
computer simulations of this process, which provides a great deal of flexibility in
representing data constraints and dependence structures.
In this Chapter we begin by defining the basic model.
We then explicate a new model for nondirected ties, including
several options for the specification of how pairs of actors coordinate tie changes. Next,
we describe coevolution models. These can be used to model the dynamics of several
interdependent sets of variables, such as the analysis of panel data on a network and the
behavior of the actors in the network, or panel data on two or more networks. We finish
by discussing the differences between Stochastic Actor Oriented Models and some other
longitudinal network models. A major distinguishing feature is the treatment of time,
which allows straightforward application of the model to panel data with different time
lags between waves. We provide a variety of applications in political science throughout.
ISBN 978-3-319-24518-8
ISBN 978-3-319-24520-1 (eBook)
DOI: 10.1007/978-3-319-24520-1
Abstract
At least two methodologies have helped social scientists of the past two
generations in overcoming the traditional divide between individualistic and holistic
approaches in the social sciences: multilevel analysis and social network analysis.
The purpose of this book is to provide an exploration of the diverse ways in
which these two methodologies can be brought together in statistical approaches
to multilevel network analysis, specifically their combination in the development
of three areas: theory, techniques, and empirical applications in the social sciences.
|
 |
Table of contents
- Emmanuel Lazega and Tom A.B. Snijders: Introduction
- Tom A.B. Snijders :
The Multiple Flavours of Multilevel Issues for Networks
- Emmanuel Lazega : Synchronization Costs in the Organizational Society:
Intermediary Relational Infrastructures in the Dynamics of Multilevel Networks
- Filip Agneessens and Johan Koskinen :
Modeling Individual Outcomes Using a Multilevel Social
Influence (MSI) Model: Individual Versus Team Effects
of Trust on Job Satisfaction in an Organisational Context
- Mark Tranmer and Emmanuel Lazega:
Multilevel Models for Multilevel Network Dependencies
- Peng Wang, Garry Robins, and Petr Matous:
Multilevel Network Analysis Using ERGM and Its Extension
-
Mengxiao Zhu, Valentina Kuskova, Stanley Wasserman,
and Noshir Contractor:
Correspondence Analysis of Multirelational Multilevel Networks
-
Aleš Žiberna and Emmanuel Lazega:
Role Sets and Division of Work at Two Levels of Collective
Agency: The Case of Blockmodeling a Multilevel
(Inter-individual and Inter-organizational) Network
-
Elisa Bellotti, Luigi Guadalupi, and Guido Conaldi:
Comparing Fields of Sciences: Multilevel Networks
of Research Collaborations in Italian Academia
-
Julien Brailly, Guillaume Favre, Josiane Chatellet,
and Emmanuel Lazega:
Market as a Multilevel System
-
Julia Brennecke and Olaf N. Rank:
Knowledge Networks in High-Tech Clusters: A Multilevel
Perspective on Interpersonal and Inter-organizational Collaboration
-
Guillaume Favre, Julien Brailly, Josiane Chatellet,
and Emmanuel Lazega:
Inter-organizational Network Influence on Long-Term
and Short-Term Inter-individual Relationships: The
Case of a Trade Fair for TV Programs Distribution
in Sub-Saharan Africa
-
James Hollway and Johan Koskinen:
Multilevel Bilateralism and Multilateralism: States’
Bilateral and Multilateral Fisheries Treaties and Their
Secretariats
-
Paola Zappa and Alessandro Lomi:
Knowledge Sharing in Organizations: A Multilevel
Network Analysis
-
Emmanuel Lazega and Tom A.B. Snijders:
General Conclusion
ISBN 978-3-319-24518-8
ISBN 978-3-319-24520-1 (eBook)
DOI: 10.1007/978-3-319-24520-1
Abstract
First, an overview is given of the current state of multilevel modeling and of
statistical modeling of social networks, the latter with some extra attention
for network models with a multilevel flavour.
It is argued that both of these modeling approaches can be seen as responses
to Coleman's (1959) plea for a methodology that is oriented
to social organization rather than treating society as a collection
of atomized individuals.
Second, the combination of these approaches is discussed.
'Multilevel' and 'network analysis' can be combined in several ways.
'Multilevel Network Analysis' is defined as the combined network analysis
of several 'parallel' groups; a two-step approach was taken by
Snijders and Baerveldt (2003), integrated approaches were taken for
the p2 model bij Zijlstra, van Duijn, and Snijders (2006),
for the latent space model by Sweet, Thomas, and Junker (2013).
The 'Analysis of Multilevel Networks', on the other hand, is defined
– following Wang, Robins, Pattison, and Lazega (2013) – as
the analysis of a network structure with nodes of different types,
where the meaning of ties depends on the types of nodes they connect.
The implementation of such models is presented for Exponential Random Graph Models
and for Stochastic Actor-Oriented Models.
Other chapters in this volume contain examples; some of Multilevel Network
Analysis, others of the Analysis of Multilevel Networks.
Abstract
The stochastic actor-oriented model (Snijders, Sociological Methodology, 2001)
models the evolution of
networks over time, given panel data in a fixed group of actors,
where at each panel wave the network between these
actors (a digraph structure) as well as attribute variables for
these actors are observed. The parameters of this model
usually are estimated by a stochastic approximation version of
the method of moments. Statistics that correspond to
the parameters in a natural way are used for fitting the model.
Here we present an estimator based on the generalized
method of moments, i.e., using more statistics than parameters,
for minimizing the distance between observed statistics
and their expected values. Again, the resulting equation is solved
by stochastic approximation. Several algorithmic
issues arise that have to be solved in order to obtain a stable procedure.
For some examples we study the resulting gain
in statistical efficiency.
Abstract
This paper examines the collaboration structures and dynamics of the coauthorship
network of all Slovenian researchers. Its goal is to identify the key factors
driving collaboration and the main differences in collaboration behavior across scientific
fields and disciplines.
Two approaches to modelling network dynamics are combined in
this paper: the small-world model and the mechanism of preferential attachment, also
known as the process of cumulative advantage.
Stochastic-actor-based modelling of coauthorship
network dynamics uses data for the complete longitudinal co-authorship networks
for the entire Slovenian scientific community from 1996 to 2010. We confirmed the
presence of clustering in all fields and disciplines. Preferential attachment is far more
complex than a single global mechanism. There were two clear distinctions regarding
collaboration within scientific fields and disciplines. One was that some fields had an
internal national saturation inhibiting further collaboration. The second concerned the
differential impact of collaboration with scientists from abroad on domestic collaboration.
In the natural, technical, medical, and biotechnical sciences, this promotes collaboration
within the Slovenian scientific community while in the social sciences and humanities this
inhibits internal collaboration.
Tom A.B. Snijders and Christian E.G. Steglich (2015).
Representing Micro-Macro Linkages by Actor-Based Dynamic Network Models.
Sociological Methods & Research, 44, 222-271.
DOI: http://dx.doi.org/10.1177/0049124113494573
Abstract
Stochastic actor-based models for network dynamics have the primary aim
of statistical inference about processes of network change, but
may be regarded as a kind of agent-based models. Similar to many other
agent-based models, they are based on local rules for actor behavior.
Different from many other agent-based models, by including elements of generalized
linear statistical models they aim to be realistic detailed representations
of network dynamics in empirical data sets. Statistical parallels to
micro-macro considerations can be found in the estimation of parameters
determining local actor behavior from empirical data, and the
assessment of goodness of fit from the correspondence with network-level
descriptives.
This article studies several network-level consequences of
dynamic actor-based models applied to represent cross-sectional network data.
Two examples illustrate how network-level characteristics can be obtained
as emergent features implied by micro-specifications of actor-based models.
Abstract
The complex interplay between bullying/victimization and defending was
examined using a longitudinal social network approach (stochastic actor-based
models). The (co)evolution of these relations within three elementary schools
(Grades 2–5 at Time 1, ages 8–11, N = 354 children) was investigated across
three time points within a year. Most bullies and defenders were in the
same grade as the victims, although a substantial number of bullies and defenders
were in other grades (most often one grade higher). Defenders were usually
of the same gender as the victims, whereas most bullies were boys, with boys
bullying both boys and girls. In line with goal-framing theory, multiplex
network analyses provided evidence for the social support hypothesis (victims
with the same bullies defended each other over time) as well as the
retaliation hypothesis (defenders run the risk of becoming victimized by
the bullies of the
victims they defend). In addition, the analysis revealed that bullies with
the same victims defended each other over time and that defenders of bullies
initiated harassment of those bullies' victims. This study can be seen as a
starting point in unraveling the relationship dynamics among bullying, victimization,
and defending networks in schools.
Tom A.B. Snijders (2014).
Siena: Statistical Modeling of Longitudinal Network Data.
Pp. 1718-1725 in
Reda Alhajj and Jon Rokne (eds.) Encyclopedia of Social Network Analysis and Mining.
New York: Springer.
DOI:
http://dx.doi.org/10.1007/978-1-4614-6170-8_312.
John M. Light, Julie C. Rusby, Kimberley M. Nies, and Tom A.B. Snijders (2014).
Antisocial behavior trajectories and social victimization
within and between school years in early adolescence.
Journal of Research on Adolescence, 24, 332-336.
DOI: http://dx.doi.org/10.1111/jora.12055.
Abstract
Antisocial behavior typically increases during early adolescence,
but the possibility of seasonal variation has not been examined.
In this study, trajectories of antisocial behavior were estimated
for early adolescent boys and girls. Data were obtained from a
3-year longitudinal study of 11 middle schools in the western
United States (n = 5,742), with assessments completed four times
per academic year. Antisocial behavior increased steadily throughout
6th grade, but beginning in 7th grade for boys and 8th grade for
girls it declined during the school year.
Significant increases between Grades 6–7 and 7–8 were
found for both genders. Trajectories varied by contextual
and individual-level social victimization and gender.
Implications for theoretical development and future studies are discussed.
DOI: http://dx.doi.org/10.1016/j.jmp.2013.05.004.
Abstract
A complete survey of a network in a large population may be prohibitively difficult and costly. So it is
important to estimate models for networks using data from various network sampling designs, such
as link-tracing designs. We focus here on snowball sampling designs, designs in which the members
of an initial sample of network members are asked to nominate their network partners, their network
partners are then traced and asked to nominate their network partners, and so on. We assume an
exponential random graph model (ERGM) of a particular parametric form and outline a conditional
maximum likelihood estimation procedure for obtaining estimates of ERGM parameters. This procedure
is intended to complement the likelihood approach developed by Handcock and Gile (2010) by providing
a practical means of estimation when the size of the complete network is unknown and/or the complete
network is very large. We report the outcome of a simulation study with a known model designed to
assess the impact of initial sample size, population size, and number of sampling waves on properties of
the estimates. We conclude with a discussion of the potential applications and further developments of
the approach.
John M. Light, Charlotte C. Greenan, Julie C. Rushby, Kimberly M. Nies, and
Tom A.B. Snijders (2013).
Onset to First Alcohol Use in Early Adolescence: A Network Diffusion Model.
Journal of Research on Adolescence, 23, 487-499.
DOI:
http://dx.doi.org/10.1111/jora.12064.
Abstract
A novel version the stochastic actor-based modeling (SABM) framework
is applied to model the diffusion of
first alcohol use through middle school–wide longitudinal networks of early
adolescents, aged approximately
11-14 years. Models couple a standard SABM for friendship network evolution
with a proportional hazard model for
first alcohol use. Meta-analysis of individual models for 12 schools found
significant effects for friendship selection
based on the same alcohol use status and for an increased rate of onset to
first use based on exposure to already-onset
peers. Onset rate was greater at higher grades and among participants who
spent more unsupervised time with
friends. Neither selection nor exposure effects interacted with grade,
adult supervision, or gender.
Jürgen Lerner, Margit Bussmann, Tom A. B. Snijders, and Ulrik Brandes (2013).
Modeling frequency and type of interaction in event networks.
Corvinus Journal of Sociology and Social Policy vol.4 (1), 3-32.
Abstract
Longitudinal social networks are increasingly given by event data,
i.e., data coding the time and type of interaction between social actors.
Examples include networks stemming from computer-mediated communication,
open collaboration in wikis, phone call data, and interaction among political actors.
In this paper we propose a general model for networks of dyadic, typed events.
We decompose the probability of events into two components:
the first modeling the frequency of interaction and the second modeling the
conditional event type, i.e., the quality of interaction, given that interaction takes place.
While our main contribution is methodological, for illustration
we apply our model to data about political cooperation and conflicts collected
with the Kansas Event Data System. Special emphasis is given to the fact that
some explanatory variables
affect the frequency of interaction while others rather determine the
level of cooperativeness vs. hostility, if interaction takes place.
Furthermore, we analyze if and how model components controlling for
network dependencies affect findings on the effects of more traditional
predictors such as geographic proximity or joint alliance membership.
We argue that modeling the conditional event type is a valuable - and in some cases superior -
alternative to previously proposed models for networks of typed events.
Rafael Wittek, Tom A.B. Snijders, and Victor Nee (eds.),
The Handbook of Rational Choice Social Research.
Stanford, CA: Stanford University Press (2013).
Tom A.B. Snijders, "Network Dynamics",
pp. 252-279 in Rafael Wittek, Tom A.B. Snijders, and Victor Nee (eds.),
The Handbook of Rational Choice Social Research.
Stanford, CA: Stanford University Press (2013).
Abstract
This chapter considers networks as relational structures
in a given set of social actors, and provides an overview of
models and some empirical results for dynamics of social networks,
considered in a setting of social actors optimizing a utility
function that is based, among other things, on their network
embeddedness (excluding purely rule-based models).
Some attention is also paid to network equilibrium,
this being relevant for network dynamics as a potential final state.
The chapter starts with discussing some basic empirical
regularities for social networks,
and how they can be understood from a rational actor perspective.
This is a background to the rest of the chapter,
focusing on models for representing networks.
It then turns to a discussion of game-theoretic models for network
equilibrium and network dynamics. The game-theoretic approach has
difficulties in becoming well aligned to empirical reality;
the latter is the purpose of the statistical models treated next.
Network dynamics is important especially in studies where not only
the network but also actor properties are endogenized.
Therefore, the second part of the chapter discusses models for
the joint dynamics of networks and actor characteristics,
both in a game-theoretic and in a statistical framework.
D. Lusher, J. Koskinen, and G. Robins,
Exponential Random Graph Models for Social Networks,
Cambridge University Press, 2013.
To this book I contributed three chapters, with co-authors:
-
Tom Snijders and Johan Koskinen, Longitudinal Models.
Chapter 11 (pp. 130-140).
-
Johan Koskinen and Tom Snijders, Simulation, Estimation, and Goodness of Fit.
Chapter 12 (pp. 141-166).
-
Pip Pattison and Tom Snijders, Modeling Social Networks: Next Steps.
Chapter 22 (pp. 287-301).
DOI: http://dx.doi.org/10.1016/j.socnet.2012.05.005.
Abstract
We propose a new stochastic actor-oriented model
for the co-evolution of two-mode and one-mode networks.
The model posits that activities of a set of actors, represented in the
two-mode network, co-evolve with exchanges and interactions between the actors,
as represented in the one-mode network.
The model assumes that the actors, not the activities, have agency.
The empirical value of the model is demonstrated by examining
how employment preferences co-evolve
with friendship and advice relations in a group of seventy-five MBA students.
The analysis shows that activity in the two-mode network,
as expressed by number of employment preferences,
is related to activity in the friendship network,
as expressed by outdegrees.
Further, advice ties between students lead to
agreement with respect to employment preferences.
In addition, considering the multiplexity of
advice and friendship ties yields a better understanding of the
dynamics of the advice relation:
tendencies to reciprocation and homophily
in advice relations are mediated to an important extent
by friendship relations.
The discussion pays attention to the implications of this
study in the broader context of current efforts to model the
co-evolutionary dynamics of social networks and individual behavior.
Gijs Huitsing, Marijtje A.J. van Duijn, Tom A.B. Snijders, Peng Wang, Miia Sainiod,
Christina Salmivalli, René Veenstra (2012).
Univariate and multivariate models of positive and negative networks: Liking,
disliking, and bully-victim relationships.
Social Networks, 34, 645-657.
DOI:
http://dx.doi.org/10.1016/j.socnet.2012.08.001
Abstract
Three relations between elementary school children were investigated: networks of general dislike and
bullying were related to networks of general like. These were modeled using multivariate cross-sectional
(statistical) network models. Exponential random graph models for a sample of 18 classrooms, numbering
393 students, were summarized using meta-analyses. Results showed (balanced) network structures
with positive ties between those who were structurally equivalent in the negative network. Moreover,
essential structural parameters for the univariate network structure of positive (general like) and negative
(general dislike and bullying) tie networks were identified. Different structures emerged in positive and
negative networks. The results provide a starting point for further theoretical and (multiplex) empirical
research about negative ties and their interplay with positive ties.
Paulina Preciado, Tom A.B. Snijders, William J. Burk, Håkan Stattin, and Margaret Kerr (2012).
Does proximity matter? Distance dependence of adolescent friendships.
Social Networks, 34, 18-31.
DOI:
http://dx.doi.org/10.1016/j.socnet.2011.01.002.
Abstract
Geographic proximity is a determinant factor of friendship. Friendship datasets that include detailed geographic information are scarce, and when this information is available, the dependence of friendship on distance is often modelled by pre-specified parametric functions or derived from theory without further empirical assessment. This paper aims to give a detailed representation of the association between distance and the likelihood of friendship existence and friendship dynamics, and how this is modified by a few basic social and individual factors. The data employed is a three-wave network of 336 adolescents living in a small Swedish town, for whom information has been collected on their household locations.
The analysis is a three-step process that combines (1) nonparametric logistic regressions to unravel the overall functional form of the dependence of friendship on distance, without assuming it has a particular strength or shape; (2) parametric logistic regressions to construct suitable transformations of distance that can be employed in (3) stochastic models for longitudinal network data, to assess how distance, individual covariates, and network structure shape adolescent friendship dynamics. It was found that the log-odds of friendship existence and friendship dynamics decrease smoothly with the logarithm of distance.
For adolescents in different schools the dependence is linear, and stronger than for adolescents in the same school. Living nearby accounts, in this dataset, for an aspect of friendship dynamics that is not explicitly modelled by network structure or by individual covariates. In particular, the estimated distance effect is not correlated with reciprocity or transitivity effects.
Snijders, Tom A.B., and Bosker, Roel J.
Multilevel Analysis: An Introduction to Basic and Advanced
Multilevel Modeling, second edition.
London etc.: Sage Publishers, 2012
ISBN 9781849202008 (hardcover), ISBN 9781849202015 (pbk).
xii + 368 p.
The second edition of an extensive textbook on multilevel analysis.
Material about this book is available
at a separate web page.
Chapters
- Introduction
- Multilevel theories, multi-stage sampling, and multilevel models
- Statistical treatment of clustered data
- The random intercept model
- The hierarchical linear model
- Testing and model specification
- How much does the model explain?
- Heteroscedasticity
- Missing data
- Assumptions of the hierarchical linear model
- Designing multilevel studies
- Other methods and models
- Imperfect hierarchies
- Survey weights
- Longitudinal data
- Multivariate multilevel models
- Discrete dependent variables
- Software
Abstract.
The issue of the influence of norms on behavior is as old as sociology itself. This paper explores the
effect of normative homophily (i.e. "sharing the same normative choices") on the evolution of the advice
network among lay judges in a courthouse. Blau's (1955, 1964) social exchange theory suggests that
members select advisors based on the status of the advisor. Additional research shows that members of an
organization use similarities with others in ascribed, achieved or inherited characteristics, as well as other
kinds of ties, to mitigate the potentially negative effects of this strong status rule. We elaborate and test
these theories using data on advisor choice in the Commercial Court of Paris.Weuse a jurisprudential case
about unfair competition (material and "moral" damages), a case thatwesubmitted to all the judges of this
court, to test the effect of normative homophily on the selection of advisors, controlling for status effects.
Normative homophily is measured by the extent to which two judges are equally "punitive" in awarding
damages to plaintiffs. Statistical analyses combine longitudinal advice network data collected among
the judges with their normative dispositions. Contrary to what could be expected from conventional
sociological theories, we find no pure effect of normative homophily on the choice of advisors. In this
case, therefore, sharing the same norms and values does not have, by itself, a mitigating effect and does
not contribute to the evolution of the network. We argue that status effects, conformity and alignments on
positions of opinion leaders in controversies still provide the best insights into the relationship between
norms, structure and behavior.
Key words:Advice networks, Longitudinal analysis, Homophily, Norms, Social selection,
Status, Learning.
Alessandro Lomi, Tom A.B. Snijders, Christian E.G. Steglich, and Vanina Jasmine Torló (2011).
Why Are Some More Peer Than Others?
Evidence from a Longitudinal Study of Social Networks and Individual Academic Performance.
Social Science Research, 40 (2011), 1506-1520.
DOI:
http://dx.doi.org/10.1016/j.ssresearch.2011.06.010 .
Abstract.
Studies of peer effects in educational settings confront two main problems.
The first is the presence of endogenous sorting which confounds the effects
of social influence and social selection on individual attainment.
The second is how to account for the local network dependencies through
which peer effects influence individual behavior. We empirically address
these problems using longitudinal data on academic performance, friendship,
and advice seeking relations among students in a full-time graduate academic
program. We specify stochastic agent-based models that permit estimation of
the interdependent contribution of social selection and social influence
to individual performance. We report evidence of peer effects.
Students tend to assimilate the average performance of their friends
and of their advisors. At the same time, students attaining similar
levels of academic performance are more likely to develop friendship and advice ties.
Together, these results imply that processes of social influence and
social selection are sub-components of a more general a co-evolutionary process
linking network structure and individual behavior.
We discuss possible points of contact between our findings and
current research in the economics and sociology of education.
Key words:
Peer effects; Stochastic actor-oriented models; Social networks; Network dynamics; Education
DOI:
http://dx.doi.org/10.1146/annurev.soc.012809.102709.
Abstract.
Statistical models for social networks as dependent variables must represent
the typical network dependencies between tie variables such as
reciprocity, homophily, transitivity, etc. This review first treats models
for single (cross-sectionally observed) networks and then for network
dynamics.For single networks, the older literature concentrated on conditionally
uniform models. Various types of latent space models have
been developed: for discrete, general metric, ultrametric, Euclidean,
and partially ordered spaces. Exponential random graph models were
proposed long ago but now are applied more and more thanks to the
non-Markovian social circuit specifications that were recently proposed.
Modeling network dynamics is less complicated than modeling single
network observations because dependencies are spread out in time. For
modeling network dynamics, continuous-time models are more fruitful.
Actor-oriented models here provide a model that can represent many
dependencies in a flexible way. Strong model development is now going
on to combine the features of these models and to extend them to more
complicated outcome spaces.
Key words:
Social networks, Statistical modeling, Inference
Miranda J. Lubbers, Tom A.B. Snijders, and Margaretha P.C. van der Werf (2011).
Dynamics of Peer Relationships Across the First Two Years of Junior High
as a Function of Gender and Changes in Classroom Composition.
Journal of Research on Adolescence, 21, 488-504.
DOI:
http://dx.doi.org/10.1111/j.1532-7795.2010.00688.x .
Abstract.
This article examines the dynamics of peer relationships across the first
2 grades of Dutch junior high schools (average age
13 - 14). Specifically, we studied how gender and compositional
changes in classrooms structured the changes in peer
relationships between the 2 grades.
Expectations were derived from past research, and we tested whether these held when
methods for data analysis were applied that control appropriately for
the dependence structure of the data (more specifically,
multilevel analysis and a multilevel application of actor-oriented models
for network evolution). Analyses revealed
that the stability of peer acceptance was relatively low and that it
was affected neither by the level of classroom
stability nor by gender. Dyadic relationships were moderately stable.
Tendencies toward reciprocity, network closure, and
gender similarity shaped the changes in networks of peer relationships
within classes. Contrary to past findings, female
newcomers in classrooms were equally as well accepted as male newcomers
or established class members.
Key words:
Social networks, Statistical modeling, Inference
Tom A.B. Snijders. (2011).
Network Dynamics.
Chapter 33 (pp. 501-513) in John Scott and Peter J. Carrington (eds.),
The SAGE Handbook of Social Network Analysis. London: Sage, 2011.
Abstract.
This article examines the use of various research designs in the social sciences as
well as the choices that are made when a quasi-experimental design is used. A content analysis
was carried out on articles published in 18 social science journals with various impact
factors. The presence of quasi-experimental studies was investigated as well as choices in the
design and analysis stage. It was found that quasi-experimental designs are not very often
used in the inspected journals, and when they are applied they are not very well designed
and analyzed. These findings suggest that the literature on how to deal with selection bias
has not yet found its way to the practice of the applied researcher.
Key words:
Quasi-experiments, Social science, Selection bias, Research designs, Content analysis.
DOI:
http://dx.doi.org/10.1080/0022250X.2010.485707.
Abstract.
For exponential random graph models, under quite general conditions, it is
proved that induced subgraphs on node sets disconnected from the other
nodes still have distributions from an exponential random graph model.
This can help in the theoretical interpretation of such models. An
application is that for saturated snowball samples from a potentially larger
graph which is a realization of an exponential random graph model, it is
possible to do the analysis of the observed snowball sample within the
framework of exponential random graph models without any knowledge of
the larger graph.
Key words:
Connected component, network delineation, network boundary,
random graphs, snowball sample.
Snijders, Tom A.B., Koskinen, Johan, and Schweinberger, Michael (2010).
Maximum Likelihood Estimation for Social Network Dynamics
.
Annals of Applied Statistics, 4, 567-588.
DOI:
http://dx.doi.org/10.1214/09-AOAS313.
Abstract.
A model for network panel data is discussed, based on the assumption
that the observed data are discrete observations of a
continuous-time Markov process on the space of all directed graphs on a given
node set, in which changes in tie variables are independent
conditional on the current graph. The model for tie changes is
parametric
and designed for applications to social network analysis, where the
network dynamics can be interpreted as being generated by choices
made by the social actors represented by the nodes of the graph. An
algorithm for calculating the Maximum Likelihood estimator is
presented, based on data augmentation and stochastic approximation.
An application to an evolving friendship network is given and a small
simulation study is presented which suggests that for small data sets
the Maximum Likelihood estimator is more efficient than the earlier
proposed Method of Moments estimator.
Key words:
Graphs, Longitudinal data, Method of moments,
Stochastic approximation, Robbins-Monro algorithm.
Abstract.
A recurrent problem in the analysis of behavioral dynamics, given a simultaneously evolving
social network, is the difficulty of separating effects of partner selection from effects of social
influence. Because misattribution of selection effects to social influence, or vice versa, suggests
wrong conclusions about the social mechanisms underlying the observed dynamics, special
diligence in data analysis is advisable. While a dependable and valid method would benefit
several research areas, according to the best of our knowledge, it has been lacking in the extant
literature. In this paper, we present a recently developed family of statistical models that enables
researchers to separate the two effects in a statistically adequate manner. To illustrate our
method, we investigate the roles of homophile selection and peer influence mechanisms in the
joint dynamics of friendship formation and substance use among adolescents. Making use of a
three-wave panel measured in the years 1995-97 at a school in Scotland, we are able to assess the
strength of selection and influence mechanisms and quantify the relative contributions of
homophile selection, assimilation to peers, and control mechanisms to observed similarity of
substance use among friends.
Key words:
statistical modeling, social networks, graphs, longitudinal, network dynamics,
smoking, alcohol consumption.
The methods proposed in this paper are implemented in the
SIENA program .
Ulrik Brandes, Jürgen Lerner, and Tom A. B. Snijders:
Networks Evolving Step by Step: Statistical Analysis of Dyadic Event Data.
Proc. 2009 Intl. Conf. Advances in Social Network Analysis and Mining
(ASONAM 2009), pp.200-205. IEEE Computer Society, 2009.
Abstract.
With few exceptions, statistical analysis of social networks
is currently focused on cross-sectional or panel data. On
the other hand, automated collection of network-data often
produces event data, i. e., data encoding the exact time of
interaction between social actors. In this paper we propose
models and methods to analyze such networks of dyadic
events and to determine the factors that influence the frequency
and quality of interaction. We apply our methods to
empirical datasets about political conflicts and test several
hypotheses concerning reciprocity and structural balance
theory.
Software implementation.
This method is implemented in visone
(release of September 5, 2012).
There are some tutorials in the visone wiki on how to use it:
an example-based illustration using Wikipedia edit data can be found on
http://visone.info/wiki/index.php/Wikipedia_edit_networks_(tutorial)
and an example with the political event data is on
http://visone.info/wiki/index.php/Event_networks_(tutorial).
The implementation can deal with general event types and weights and with general date/time formatting
Snijders, T.A.B., Doreian, P. (2010).
Introduction to the special issue on network dynamics.
Social Networks, 32, 1-3.
http://dx.doi.org/10.1016/j.socnet.2009.12.002.
Abstract.
This journal issue contains the first of two connected special
issues on Dynamics of Social Networks. This second special
issue will appear later this year. For a rather long time, attention
to dynamic aspects in Social Network Analysis took the form of
descriptive studies. However, over the last fifteen years model-based
approaches to studying network change have been flowering.
Landmarks were three special issues on Network Evolution of
the Journal of Mathematical Sociology, edited by Frans Stokman
and Patrick Doreian, in 1996 (with a book version: Doreian and
Stokman, 1997), 2001, and 2003. These three special issues demonstrated
how formal and statistical modeling and empirical analysis
were coming together. The 2001 and 2003 special issues were
focused on joining of theoretical developments with the analysis of
empirical data using advanced modeling. This special issue presents
a continuation of jointly using theories and modeling to understand
social network phenomena.
Abstract.
Stochastic actor-based models are models for network dynamics that can
represent a wide variety of influences on network change, and allow to estimate
parameters expressing such influences, and test corresponding hypotheses.
The nodes in the network represent social actors, and the collection
of ties represents a social relation. The assumptions posit that the network
evolves as a stochastic process "driven by the actors', i.e., the model
lends itself especially for representing theories about how actors change
their outgoing ties. The probabilities of tie changes are in part endogenously
determined, i.e., as a function of the current network structure itself, and in
part exogenously, as a function of characteristics of the nodes ('actor covariates')
and of characteristics of pairs of nodes ('dyadic covariates'). In an
extended form, stochastic actor-based models can be used to analyze longitudinal
data on social networks jointly with changing attributes of the actors:
dynamics of networks and behavior.
This paper gives an introduction to stochastic actor-based models for
dynamics of directed networks, using only a minimum of mathematics. The
focus is on understanding the basic principles of the model, understanding
the results, and on sensible rules for model selection.
Key words: statistical modeling, longitudinal, Markov chain, agent-based model,
peer selection, peer influence.
Abstract.
The mutual influence of smoking behavior and friendships in adolescence is studied.
It is attempted to disentangle influence and selection processes in reciprocal and
non-reciprocal friendships. An actor-based model is described for
the co-evolution of friendship networks and smoking behavior.
This model considers alternative selection and influence mechanisms,
and models continuous-time changes in network and behavior.
The data consists of a longitudinal sample of 1326 Finnish adolescents
in 11 high schools. Findings suggest that selection as well as
influence processes play an important role in adolescent smoking behavior.
Selection had a relatively stronger role than influence,
in particular when selecting non-reciprocal friends.
The strength of both influence and selection processes decreased over time.
Key words: Smoking; Adolescents; Selection; Influence; Friends; Reciprocity; Siena
Zijlstra, B.J.H., van Duijn, M.A.J., & Snijders, T.A.B. (2009).
MCMC estimation for the p2 network regression model with crossed random effects.
British Journal of Mathematical and
Statistical Psychology, 62, 143-166.
Abstract.
The p2 model is a statistical model for the analysis of binary relational data with
covariates, as occur in social network studies. It can be characterized as a multinomial
regression model with crossed random effects that reflect actor heterogeneity and
dependence between the ties from and to the same actor in the network. Three
Markov chain Monte Carlo (MCMC) estimation methods for the p2 model are
presented to improve iterative generalized least squares (IGLS) estimation developed
earlier, two of which use random walk proposals. The third method, an independence
chain sampler, and one of the random walk algorithms use normal approximations of
the binary network data to generate proposals in the MCMC algorithms. A large-scale
simulation study compares MCMC estimates with IGLS estimates for networks with 20
and 40 actors.
Emmanuel Lazega, Lise Mounier, Tom Snijders (guest editors).
Revue Française de Sociologie,
Special issue on Dynamics of Networks
(Numéro Spécial sur la Dynamique des Réseaux),
vol. 49 (2008) no. 3.
Emmanuel Lazega, Lise Mounier, Tom Snijders.
Presentation of the special issue, pp. 463-465.
Emmanuel Lazega, Lise Mounier, Tom Snijders, and Paola Tubaro.
Réseaux, normes et controverses. pp. 467-498.
Martin Van der Gaag, Tom A. B. Snijders, and Henk Flap.
Position Generator Measures and their Relationship to other
Social Capital Measures.
Chapter 2 in Nan Lin & Bonnie Erickson (eds.),
Social Capital: Advances in Research.
New York: Aldine de Gruyter (2008).
Snijders, Tom A.B., and Berkhof, Johannes,
Diagnostic checks for multilevel models.
Chapter 3 (pp. 141-175) in Jan de Leeuw & Erik Meijer (eds.),
Handbook of Multilevel Analysis, Springer (2008).
By clicking here you can view or download
a preprint of this chapter in .pdf format.
Abstract.
This chapter is about diagnostics for the two-level
Hierarchical Linear Model (HLM).
It treats various types of residuals and influence diagnostics.
The choice between fixed and random effects (which some authors like
to base on the Hausman test) is discussed.
Methods to estimate and test non-linear fixed effects of explanatory variables
are discussed extensively.
Key words: Residuals, Hausman test, empirical Bayes, spline functions,
deletion residuals, influence diagnostics, non-linear transformations,
mixed models, Hierarchical Linear Model.
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.
Key words:MRQAP,
Mantel tests, permutation tests, social networks, network autocorrelation,
collinearity, dyadic data.
Koskinen, Johan H., and Snijders, Tom A.B., (2007).
Bayesian inference for dynamic social network data.
Journal of Statistical Planning and Inference, 137 (2007), 3930-3938.
DOI:
http://dx.doi.org/10.1016/j.jspi.2007.04.011.
Abstract.
We consider a continuous-time model for the evolution of social networks.
A social network is here conceived as a (di-)graph
on a set of vertices, representing actors, and the changes of interest are
creation and disappearance over time of (arcs) edges in the graph.
Hence we model a collection of random edge indicators that are not,
in general, independent. We explicitly model the
interdependencies between edge indicators that arise from interaction between social entities.
A Markov chain is defined in terms of an embedded chain with holding times and
transition probabilities. Data are observed at fixed points in time and hence we are
not able to observe the embedded chain directly.
Introducing a prior distribution for the parameters we may implement an MCMC
algorithm for exploring the posterior distribution of the
parameters by simulating the evolution of the embedded process between observations.
Key words:
Longitudinal social networks; Data augmentation; Bayesian inference; Random graphs.
Abstract.
This paper describes an empirical comparison of four specifications of the exponential family of random
graph models (ERGM), distinguished by model specification (dyadic independence, Markov, partial
conditional dependence) and, for the Markov model, by estimation method (Maximum Pseudolikelihood,
Maximum Likelihood). This was done by reanalyzing 102 student networks in 57 junior high school classes.
At the level of all classes combined, earlier substantive conclusions were supported by all specifications.However,
the different specifications led to different conclusions for individual classes. PL produced unreliable
estimates (whenMLis regarded as the standard) and had more convergence problems than ML. Furthermore,
the estimates of covariate effects were affected considerably by controlling for network structure, although
the precise specification of the structural part (Markov or partial conditional dependence) mattered less.
Key words: Social networks; ERGM; Dependence structure
Abstract.
Actor-oriented models are described as a longitudinal strategy for examining the co-evolution of
social networks and individual behaviors.We argue that these models provide advantages over conventional
approaches due to their ability to account for inherent dependencies between individuals
embedded in a social network (i.e., reciprocity, transitivity) and model interdependencies between
network and behavioral dynamics. We provide a brief explanation of actor-oriented processes,
followed by a description of parameter estimates, model specification, and selection procedures used
by the Simulation Investigation for Empirical Network Analyses (SIENA) software program
(Snijders, Steglich, Schweinberger, & Huisman, 2006).To illustrate the applicability of these models,
we provide an empirical example investigating the co-evolution of friendship networks and delinquent
behaviors in a longitudinal sample of Swedish adolescents with the goal of simultaneously
assessing selection and influence processes. Findings suggest both processes play a substantive role
in the observed dynamics of delinquent behaviors, with influence having a relatively stronger role
than selection (especially in reciprocated friendships).
Key words: delinquency; friendship networks; interdependence; SIENA
DOI:
http://dx.doi.org/10.1111/j.1467-9531.2006.00176.x.
Abstract.
This article reviews new specifications for exponential random graph models proposed by
Snijders et
al. (2006) and demonstrates their improvement over homogeneous
Markov random graph models in fitting empirical network data. Not only do the new specifications show
improvements in goodness of fit for various data sets, but they also help to avoid the problem of neardegeneracy
that often afflicts the fitting of Markov random graph models in practice, particularly to network
data exhibiting high levels of transitivity. The inclusion of a new higher order transitivity statistic allows
estimation of parameters of exponential graph models for many (but not all) cases where it is impossible
to estimate parameters of homogeneous Markov graph models. The new specifications were used to model
a large number of classical small-scale network data sets and showed a dramatically better performance
than Markov graph models. We also review three current programs for obtaining maximum likelihood
estimates of model parameters and we compare these Monte Carlo maximum likelihood estimates with
less accurate pseudo-likelihood estimates. Finally, we discuss whether homogeneous Markov random graph
models may be superseded by the new specifications, and how additional elaborations may further improve
model performance.
Key words: Exponential random graph models; p* models; Statistical models for social networks
Michael Schweinberger and Tom A.B. Snijders (2007).
Markov models for digraph panel data: Monte Carlo-based
derivative estimation.
Computational Statistics and Data Analysis 51, 4465-4483.
Abstract.
A parametric, continuous-time Markov model for digraph panel data is considered.
The parameter is estimated by the method of moments.
A convenient method for estimating the variance-covariance matrix of the moment estimator relies on the delta method, requiring the Jacobian matrix - that is, the matrix of partial derivatives - of the estimating function.
The Jacobian matrix was estimated hitherto by Monte Carlo methods based on finite differences.
Three new Monte Carlo estimators of the Jacobian matrix are proposed,
which are related to the likelihood ratio / score function method of derivative estimation
and have theoretical and practical advantages compared to the finite differences method.
Some light is shed on the practical performance of the methods
by applying them
in a situation where the true Jacobian matrix is known and in a situation where the true Jacobian matrix is unknown.
Key words:
digraphs, continuous-time Markov processes, gradient
estimation,
likelihood ratio / score function method, variance
reduction, control variates.
The methods proposed in this paper are implemented in the
SIENA program .
Tom A.B. Snijders, Christian E.G. Steglich, and Michael
Schweinberger.
Modeling the co-evolution of networks and behavior.
Pp. 41-71 in Longitudinal models in the behavioral and related sciences,
edited by Kees van Montfort, Han Oud and Albert Satorra;
Lawrence Erlbaum, 2007.
Abstract.
A deeper understanding of the relation
between individual behavior and individual actions on one hand and
the embeddedness of individuals in social structures on the other
hand can be obtained by empirically studying the dynamics of
individual outcomes and network structure, and how these mutually
affect each other. In methodological terms, this means that behavior
of individuals -- indicators of performance and success, attitudes
and other cognitions, behavioral tendencies -- and the ties between
them are studied as a social process evolving over time, where
behavior and network ties mutually influence each other. We propose
a statistical methodology for this type of investigation and
illustrate it by an example.
Key words: statistical
modeling, social networks, graphs, longitudinal, network dynamics.
The methods proposed in this paper are implemented in the SIENA program .
Abstract.
We analyse the co-evolution of social networks and substance use behaviour of adolescents
and address the problem of separating the effects of homophily and assimilation. Adolescents
who prefer friends with the same substance-use behaviour exhibit the homophily principle.
Adolescents who adapt their substance use behaviour to match that of their friends display
the assimilation principle. We use the Siena software to illustrate the co-evolution of
friendship networks, smoking, cannabis use and drinking among sport-active teenagers.
Results indicate strong network selection effects occurring with a preference for same sex
reciprocated relationships in closed networks. Assimilation occurs among cannabis and
alcohol but not tobacco users. Homophily prevails among tobacco and alcohol users.
Cannabis use influences smoking behavior positively (i.e., increasing cannabis increases
smoking). Weaker effects include drinkers smoking more and cannabis users drinking more.
Homophily and assimilation are not significant mechanisms with regard to sporting activity
for any substance. There is, however, a significant reduction of sporting activity among
smokers. Also, girls engaged in less sport than boys. Some recommendations for health
promotion programmes are made.
Key words: statistical
modeling, social networks, graphs, longitudinal, network dynamics.
The methods proposed in this paper are implemented in the
SIENA program .
Tom A.B. Snijders (2006).
Statistical Methods for Network Dynamics.
In: S.R. Luchini et al. (eds.), Proceedings of the XLIII Scientific Meeting,
Italian Statistical Society, pp. 281-296. Padova: CLEUP.
Abstract.
Social networks can be defined as the patterns of ties between social
actors.
This paper gives a review of recently developed statistical models
and estimation methods for the analysis of social network
panel data.
To represent the feedback processes inherent in network dynamics,
it is helpful to regard such panel data as momentary observations
on a continuous-time process on the space of directed graphs.
Tie-oriented and actor-oriented stochastic models are presented,
which can reflect endogenous network dynamics
as well as effects of exogenous variables.
These models do not allow explicit calculations, but they can be implemented
as computer simulation models.
Stochastic approximation methods can be used to estimate the parameters.
An example is given where the models are applied to an early precursor
of email communication.
M.J. Lubbers, M.P.C. Van Der Werf, T.A.B. Snijders, B.P.M. Creemers, and H. Kuyper. The impact of peer relations on academic progress in junior high.
Journal of School Psychology 44 (2006), 491-512.
Abstract.
Abstract
The purpose of this study is to examine whether peer relations within classrooms were related
to students' academic progress, and if so, whether this can be explained by students' relatedness
and engagement, in line with Connell and Wellborn's self-system model. We analyzed data of
18,735 students in 796 school classes in Dutch junior high schools, using multilevel analysis.
Academic progress, conceptualized as regular promotion to the next year versus grade retention,
moving upward, and moving downward in the track system, was measured at the time of
transition between Grades 1 and 2 (equivalent to US Grades 7 and 8). The results indicated that
students who were accepted by their peers had lower probabilities to retain a grade or to move
downward in the track system. Although peer acceptance was associated with relatedness and
engagement, these variables did not explain why peer acceptance was associated to academic progress. Furthermore, peer acceptance and relatedness were more strongly related in classes with
more negative class climates.
Snijders, Tom A.B.,
Multi-level event history analysis for a sibling design: The choice of predictor variables.
In F.J. Yammarino and F. Dansereau (eds.),
Research in Multi-level issues, vol. 5.
Multi-level issues in social systems, p. 243-251 (2006).
Abstract.
The chapter in this volume
by Dronkers and Hox presents an interesting multilevel event history analysis of divorce risks.
The sibling design gives excellent opportunities for studying the similarity between
brothers and sisters in the risks of divorce. Various discussion points are raised,
all of which bear in some way upon the choice of predictor variables in the multilevel
logistic regression. Questions are posed about
the level of detail of modeling time trends; about the fact that sampling weights are a
function of number of siblings; and about the inclusion in the fixed part of the model
of the fraction of previously divorced siblings, which is correlated with the family-level
random intercept.
Christian E.G. Steglich, Tom A.B. Snijders, and Patrick West (2006).
Applying SIENA: An Illustrative Analysis of the Coevolution of
Adolescents' Friendship Networks, Taste in Music, and Alcohol Consumption.
Methodology, 2 (2006), 48-56.
Abstract.
We give a nontechnical introduction into recently developed methods for analyzing the coevolution of social networks and behavior(s) of the network actors. This coevolution is crucial for a variety of research topics that currently receive a lot of attention, such as the role of peer groups in adolescent development. A family of dynamic actor-driven models for the coevolution process is sketched, and it is shown how to use the SIENA software for estimating these models. We illustrate the method by analyzing the coevolution of friendship networks, taste in music, and alcohol consumption of teenagers.
Key words:
network dynamics, longitudinal, social networks, stochastic modeling.
The methods proposed in this paper are implemented in the
SIENA program .
Tom A.B. Snijders,
Philippa E. Pattison, Garry L. Robins, and Mark S. Handcock.
New specifications for exponential random graph models.
Sociological Methodology, 36 (2006), 99-153.
DOI:
http://dx.doi.org/10.1111/j.1467-9531.2006.00176.x
Abstract.
The most
promising class of statistical models for expressing structural
properties of social networks observed at one moment in time, is the
class of Exponential Random Graph Models (ERGMs), also known as p*
models. The strong point of these models is that they can represent
a variety of structural tendencies, such as transitivity, that
define complicated dependence patterns not easily modeled by more
basic probability models. Recently, MCMC algorithms have been
developed which produce approximate Maximum Likelihood estimators.
Applying these models in their traditional specification to observed
network data often has led to problems, however, which can be traced
back to the fact that important parts of the parameter space
correspond to nearly degenerate distributions, which may lead to
convergence problems of estimation algorithms, and a poor fit to
empirical data.
This paper proposes new specifications of
Exponential Random Graph Models. These specifications represent
structural properties such as transitivity and heterogeneity of
degrees by more complicated graph statistics than the traditional
star and triangle counts. Three kinds of statistic are proposed:
geometrically weighted degree distributions, alternating
k-triangles, and alternating independent two-paths. Examples are
presented both of modeling graphs and digraphs, in which the new
specifications lead to much better results than the earlier existing
specifications of the ERGM. It is concluded that the new
specifications increase the range and applicability of the ERGM as a
tool for the statistical analysis of social networks.
Key
words: statistical modeling, social networks, graphs,
transitivity, clustering, maximum likelihood, MCMC, p* model.
Also see Snijders (2002).
The methods
proposed in this paper are implemented in the SIENA program , part of the StOCNET package.
Zijlstra, B.J.H., van Duijn, M.A.J., & Snijders, T.A.B.
The multilevel p2 model - A random effects model for the analysis of multiple
social networks.
Methodology, 2 (2006), 42-47.
Abstract.
This paper proposes a multilevel extension to the p2 model for the analysis of social networks. In the p2 model dichotomous tie observations between actors in a given set can be regressed on explanatory variables. The multilevel p2 model is a model for social networks with a multilevel data structure, e.g., networks observed in multiple schools. It defines an identical model for the independent observations of the same type of social network, where the parameters can be allowed to vary across the social networks using random effects. For the multilevel p2 model a Bayesian MCMC algorithm has been developed, which is briefly described here. The model is applied to investigate reported received practical support among Dutch high school pupils of different ethnic backgrounds.
The methods proposed in this paper are implemented
in
the StOCNET package.
Snijders, Tom A.B. (2005). Entries in Wiley Encyclopedia of
Statistics in Behavioral Science.
The following entries in
B.S. Everitt and D.C. Howell (eds.),
Encyclopedia
of Statistics in Behavioral Science. Chicester (etc.):
Wiley, 2005:
Abstract.
This chapter treats statistical methods for network evolution. It
is argued that it is most fruitful to consider models where network
evolution is represented as the result of many (usually non-observed)
small changes occurring between the consecutively observed networks.
Accordingly, the focus is on models where a continuous-time network
evolution is assumed although the observations are made at discrete
time points (two or more).
Three models are considered in detail, all based on the assumption
that the observed networks are outcomes of a Markov process
evolving in continuous time. The independent arcs model is a trivial
baseline model. The reciprocity model expresses effects of reciprocity,
but lacks other structural effects. The actor-oriented model is based
on a model of actors changing their outgoing ties as a consequence of
myopic stochastic optimization of an objective function. This framework
offers the flexibility to represent a variety of network effects. An
estimation algorithm is treated, based on a Markov chain Monte Carlo
implementation of the method of moments.
Key words:
network evolution, Markov process, stochastic actor-oriented
network model.
Also see Snijders (2001).
The methods proposed in this paper are implemented
in the SIENA program.
Van der Gaag, Martin P.J. and Snijders, Tom A.B. (2005).
The Resource Generator: Social capital quantification with concrete items.
Social Networks, 27, 1-27.
DOI:
http://dx.doi.org/10.1016/j.socnet.2004.10.001.
Abstract.
In research on the social capital of individuals,
there has been little standardisation of measurement instruments.
In this paper we propose two innovations. First, a new
measurement method: the Resource Generator; an instrument with concretely
worded items covering `general' social capital in a population, that
combines advantages of earlier techniques. Construction, use, and first
empirical findings are discussed for a representative sample (N = 1,004)
for the Dutch population in 1999-2000. Second, we propose to
investigate social capital by latent trait analysis, and we
identify separately accessed portions of social capital: prestige and
education related social capital, entrepreneurial social capital, skills
social capital, and personal support social capital. This underlines that
social capital measurement needs multiple measures, and cannot be reduced
to one total measure of indirectly `owned' resources. Constructing a theory-based
Resource Generator can be a challenge for different
contexts of use, but also retrieve meaningful
information for investigating the productivity and goal specificity of social capital.
This paper is part of the
Ph.D. research by Martin van der Gaag on measurement of social capital.
Abstract.
A measure for explained variation is proposed for stochastic actor-driven models
for data on social networks. The measure is based on the entropy of the distribution of the choices
made by the actors during the network evolution process. This measure can be helpful in the
specification and interpretation of statistical models for longitudinal network data.
Key words:
Complete network, Longitudinal study, Dynamics, Explained variation,
Coefficient of Determination, Entropy.
Also see Snijders (2001).
The methods proposed in this paper are implemented
in the SIENA program .
Abstract.
A random effects model is proposed for the analysis of binary dyadic data that
represent a social network or directed graph, using nodal and/or dyadic attributes
as covariates. The network structure is reflected by modeling the dependence
between the relations to and from the same actor or node. Parameter estimates
are proposed that are based on an iterated generalized least squares procedure.
An application is presented to a data set on friendship relations between American
lawyers.
The methods proposed in this paper are implemented
in
the StOCNET package.
Snijders, Tom A.B. (2003). Entries in SAGE Encyclopedia of
Social Science Research Methods.
The following entries in M.
Lewis-Beck, A.E. Bryman, and T.F. Liao (eds.),
The SAGE Encyclopedia of Social Science Research Methods.
Thousand Oaks, CA: Sage, 2003:
- Fixed-Effects Model
(Volume I, 390)
- Hierarchical (Non-)Linear Model (Volume II,
460-461)
- Mixed-Effects Model (Volume II, 652-653)
- Multilevel Analysis
(Volume II, 673-677)
- Random-Effects Model (Volume II,
915-916).
Schweinberger, Michael, and Snijders, Tom A.B. (2003).
Settings in Social Networks: A Measurement Model
Pp. 307-341 in Sociological Methodology - 2003, edited by
R.M. Stolzenberg. Boston and London: Basil Blackwell.
Abstract.
A class of statistical models is proposed which aims to recover latent
settings structures in social networks. Settings may be regarded as
clusters of vertices. The measurement model builds on two assumptions.
The observed network is assumed to be generated by hierarchically
nested latent transitive structures, expressed by ultrametrics.
It is assumed that expected tie strength decreases with ultrametric
distance. The approach could be described as model-based clustering
with an ultrametric space as the underlying metric to capture the dependence
in the observations. Maximum likelihood methods as well
as Bayesian methods are applied for statistical inference. Both approaches
are implemented using Markov chain Monte Carlo methods.
The methods proposed in this paper are implemented
in
the StOCNET package.
Abstract.
Markov chains can be used for the modeling of complex longitudinal network
data. One class of probability models to model the evolution of social
networks are stochastic actor-oriented models for network change, proposed
by Snijders (1996, 2001).
These models are continuous-time Markov chain
models that are implemented as simulation models. In this paper an extension
of the simulation algorithm of stochastic actor-oriented models is proposed
to include networks of changing composition. In empirical research,
the composition of networks may change due to actors joining or leaving the
network at some points in time. The composition changes are modeled as
exogenous events that occur at given time points and are implemented in the
simulation algorithm. The estimation of the network effects and the effects
of actor and dyadic attributes that influence the evolution of the network,
is based on the simulation of Markov chains.
Key words:
network evolution, Markov process, stochastic actor-oriented
network model, changing composition.
Also see Snijders (2001).
The methods proposed in this paper are implemented
in the SIENA. program.
Van der Gaag, Martin P.J. and Snijders, Tom A.B.
An approach to the measurement
of individual social capital.
Pp. 199-218 in H. Flap and B. Völker (eds.),
Creation and Returns of Social Capital. London: Routledge, 2003.
Abstract.
This is a chapter in the volume on
the 1999 SCALE conference on social capital (Amsterdam, december 9-11, 1999).
The chapter presents a conceptual approach to the measurement of social
capital as defined on the level of individuals, with the aim to develop a
yardstick for social capital that can be used in prospective studies
investigating its productivity and goal specificity. It discusses several
theoretical choices that should be made before starting
measurements, and introduces an empirical approach to the
construction of domain specific social capital measures.
This paper is part of the
Ph.D. research by Martin van der Gaag on measurement of social capital.
Wittek, Rafael, van Duijn, Marijtje A.J., and Snijders, Tom A.B.,
Frame decay, informal power, and the escalation of social control
in a management team. A Relational Signaling Perspective.
Research in the Sociology of Organizations,
20 (2003), 355-380.
Abstract.
In a study of conflict in organizations, Lindenberg's relational signaling theory
is used to develop hypotheses on the impact of relationship strength,
network embeddedness, and organizational change on social escalation.
Social escalation is defined as the involvement of one or more third
parties in a conflict. An empirical test is conducted with data on
67 conflicts involving 22 managers, gathered during three years of
ethnographic fieldwork and a longitudinal network study in a
management team of a German Paper Factory.
Multilevel analysis indicates that strong ties between conflicting
parties decrease the level of social escalation, whereas informal
power advantage of one party increases the chances for social escalation.
Both effects disappear over time. It is argued that the dissolving impact
of relationships and networks is due to the disappearance of so-called
solidarity frame-stabilizing activities in the firm.
The results highlight the context-dependence of network
effects and escalation processes.
Maas, Cora J.M., and Snijders, Tom A.B.,
The multilevel approach to repeated measures
for complete and incomplete data
Quality and Quantity, 37 (2003), 71-89.
Abstract.
Repeated measurements often are analyzed by multivariate analysis of
variance (MANOVA). An alternative approach is provided by multilevel
analysis, also called the hierarchical linear model (HLM), which makes use
of random coefficient models. This paper is a tutorial which indicates that
the HLM can be specified in many different ways, corresponding to
different sets of assumptions about the covariance matrix of the repeated
measurements. The possible assumptions range from the very restrictive
compound symmetry model to the unrestricted multivariate model, and
include polynomial and other types of trend models between these two
extremes. Thus, the HLM can be used to steer a useful middle road
between the two traditional methods for analyzing repeated measurements.
Another important advantage of the multilevel approach to analyzing
repeated measures is the fact that it can be easily used also if the data are
incomplete. Thus it provides, e.g., a way to achieve a fully multivariate
analysis of repeated measures with incomplete data. It is discussed also
how the multilevel approach can be used for trend tests.
Key words: MANOVA, incomplete data, missing at random, hierarchical
linear model, Hotelling's test, Wald test, Lawley - Hotelling trace criterion,
trend tests, compound symmetry model.
Snijders, Tom A.B,
Accounting for Degree Distributions
in Empirical Analysis of Network Dynamics.
Pp. 146-161 in: R. Breiger, K. Carley, and P. Pattison (eds.),
Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers.
National Research Council of the National Academies, 2003.
Washington, DC: The National Academies Press.
Abstract.
Degrees (the number of links attached to a given node) play a particular
and important role in empirical network analysis because of their obvious
importance for expressing the position of nodes.
It is argued here that there is no general straightforward relation
between the degree distribution on one hand and structural aspects on
the other hand, as this relation depends on further characteristics of
the presumed model for the network. Therefore empirical inference from
observed network characteristics to the processes that could be responsible
for network genesis and dynamics cannot be based only, or mainly, on the
observed degree distribution.
As an elaboration and practical implementation of this point,
a statistical model for the dynamics of networks, expressed as digraphs
with a fixed vertex set,
is proposed in which the outdegree distribution is governed by parameters
that are not connected to the parameters for the structural dynamics.
The use of such an approach in statistical modeling
minimizes the influence of the observed degrees on the conclusions
about the structural aspects of the network dynamics.
The model is a stochastic actor-oriented model, and deals
with the degrees in a manner resembling Tversky's
Elimination by Aspects approach.
A statistical procedure for parameter estimation in
this model is proposed, and an example is given.
Also see
Snijders (2001).
The methods proposed in this paper are implemented
in the SIENA. program.
Abstract.
A multilevel approach is proposed to the study of the evolution
of multiple networks. In this approach, the basic evolution process
is assumed to be the same, while parameter values may differ
between different networks.
For the network evolution process,
stochastic actor-oriented models are used, of which the parameters
are estimated by Markov chain Monte Carlo methods.
This is applied to the study of effects of delinquent behavior
on friendship formation, a question of long standing in criminology.
The evolution of friendship is studied empirically in 19 school classes.
It is concluded that there is evidence for an effect of
similarity in delinquent behavior on friendship evolution.
Similarity of the degree of
delinquent behavior has a positive effect on tie formation
but also on tie dissolution.
The last result seems to contradict criminological theories, and deserves
further study.
Key words: actor-oriented model; longitudinal data;
social networks; criminology; adolescents.
Also see
Snijders (2001).
Snijders, Tom A.B., and van Duijn, Marijtje A.J. (2002).
Conditional Maximum Likelihood Estimation under Various Specifications of
Exponential Random Graph Models.
Pp. 117-134 in Jan Hagberg (ed.),
Contributions to Social Network Analysis, Information Theory,
and Other Topics in Statistics; A Festschrift in honour of Ove Frank.
University of Stockholm, Department of Statistics.
Abstract.
Markov graphs and exponential random graph models are an important family
of probability distributions for graphs and digraphs because they allow
the kind of dependency that is often considered in social network
analysis, e.g., transitivity of choice. To estimate parameters in these
statistical models, pseudo-likelihood methods have been proposed, but they
are of doubtful value. Maximum likelihood (ML) estimates would be better
but are hard to calculate.
These can be approximated, however, by MCMC methods that solve the moment
equation. The use of MCMC methods in these models often is hampered by
convergence problems, of which the cause can be traced to steepness of the
moments as functions of the parameters;
moreover, in the region where this steepness
occurs, the distribution can have a bimodal shape, which in itself already
leads to serious convergence problems.
A possible way out of these problems is to model the degrees more
carefully. On one hand, precisely modeling the degrees may confine the
algorithm to a region in the parameter space where the moment function is
well-behaved and where the distribution has a unimodal shape. On the other
hand, modeling the degrees may lead to a better fitting model, which also
can lead to a better-behaving algorithm.
Three types of specification of exponential random digraph models are
considered: (1) conditional on the number of ties; (2) conditional on all
in- and out-degrees; (3) conditional on the number of ties, and icluding
incidental vertex parameters. In some examples, it is investigated
how well it is possible to achieve convergence in the MCMC parameter
estimation, and how the parameter estimates differ between these
specifications.
Also see Snijders (JoSS, 2002).
The methods proposed in this paper are implemented
in
the StOCNET package.
David, Beata, and Tom A.B. Snijders.
Estimating the size of the homeless population in Budapest, Hungary.
Quality and Quantity, 36 (2002), 291-303.
Abstract.
In this study we try to estimate the size of the homeless population
in Budapest by using
two "non-standard" sampling methods: snowball sampling and the
capture-recapture method. Using
two methods and three different data sets we are able to compare
the methods as well as the results,
and we also suggest some further applications. Apart from the
practical purpose of our study there
is a methodological one as well: to use two relatively unknown
methods for the estimations of this
very peculiar kind of population.
Key words:
snowball sampling, capture-recapture, hidden population, homeless.
Abstract.
A number of estimation methods of the variance components
in Wing & Kristofferson's model for inter-response times are examined
and compared by means of a simulation study.
The estimation methods studied are the method of moments,
maximum likelihood, and an alternative approach in which
the WK-model is recognized as a moving average model.
Key words:
discrete motor responses, moving average model, EM, maximum likelihood,
method of moments.
Here is a direct link to this internet publication.
By clicking here you can run the JAVA applet
that is used in this paper to demonstrate proprties of the
treated probability model.
The estimation procedure in this publication is available
in the program SIENA.
Abstract.
This paper is about estimating the parameters of the
exponential random graph model, also known as the p* model,
using frequentist Markov chain Monte Carlo (MCMC) methods.
The exponential random graph model is simulated using Gibbs
or Metropolis-Hastings sampling.
The estimation procedures considered are based on
the Robbins-Monro algorithm for approximating
a solution to the likelihood equation.
A major problem with exponential random
graph models resides in the fact that such models
can have, for certain parameter values, bimodal
(or multimodal) distributions
for the sufficient statistics such as the number of ties.
The bimodality of the exponential graph distribution
for certain parameter values seems a severe limitation
to its practical usefulness.
The possibility of bi- or multimodality is reflected in the possibility that the
outcome space is divided into two (or more) regions
such that the more usual type of MCMC algorithms,
updating only single relations, dyads, or triplets,
have extremely long sojourn times
within such regions, and a negligible probability to
move from one region to another.
In such situations, convergence to the target distribution
is extremely slow.
To be useful, MCMC algorithms must be able to make transitions
from a given graph to a very different graph.
It is proposed to include transitions to the graph complement
as updating steps
to improve the speed of convergence to the target distribution.
Estimation procedures implementing these ideas work satisfactorily for some
data sets and model specifications, but not for all.
Key words: p* model; Markov graph; digraphs;
exponential family; maximum likelihood;
method of moments; Robbins-Monro algorithm;
Gibbs sampling; Metropolis-Hastings algorithm.
Also see Snijders, Pattison, Robins, and Handcock (2006).
The methods proposed in this paper are implemented in
the SIENA program in
the StOCNET package,
and also in the standalone pnet program and the R package statnet.
Berkhof, Johannes, and Snijders, Tom A.B.,
Variance Component Testing in Multilevel Models.
Journal of Educational and Behavioral Statistics, 26 (2001), 133-152.
Abstract.
Available variance component tests are reviewed and three new score tests are presented.
In the first score test, the asymptotic normal distribution of the test statistic
is used as a reference distribution.
In the other two score tests, a Satterthwaite approximation is used
for the null distribution of the test statistic.
We evaluate the performance of the score tests and other available tests
by means of a Monte Carlo study.
The new tests are computationally relatively cheap and have
good power properties.
Key words: multilevel models; variance components;
random coefficients; score tests; Monte Carlo study.
DOI:
http://dx.doi.org/10.1111/0081-1750.00099.
Abstract.
A class of statistical models is proposed for longitudinal network data.
The dependent variable is the changing (or evolving) relation network,
represented by two or more observations of a directed graph
with a fixed set of nodes.
The nodes are modeled as actors whose choices determine the network.
Individual and dyadic exogenous variables can be used as covariates.
The change in the network is modeled as the stochastic result of
network effects (reciprocity, transitivity, etc.) and these covariates.
The existing network structure is a dynamic constraint for the
evolution of the structure itself.
The models are continuous time Markov chain models that
can be implemented as simulation models.
The network evolution is modeled as the consequence of the actors
making new choices, or withdrawing existing choices, on the basis
of functions, with fixed and random components, that the actors
try to maximize.
The models parameters must be estimated from observed data.
For estimating and testing these models,
statistical procedures are proposed which are based on the method of moments.
The statistical procedures are implemented
using a stochastic approximation algorithm based on
computer simulations of the network evolution process.
Key words: actor-oriented model; longitudinal data;
continuous-time Markov process;
Robbins-Monro algorithm; simulation models; method of moments;
stochastic approximation;
simulated moments; random utility; Markov chain Monte Carlo.
This paper is related to various other papers;
these can be found by searching in this publication list for the key word SIENA.
The methods proposed in this paper are implemented
in the program SIENA.
Nowicki, Krzysztof, and Snijders, Tom A.B,
Estimation and prediction for stochastic blockstructures.
Journal of the American Statistical Association, 96 (2001),
1077-1087.
The estimation procedure in this publication is available
in the program BLOCKS.
Abstract.
A statistical approach to a posteriori
blockmodeling for digraphs and
valued digraphs is proposed.
The probability model assumes that the vertices
of the digraph are partitioned
into several unobserved (latent) classes and that the
probability of a relationship between two vertices
depends only on the classes to which they belong.
A Bayesian estimator, based on Gibbs sampling, is proposed.
The basic model is not identified, because class labels are arbitrary.
The resulting identifiability problems are solved by restricting inference to
the posterior distributions of invariant functions of the
parameters and the vertex class membership.
In addition, models are considered where class labels are identified by
prior distributions for the class membership of some of the vertices.
The model is illustrated by an example from the social networks literature
(Kapferer's tailor shop).
Key words: Colored graph; Gibbs sampling; latent class model; social
network; cluster analysis; mixture model.
This paper continues earlier work published as
Nowicki and Snijders (1997).
The methods proposed in this paper are implemented
in
the StOCNET package.
Snijders, T.A.B.,
Hypothesis Testing.
International Encyclopedia of the
Social and Behavioral Sciences, vol. 10, 7121-7127.
Amsterdam, etc.: Elsevier, 2001.
Abstract.
A test is a statistical procedure to obtain a statement on the
truth or falsity of a proposition, on the basis of
empirical evidence.
After a brief historical account, attention is given to the
contrasting approaches of R.A. Fisher's significance tests and
J. Neyman and E. Pearson's tests of a null against an alternative hypothesis.
The t-test is treated as a paradigmatic example and used to
illustrate the role of assumptions. The p-value and the
confidence interval are statistical procedures which are more
informative than a test leading merely to the
"reject"/"do not reject" dichotomy.
A discussion is presented about problems in the interpretation and use of
hypothesis tests, and it is argued that these result to a great extent
from our human limitations in reasoning with
uncertain evidence.
Research results do usually not stand on their own
(as assumed in the model of a hypothesis test),
but are to be combined with other results, e.g., by meta-analysis.
Snijders, T.A.B.,
Asymptotic null distribution of person fit statistics with estimated person
parameters.
Psychometrika, 66 (2001), 331-342.
Abstract.
Person fit statistics are considered for dichotomous item response
models. The asymptotic null distribution is derived for
statistics which are linear in the item responses, and in which the
ability parameter is replaced by an estimate.
This allows the asymptotically correct standardization of linear person
fit statistics with estimated ability parameter.
The fact that the ability parameter is estimated
decreases the variance of this distribution.
Key words: item response theory, person fit,
asymptotic approximations.
Snijders, T.A.B., and Hagenaars, J.
Guest editors' introduction to the Special Issue on Causality at work.
Sociological Methods & Research, 30 (2001), 3-10.
This is the introduction to an issue of SMR
on various issues of causal interpretations from
statistical analyses.
The issues contains papers by Willem Saris on
the causal relationship between living conditions
and satisfaction;
by Johannes van der Zouwen and Theo van Tilburg on reactivity
in a panel study on personal networks;
by Peter Abell on causality and low-frequency complex events;
and by Patrick Doreian on causality in social network analysis.
Snijders, T.A.B.,
Sampling.
Chapter 11 (p. 159-174) in
A. Leyland and H. Goldstein (eds.) (2001)
Multilevel Modelling of Health Statistics,
Chichester etc.: Wiley.
Abstract.
The relation between multilevel analysis and multistage sampling
is discussed.
After this, much attention is paid to the determination of sample sizes
in multilevel analysis.
van Baarsen, B., Snijders, T.A.B., Smit, J.H., and van Duijn, M.A.J.,
Lonely but not alone: Emotional isolation and social isolation as
two distinct dimensions of loneliness in older people.
Educational and Psychological Measurement, 61, 2001, 119-135.
Abstract.
This study addresses the validity of the De Jong-Gierveld loneliness scale.
The internal properties of the scale scores were studied
using item response theory (IRT), supplemented by an
external validity study.
Tests of the Rasch model using PML did not support the cumulativeness
and unidimensionality of the scale. Correlational analyses
investigating the relationships between concepts of loneliness and
theoretically relevant external measures supported the bidimensionality
of the loneliness scale. In line with attachment theory and the theory
of relational loneliness, the results stress the significance of
distinguishing between emotional loneliness and social loneliness.
Key words:
loneliness, item response theory, Rasch model, dimensionality, aging.
A volume of essays on IRT, dedicated to Ivo W. Molenaar on the occasion
of his 65th birthday.
Snijders, T.A.B., Two-level non-parametric scaling for dichotomous data.
Pp. 319-338 in
A. Boomsma, M.A.J. van Duijn, and T.A.B. Snijders (eds.),
Essays on Item Response Theory.
Lecture Notes in Statistics, 157. New York: Springer, 2001.
This paper and the accompanying program can be downloaded from my
multilevel page.
Abstract.
This paper considers a design where the
objects to be scaled are the higher level units; nested within each
object are lower level units, called `subjects';
and a set of dichotomous items is administered to each subject.
The subjects are regarded as strictly parallel tests
for the objects.
Examples are the scaling of teachers on the basis of their pupils' responses,
or of neighborhoods on the basis of responses by inhabitants.
A two-level version is elaborated of the non-parametric
scaling method first proposed by Mokken (1971).
The probabilities of positive responses to the items are assumed to be
increasing functions of the value on a latent trait.
The latent trait value for each subject is
composed of an object-dependent value and a
subject-dependent deviation from this value .
The consistency of responses within, but also between objects
is expressed by two-level versions of Loevinger's H coefficients.
The availability of parallel tests is used to calculate
a reliability coefficient.
Key words: Multi-level models, item response theory,
reliability, parallel tests, ecometrics.
Van Yperen, Nico W., and Snijders, Tom A.B.,
Multilevel analysis of the Demands-Control Model,
Journal of Occupational Health Psychology,
5 (2000), 182-190.
Abstract
This study explored the extent to which negative health-related outcomes
are associated with differences between work groups
and with differences between individuals within work groups
using R.A. Karasek's (1979) demands-control model.
The sample consisted of 260 employees in 31 working groups
of a national bank in The Netherlands.
Results suggest that job demands and job control should be conceptualized
as having both group- and individual-level foundations.
Support for Karasek's demands-control model was found only when
these variables were split into the two parts, reflecting
shared perceptions and employees' subjective assessment, respectively.
One of the most appealing practical implications is that absence rates
among homogeneous work groups could be reduced by enhancing actual
control on the job.
Snijders, Tom A.B., and Bosker, Roel J.
Multilevel Analysis: An Introduction to Basic and Advanced
Multilevel Modeling
London etc.: Sage Publishers, 1999
ISBN 0-7619-5889-4 (hardcover), ISBN 0-7619-5890-8 (pbk).
ix + 266 p.
An extensive textbook on multilevel analysis.
Material about this book is available
at a separate web page.
Chapters
- Introduction
- Multilevel theories, multi-stage sampling, and multilevel models
- Statistical treatment of clustered data
- The random intercept model
- The hierarchical linear model
- Testing and model specification
- How much does the model explain?
- Heteroscedasticity
- Assumptions of the hierarchical linear model
- Designing multilevel studies
- Crossed random coefficients
- Longitudinal data
- Multivariate multilevel models
- Discrete dependent variables
- Software
Abstract.
Two procedures are proposed for calculating standard errors
for network statistics. Both are based on resampling of vertices:
the first follows the bootstrap approach,
the second the jackknife approach. In addition, we demonstrate how
to use these estimated standard errors to compare statistics using
an approximate t-test and how statistics can also be
compared by another bootstrap approach that is not based on
approximate normality.
Van Duijn, M.A.J., Van Busschbach, J.T., and Snijders, T.A.B.,
Multilevel analysis of personal networks as dependent variables.
Social Networks, 21 (1999), 187-209.
Abstract.
It is shown that multilevel methods are particularly well-suited
for the analysis of relations in personal networks and the changes in these
relations. Justice is done to the hierarchical nested structure
of the data and the resulting dependence of these observations
'within egos'.
Multilevel techniques can also give more specific insight
on why personal networks change: they allow to distinguish
between the influence of individual and tie characteristics
on the stability of personal networks as a whole and of specific
ties within a personal network.
This is illustrated by an application to changes in networks
of four Dutch samples experiencing different life events.
Abstract.
This paper is about social capital as a second-order resource of
individuals.
In spite of its growing popularity, social capital has mostly been
measured in ad hoc fashions.
This paper discusses possible approaches that could be taken to
measure the social capital of individuals.
What kinds of questions should be posed to the individual,
and how should these questions be integrated to a measure
of his or her social capital?
Several domains of well-being should be distinguished,
and social capital should be measured for these domains separately.
It is argued that aggregation over alters is not additive,
because the main distinction is between having no alter, or
at least one alter who could provide a given resource.
Aggregation over resources is necessary but debatable;
it can be based on either a common valuation, or on statistical
asociations, or on substitutability in the production of the individual's
well-being.
For studying the statistical association between second-order resources
available to a given individual, a distinction is proposed between,
on one hand, within-alter associations, and on the other,
within-ego associations.
The elaboration of these ideas into a questionnaire and a concrete
measurement instrument is being carried out in the
SCALE research programme and its 1999 survey of the
'social networks of the Dutch'.
Key words: social resources, aggregation.
This is further elaborated in
the
Ph.D. research by Martin van der Gaag on measurement of social capital.
Abstract.
Multilevel models are proposed to study relational or dyadic
data from multiple persons in families or other groups.
The variable under study is assumed to refer to a
dyadic relation between individuals in the groups.
The proposed models are elaborations of the Social Relations Model.
The different roles of father, mother, and child
are emphasized in these models.
Multilevel models provide researchers with a
method to estimate the variances and correlations of the
Social Relations Model, as well as to incorporate the effects of
covariates and to test specialized models, even for possibly incomplete data.
MLn/MLwiN macros for fitting these models can be obtained
from my macro page.
Snijders, T.A.B.,
The transition probabilities of the reciprocity model
Journal of Mathematical Sociology, 23 (1999), 241-253.
Abstract.
The reciprocity model is a continuous-time Markov chain model
used for modeling longitudinal network data.
A new explicit expression is derived for its transition probability matrix.
This expression can be checked relatively easily.
Some properties of the transition probabilities are given,
as well as a chi-squared goodness of fit test.
Key words: network dynamics, longitudinal social network data,
continuous-time Markov chain.
Van De Bunt, G.G., Van Duijn, M.A.J., and Snijders, T.A.B.,
Friendship networks through time:
An actor-oriented dynamic statistical network model.
Computational and Mathematical Organization Theory,
5 (1999), 167-192.
Abstract.
We propose a class of actor-oriented statistical models for closed
social networks in general, and friendship networks in particular.
The models are random utility models developed within a rational
choice framework. Based on social psychological and sociological
theories about friendship, mathematical functions capturing
expected utility of individual actors with respect to friendship
are constructed. Expected utility also contains a random
(i.e., unexplained) component. We assume that, given their restrictions
and contact opportunities, individuals evaluate their utility function
and behave such that they maximize the expected amount of utility.
The behavior under consideration is the expression of like and dislike
(choice of friends). Theoretical mechanisms modelled are, e.g.,
the principle of diminishing returns, the tendency towards reciprocated
choices, and the preference for friendship relations with similar others.
Constraints imposed on individuals are, e.g., the structure of the
existing network, and the distribution of personal characteristics
over the respondents. The models are illustrated by means of
a data set collected among university freshmen at 7 points in time
during 1994 and 1995.
The methods used in this paper are implemented
in the SIENA. program.
Key words: rational choice, friendship, Markov processes,
random utility models, simulation, empirical test.
Abstract.
This study employs a multidisciplinary model to explain the adoption of agricultural
innovations in developing economies with reference to hybrid cocoa in Ghana. The empirical
evidence shows that, in the adoption of hybrid cocoa, the support that small-scale
farmers obtain via their social networks is more relevant than the advantage of farm size
enjoyed by large-scale farmers. However, for large-scale farmers, access to a bank loan
strongly increases their chance of adoption compared with small-scale farmers. Contacts
with extension agents, education, and availability of hired labor also have positive effects
on adoption. The social status of the farmers has only an indirect effect on adoption:
farmers with higher social status are more likely to obtain a bank loan, and a bank loan
has a positive impact on adoption.
Bonacich, Ph., Oliver, A., and Snijders, T.A.B.,
Controlling for size in centrality scores.
Social Networks, 20 (1998), 135-141.
Abstract.
All measures of centrality in graphs seem to be correlated with degree,
the sheer number of connections of a position.
There are occasions in which one wants a measure that is not
necessarily related to degree but whose relationship to degree is an
empirical finding. Existing corrections, which force a lack of
correlation, or which have no statistical justification, are inadequate
for this purpose. Based on an
algorithm
developed by Snijders (1991)
for generating random graphs with fixed marginals, we suggest
a measure of centrality that is logically but not necessarily
empirically independent of degree.
Snijders, T.A.B.,
Methodological issues in studying effects of networks in organizations.
Computational and Mathematical Organization Theory,
4 (1998), 205-215.
Abstract.
Three methodological issues are discussed that are important for the analysis
of data on networks in organizations. The first is the two-level nature
of the data: individuals are nested in organizations.
This can be dealt with by using multilevel statistical methods.
The second is the complicated nature of statistical methods for network
analysis. The third issue is the potential of mathematical modeling
for the study of network effects and network evolution in organizations.
Two examples are given of mathematical models for gossip in organizations.
The first example is a model for cross-sectional data, the second is a model
for longitudinal data tha reflect the joint development of network
structure and individual behavior tendencies.
Key words: Multilevel analysis, network analysis,
longitudinal models, mathematical modeling, gossip.
Snijders, T.A.B. & Van Duijn, M.A.J.,
Simulation for statistical inference in dynamic network models.
In: Conte, R., Hegselmann, R. Terna, P. (eds.),
Simulating social phenomena , 493-512. Berlin: Springer (1997).
Abstract.
Actor-oriented models are proposed for the statistical analysis of
longitudinal social network data. These models are implemented as
simulation models, and the statistical evaluation
is based on the method of moments and the Robbins-Monro process
applied to computer simulation outcomes.
In this approach,
the calculations that are required for statistical inference are too
complex to be carried out analytically, and therefore they are replaced
by computer simulation.
The statistical models are continuous-time Markov chains.
It is shown how the reciprocity model of Wasserman
and Leenders can be formulated as a special case of the actor-oriented model.
Key words: Social networks, statistical modeling, actor-oriented model,
continuous-time Markov chain, Robbins-Monro process.
Also see Snijders (2001) and the
SIENA program.
Abstract.
A statistical approach to a posteriori blockmodeling for graphs is
proposed.The model assumes that the vertices of the graph are
partitioned into two unknown blocks and that the probability of an edge
between two vertices depends only on the blocks to which they belong.
Statistical procedures are derived for estimating the probabilities of
edges and for predicting the block structure from observations of the
edge pattern only. ML estimators can be computed using the EM
algorithm, but this strategy is practical only for small graphs. A
Bayesian estimator, based on Gibbs sampling, is proposed. This
estimator is practical also for large graphs. When ML estimators are
used, the block structure can be predicted based on predictive
likelihood. When Gibbs sampling is used,the block structure can be
predicted from posterior predictive probabilities.
A side result is that when the number of vertices tends to infinity while
the probabilities remain constant, the block structure can be recovered
correctly with probability tending to 1.
Key words: Colored graph, EM algorithm, Gibbs sampling, latent class
model, social network.
Also see Nowicki and Snijders (2001)
and the associated computer program
BLOCKS.
Abstract
Approaches to studying sexuality have frequently been based
on examination of individuals.
This book argues that explanations of sexual behaviour
should move away from individualistic approaches.
This chapter proposes a view of sexual life as relational:
the behaviour of partners is not only restricted by social context,
but simultaneously influences and shapes the social context.
There are five main postulates: that sexuality is dyadic
(there is a focus on interactions between pairs of people);
that a relation is viewed as a sequence of interactions,
with an individual acting
on the basis of the expected and perceived answers of another individual;
that there is bargaining and change in the
relationship over time; that relations are embedded in social networks
(the relation is an element in each individual's
system of interpersonal relationships as well as linking the two individuals);
that norms and values are flexible (social norms are not rigid,
and change in response to changes in situations).
These postulates are explained in detail.
Gerlsma, C., Snijders, T.A.B., Van Duijn, M.A.J., & Emmelkamp, P.M.G.,
Parenting and Psychopathology: Differences in Family Members Perceptions
of Parental Rearing Styles.
Personality and Individual Differences, 23 (1997), 271-282.
Abstract
Psychiatric patients generally report more adverse recollections of their
parents' rearing behaviour than individuals from the general community.
It is, however, as yet unclear whether we can infer from this finding
that the families of psychiatric patients differ from the families of
healthy controls, that is, whether patients' adverse views are shared
by their family members. This issue bears on the construct validity of
reports about parental rearing styles: should these reports be
interpreted to reflect characteristics of the family, of the
parent-child relationship, or of the individual providing the reports?
In this study, patterns of agreement and variability within families
with regard to recalled parental behaviour were analysed in order to
examine this aspect of the validity of parental representations. We
examined whether families of psychiatric patients report less favourable
parenting styles than families of healthy controls. Furthermore, we
examined the level of agreement between all family members participating
in the study, between the two members reporting on the same parent-child
relationship, between parents, and between siblings. Finally, we examined
what factors might be accountable for differences of opinion between
family members. Results suggested that perceptions of parental rearing
styles are primilary tales by individuals, and to a much smaller extend
tales about families, parents of relationships. The implications of these
findings for research with regard to the relationship between parental
rearing behaviour and adult psychopathology are discussed.
T.A.B. Snijders, E.P.H. Zeggelink, and F.N. Stokman, Parameters in
collective decision making models: estimation and sensitivity,
Mathématiques, Informatique et Sciences Humaines,
137 (1997), 81-99.
Abstract.
Simulation models for collective decision making are based on
theoretical and empirical insight in the decision making process, but
still contain a number of parameters of which the values are
determined ad hoc. For the dynamic access model, some of such
parameters are discussed, and it is proposed to extend the utility
functions with a random term of which the variance also is an unknown
parameter. These parameters can be estimated by fitting model
predictions to data, where the predictions can refer to decision
outcomes but also to network structure generated as a part of the
decision making process. Given the stochastic nature of the model, this
parameter estimation can be carried out with the Robbins Monro
process. Such fitting is not completely straightforward: statistics must
be chosen on which to base the parameter estimation, it is not certain
a priori that there will be a solution to the estimating equation and
that the Robbins Monro process will converge. The method is
illustrated with data from the financial restructuring of a large company.
Key words. Dynamic access model, policy networks, computer
simulation, method of moments, Robbins Monro process.
Snijders, T.A.B., and Spreen, M.,
Segmentation in personal networks.
Mathematiques, Informatique et Sciences Humaines, 35 (1997), 25-36.
Abstract.
A concept and several measures for segmentation of personal networks
are proposed. It is argued that the implications of segmentation of
personal networks are, in a sense, the opposite of those of
segmentation of entire networks. The measures are illustrated by the
example of the trust network in a civil service department. For the case
where relations in the personal network are observed by a sample
rather than completely, estimators for the segmentation measures are
given.
Snijders, T.A.B., Stochastic actor-oriented dynamic network analysis.
Journal of Mathematical Sociology, 21 (1996), 149-172.
Also see Snijders (2001),
Snijders and Van Duijn (1997), and the
SIENA program .
Abstract.
A class of models is proposed for longitudinal network data. These
models are along the lines of methodological individualism: actors use
heuristics to try to achieve their individual goals, subject to constraints.
The current network structure is among these constraints. The models
are continuous time Markov chain models that can be implemented as
simulation models. They incorporate random change in addition to the
purposeful change that follows from the actors' pursuit of their goals,
and include parameters that must be estimated from observed data.
Statistical methods are proposed for estimating and testing these
models. These methods can also be used for parameter estimation for
other simulation models. The statistical procedures are based on the
method of moments, and use computer simulation to estimate the
theoretical moments. The Robbins-Monro process is used to deal with
the stochastic nature of the estimated theoretical moments. An example
is given for Newcomb's fraternity data, using a model that expresses
reciprocity and balance.
Key words: methodological individualism; Markov process; Newcomb
data; balance; Robbins-Monro process; simulation models; method of
moments; simulated moments; random utility.
Snijders, T.A.B., What to do with the upward bias in R²: A comment on
Huberty.
Journal of Educational and Behavioral Statistics, 21 (1996), 283-287.
Abstract.
A recent article by Huberty (1994) discusses significance testing of R²
in linear regression and the definition of a corresponding effect size
index. It recommends an adjustment to the standard null hypothesis,
rho² = 0, in order to adjust for an upward bias in the statistic R².
This note suggests that the adjustment proposed by Huberty has some
conceptual shortcomings. Existing improvements on R² are described
in some detail.
Snijders, T.A.B. (1995),
Methods for Longitudinal Social Network Data: Review and Markov Process Models.
In E.Tiit, T. Kollo, and H. Niemi (eds.),
New Trends in Probability and Statistics.
Vol. 3: Multivariate Statistics and Matrices in Statistics.
Proceedings of the 5th Tartu Conference, p. 211-227.
Vilnius, Lithuania: VSP/TEV.
Abstract.
Social network data pertain to social relations between individuals,
or between other actors (countries, firms, etc.). The facts that each
relation links two individuals, and that each individual can be
related with multiple others, result in complicated stochastic
dependence structures. Longitudinal studies about the change and
development of social networks in time are important for progress in
sociology and related disciplines, but the literature does not contain
many satisfactory statistical models for longitudinal social network
data. This is related to the difficulty of reconciling mathematical
tractability with the possibility to reflect substantively interesting
processes.
A review is given of models for longitudinal network data.
Interesting developments are taking place in
applying continuous time Markov processes to model network change.
Such models can incorporate sociologically important effects. They are
in many aspects similar to computer simulation models such as are now
common in sociological theory, but with the extra ingredient of a
component of random change. Difficulties of mathematical tractability
can be resolved by using stochastic approximation methods to obtain
estimators and tests. This is done, however, at the expense of
introducing new problems: computer intense computations and questions
about statistical efficiency. An example of this approach is given
using a time series of personal preference relations in a closed
group.
Abstract.
The hierarchical linear model is a linear model with nested random
coefficients, fruitfully used for multilevel research. A tutorial is presented
on the use of this model for the analysis of longitudinal data, i.e.,
repeated data on the same subjects. An important advantage of this
approach is that differences across subjects in the numbers and
spacings of measurement occasions do not present a problem, and
that changing covariates can easily be handled. The tutorial
approaches the longitudinal data as measurements on populations of
(subject-specific) functions.
Key words: multilevel analysis, hierarchical linear model, random
coefficients.
Abstract.
This paper explains how multilevel methods can be employed to
analyze personal network data, when the dependent variable under
consideration is a function of the relations contained in the personal
networks. These methods take into account the mutual dependence of
relations of the same respondent, and allow us to study the variability
between respondents as well as the variability between different
relations within respondents. As an illustration, multilevel models are
applied to an analysis of personal networks of cocaine users, focusing
on the significance of cocaine in their personal relations with other
cocaine users.
Key words: Personal network, snowball sample, multilevel
analysis,hierarchical linear model, random effects, cocaine.
Also see
van Duijn, van Busschbach and Snijders (1999).
Snijders, T.A.B. & Bosker, R.J.,
Modeled variance in two-level models.
Sociological Methods and Research, 22 (1994), 342-363.
Abstract.
The concept of explained or modeled proportion of variance is
reviewed in the situation of the random effects hierarchical two-level
model. It is argued that the proportional reduction in (estimated)
variance components is not an attractive parameter to represent the
joint importance of explanatory variables for modeling the dependent
variable. It is preferable instead to work with the proportional reduction
in mean squared prediction error for predicting individual values (for the
modeled variance at level 1) and for predicting group averages (for the
modeled variance at level 2). It is shown that when predictors are
added, the proportion of modeled variance defined in this way cannot
go down in the population if the model is correctly specified, but can
go down in a sample; the latter situation then points to the possibility of
misspecification. This provides a diagnostic means for identifying misspecification.
Key words: R-squared, explained variance, coefficient of determination,
multilevel analysis, misspecification.
Snijders, T.A.B., Dam, M. van & Weesie, J., Who contributes to public
goods? With an application to local economic policies in the
Netherlands.
Journal of Mathematical Sociology, 19 (1994), 149-164.
Abstract.
We present three models for the extent to which actors reduce their
contributions to the production of a public good because of expected
contributions by other actors. The first model is a simple game
theoretic model, the second a spatial autocorrelation model, and the
third is a hybrid of the first two models. Estimation of the three models
from incomplete data is discussed. The three models are applied to
data on economic policies of municipalities in the Netherlands. In
particular, it is probed whether municipalities take a free ride on the
measures of their neighbors.
Frank, O. & Snijders, T.A.B., Estimating hidden populations using
snowball sampling.
Journal of Official Statistics, 10 (1994), 53-67.
Abstract.
Snowball sampling is a term used for sampling procedures that allow
the sampled units to provide information not only about themselves but
also about other units. This might be advantageous when rare
properties are of interest. This article illustrates snowball sample
situations and discusses various modelling and estimation problems in
this context. The problem of estimating the size of a population is
discussed for both design-based and model-based approaches. An
application to a study of heroin use is included. Simulation results are
provided for comparing and evaluating various estimators.
Key words: Network sampling; random graphs; link-tracing designs.
Baerveldt, C. & Snijders, T.A.B., Influences on and from the
segmentation of networks: hypotheses and tests.
Social Networks, 16 (1994), 213-232.
Abstract.
This article discusses (a) the influence of network structure on the
diffusion of (new) cultural behavior within the network and (b) the
influence of external events, especially of social programs, on the
diffusion of (new) cultural behavior, and on the network structure.
Hypotheses are formulated and tested on data from a study on the
diffusion of petty crime in pupils' networks in high schools. To test
these hypotheses we propose and use a new measure of network
structure: the segmentation index.
Abstract.
What are essential requirements, formulated in terms of item response
theory, for unidimensional unfolding models for dichotomous data, if
one does not wish to make specific assumptions concerning the form
of the tracelines and of the population distribution of latent trait values?
Tracelines should be unimodal, of course, but this requirement is not
sufficient to derive empirically testable
consequences. Two basic postulates are formulated concerning the
inference about subjects' latent trait values on the basis of observed
responses to items. These postulates are proven to be equivalent to
total positivity of orders 2 and 3 for the traceline family. Given these
postulates, unimodality of the tracelines leads to some empirically
testable results. These are formulated as properties of the conditional
adjacency matrix and of the correlation matrix.
Key words: Unfolding, item response theory, unimodal response
models, total positivity, unidimensional scaling, measurement theory.
Abstract.
The hierarchical linear model approach to a two-level design is
considered, some variables at the lower level having fixed and others
having random regression coefficients. An approximation is derived to
the covariance matrix of the estimators of the fixed regression
coefficients (for variables at the lower and the higher level) under the
assumption that the sample sizes at either level are large enough. This
covariance matrix is expressed as a function of parameters occurring in
the model. If a research planner can make a reasonable guess as to
these parameters, this approximation can be used as a guide to the
choice of sample sizes at either level.
A PC program to carry out the calculations developed in this paper
is available from my
multilevel page.
Key words: hierarchical linear model, multilevel research, sample
design.
Abstract.
What are the possibilities of snowball sampling, if one desires valid
statistical inference without making probabilistic assumptions on the
network structure? In a critical review of the possibilities of snowball
sampling for a population of vertices connected by a network of arcs, it
is argued that the snowball method is much more suitable for the
estimation of parameters of the network structure (or parameters of the
population of arcs) than to estimate parameters of the population of
vertices. Further work needs to be done to relax the assumption of
randomness of the initial sample of the snowball.
Key words: Snowball Sampling, Weighting, Parameter Estimations,
Social Networks.
Muilwijk, J., Snijders, T.A.B. & Moors, J.J.A.,
Kanssteekproeven. Leiden: Stenfert Kroese, 1992.
(Probability Samples, in Dutch).
This is a textbook on sampling theory.
Jansen, M.G.M. & Snijders, T.A.B., Comparisons of Bayesian
estimation procedures for two-way contingency tables without
interaction.
Statistica Neerlandica, 45 (1991), 51-65.
Abstract.
Bayesian and empirical Bayesian estimation methods are reviewed and
proposed for the row and column parameters in two-way contingency
tables without interaction. Rasch's multiplicative Poisson model for
misreadings is discussed in an example. The case is treated where
assumptions of exchangeability are reasonable a priori for the
unknown parameters. Two different types of prior distributions are
compared. It appears that gamma priors yield more tractable results
than lognormal priors.
Key words: lognormal prior, Dirichlet prior, gamma prior, posterior
mode, Rasch's multiplicative Poisson model, empirical Bayes
estimation.
Snijders, T.A.B., Enumeration and simulation methods for 0-1 matrices
with given marginals.
Psychometrika, 56 (1991), 397-417.
The algorithms in this publication are available
in the program collection ZO.
Abstract.
Data in the form of zero-one matrices where conditioning on the
marginals is relevant arise in diverse fields such as social networks and
ecology; directed graphs constitute an important special case. An
algorithm is given for the complete enumeration of the family of all
zero-one matrices with given marginals and with a prespecified set of
cells with structural zero entries. Complete enumeration is
computationally feasible only for relatively small matrices. Therefore, a
more useable Monte Carlo simulation method for the uniform
distribution over this family is given, based on unequal probability
sampling and ratio estimation. This method is applied to testing
reciprocity of choices in social networks.
Key words: adjacency matrices, random digraphs, networks, ecology,
Monte Carlo methods, unequal probability sampling, reciprocity.
Abstract.
For three coefficients of similarity between pairs (dyads) of operational
taxonomic units for multivariate binary data (presence/absence of
attributes), parameters of their distribution under statistical
independence are derived. These are applied to test independence for
dyadic data. Association among attributes within operational taxonomic
units is allowed. It is also allowed that the two units in the dyad are
drawn from different populations having different presence probabilities
of attributes. The variance of the distribution of the similarity coefficients
under statistical independence is shown to be relatively large in many
empirical situations. This result implies that the practical interpretation
of these coefficients requires much care. An application using the
Jaccard index is given for the assessment of consensus between
psychotherapists and their clients.
Key words: Consensus, Dice coefficient, Jaccard coefficient, Simple
Matching coefficient, Multivariate binary data, Observer agreement,
Similarity coefficients, Beta distribution.
Snijders, T.A.B., Reliable counts: corrections of losses and gains for
unreliability.
ISOR-Methodenreeks, Utrecht (1990).
Abstract.
Reliability of counts is investigated for situations where there may be
underreporting because some of the elements to be counted are
forgotten. An example of such a count is the size of a personal
network. The reliability parameter is defined as the probability of duly
reporting an element of the set to be counted. This parameter can be
estimated from test-retest data. Estimation methods are given for the
situation where one single reliability parameter value is estimated, for
the situation where reliabilities vary randomly across individuals, and for
the situation where the reliability parameter is modeled by linear
regression on some independent variable. For data collected over time,
relative losses and relative gains with respect to these counts can be
corrected for unreliability.
Key words: Counts; personal networks; reliability; reliability of change;
binomial distribution; random effects; empirical Bayes; regression.
Snijders, T.A.B., Testing for change in a digraph at two time points.
Social Networks, 12 (1990), 359-373.
The algorithms in this publication are available
in the program collection ZO.
Abstract.
A method is presented for testing change of digraphs (representing
some binary relation) observed at two points in time, labeled I and II.
The test is conditional on the entire digraph at time I, the numbers of
new arcs to and from each actor, and the numbers of disappeared arcs
to and from each actor. A new arc is defined as an arc existing at time
II but not at time I; a disappeared arc is an arc existing at time I but not
at time II. In particular, tests are conditional simultaneously on in-
degrees and out-degrees at times I and II. The elements of the dyad
transition matrix, indicating the numbers of dyads of some particular
type (mutual, asymmetric, of null) at time I, and of some (same or
other) type at time II, are proposed as possible test statistics.
Also see
Snijders (Psychometrika, 1991).
Schweigman, C., Snijders, T.A.B. & Bakker, E.J., Operations Research
as a tool for analysis of food security problems.
European Journal of Operations Research, 49 (1990), 211-221.
Abstract.
In the first part of the paper the role of operations research in analyzing
daily life problems of farmers in developing countries is discussed.
Experiences on village studies in Tanzania are reported which formed
part of the training in operations research of students of the University
of Dar es Salaam. In the second part, two examples of food security
problems are worked out: risk of food shortage in subsistence farming
in Tanzania and the use of rainfall-yield models to predict shortages of
sorghum production at an early stage of the growing season in Burkina
Faso. At the end of the paper, some discussion points are formulated.
Key words: Subsistence agriculture, risk, early warning.
Wilmink, F.W. & Snijders, T.A.B., Polytomous logistic regression
analysis of the General Health Questionnaire and the Present State
Examination.
Psychological Medicine , 19 (1989), 755-764.
Abstract.
First, two examples of dichotomous logistic regression analysis are
presented. The probability of being a psychiatric case according to the
Present State Examination is predicted from the total score on the
General Health Questionnaire and from the general practitioner's
judgement on the presence of a mental health problem. Subjects were
292 primary care attenders. Results are compared with those from prior
studies.
Next, the extension to the polytomous case is demonstrated. The
probability of being at any given level of the Index of Definition
(computed from PSE data) is estimated from the General Health
Questionnaire total score by an ordered polytomous logistic regression
model. Several applications of the polytomous logistic regression
model are discussed. These range from estimating the proportion of
psychiatric cases among individuals who refuse to be interviewed to
the formulation of sampling schemes which can be expected to reduce
costs while at the same time yielding optimal information for testing
specific hypotheses.
Snijders, T.A.B. & Stokman, F.N., Extensions of triad counts to
networks with different subsets of points and testing the underlying
random graph distributions.
Social Networks, 9 (1987), 249-275.
Abstract.
Triad counts are defined for bipartite directed graphs, i.e., directed
graphs where the set of points is partitioned into two subsets. Triads
are considered with two points in the first, and one point in the second
subset. The means, variances, and covariances of triad counts are
given for various random digraph distributions.
Key words: bipartite graphs, conditionally uniform distribution.
Bhoj, D.S. & Snijders, T.A.B., Testing equality of correlated proportions
with incomplete data on both responses.
Psychometrika, 51 (1986), 579-588.
Abstract.
Two test statistics are proposed for testing the equality of two
correlated proportions when some observations are missing on both
responses. The performance of these tests in terms of size and power
is compared with other tests by means of Monte Carlo simulations. The
proposed tests are easily computed and compare favorably with other
tests.
Key words: combination of tests, equality of correlated proportions,
incomplete data, asymptotically most powerful test, Monte Carlo study,
antithetic variates, power comparison.
Snijders, T.A.B., Inter-station correlations and non-stationarity of Burkina
Faso rainfall.
Journal of Climate and Applied Meteorology, 25 (1986), 524-531.
Abstract.
A study is presented of the rainfall regime for central and northern
Burkina Faso over 1923-83. Interstation correlations for seasonal rainfall
totals are rather low, with median 0.31. Sums of truncated daily rainfall
values, with the number of rainy days as an extreme case, exhibit quite
larger interstation correlations. An explanation is that factors
determining the occurrence of rainfall in West Africa operate on a larger
scale than those determining exact rainfall amounts. A new method is
proposed for constructing regional rainfall indices from data for several
locations in the presence of missing data. This method is applied in a
study of (non-)stationarity of Burkina Faso rainfall. A highly significant
departure of stationarity is found, which is especially expressed in
earlier dates for the end of the rains, and smaller average rainfall
amounts per day between the start and the end of the rains.
Snijders, T.A.B. & Schweigman, C., The stochastic nature of yields (Ch.
4).
Snijders, T.A.B. & Joosten, G. & Schweigman, C., Simulation (Ch. 5).
Snijders, T.A.B., Statistics (Appendix).
These are chapters in Operations research problems in agriculture
in developing countries, ed. C. Schweigman, Khartoum University
Press, Khartoum and Tanzania Publishing House, Dar-Es-Salaam
(1985).
Snijders, T.A.B., Antithetic variates for Monte Carlo estimation of
probabilities.
Statistica Neerlandica, 38 (1984), 55-74.
Abstract.
This paper explores some possibilities for variance reduction by the use
of antithetic variates when estimating probabilities.
Key words: antithetic variates, Monte Carlo, variance reduction,
change-point test, Wilcoxon test.
Abstract.
In the analysis of empirically found graphs, the variance of the degrees
can be used as a measure for the heterogeneity of (the points in) the
graph. For several types of graphs, the maximum value of the degree
variance is given, and the mean and variance of the degree variance
under a simple stochastic null model are computed. These are used to
produce normalized versions of the degree variance, which can be
used as heterogeneity indices of graphs.
Key words: graph heterogeneity, graph centrality, random graphs,
degree variance.
Snijders, T.A.B., Rank tests for bivariate symmetry.
Annals of Statistics, 9 (1981), 1087-1095.
Abstract.
The problem is considered of testing symmetry of a bivariate
distribution L(X, Y) against "asymmetry towards high X-values," subject
to the restriction of invariance under the transformations of (x,y) to
(g(x),g(y)) for increasing bijections g. This invariance restriction
prohibits the common reduction to the differences x - y. The intuitive
concept of "asymmetry towards high X-values" is approached in several
ways, and a mathematical formulation for this concept is proposed.
Most powerful and locally most powerful invariant similar tests against
certain subalternatives are characterized by means of a Hoeffding
formula. Asymptotic normality and consistency results are obtained for
appropriate linear rank tests.
Key words: Nonparametric tests, bivariate symmetry and asymmetry,
locally most powerful tests, asymptotic normality.
ten Berge, J. M. F., Snijders, T.A.B. & Zegers, F.E., Computational
aspects of the greatest lower bound to the reliability and constrained
minimum trace factor analysis.
Psychometrika, 46 (1981), 201-213.
Abstract.
In the last decade several algorithms for computing the greatest lower
bound to reliability or the constrained minimum-trace communality
solution in factor analysis have been developed. In this paper
convergence properties of these methods are examined. Instead of
using Lagrange multipliers, a new theorem is applied that gives a
sufficient condition for a symmetric matrix to be Gramian. Whereas
computational pitfalls for two methods suggested by Woodhouse and
Jackson can be constructed it is shown that a slightly modified version
of one method suggested by Bentler and Woodward can safely be
applied to any set of data. A uniqueness proof for the solution desired
is offered.
Key words: communality, internal consistency, Heywood case, positive
definite.
This is a companion paper to "The degree variance: an index of graph heterogeneity",
published in Social Networks, 1981,
containing the proofs and derivations published there.
In the analysis of empirically found graphs, the variance of the degrees
can be used as a measure for the heterogeneity of (the points in) the
graph. For several types of graphs, the maximum value of the degree
variance is derived, and the mean and variance of the degree variance
under a simple stochastic null model are computed.
Key words: graph heterogeneity, graph centrality, random graphs,
degree variance.
Snijders, T.A.B., Asymptotic optimality theory for testing problems
with restricted alternatives.
Mathematical Centre Tracts 113,
Mathematical Centre, Amsterdam 1979.
Abstract.
This monograph develops a theory of asymptotic optimality for testing
problems where the alternative hypothesis is multidimensional and
restricted by a finite number of linear inequalities. The optimality
criterion is an asymptotic version of the "most stringent" property.
Snijders, T.A.B., Complete class theorems for the simplest empirical
Bayes decision problems.
Annals of Statistics, 5 (1977), 164-171.
Abstract.
For the problem of empirical Bayes classification into two known
probability distributions on a finite outcome space, an essentially
complete class of procedures is determined. This class is proven to be
minimal essentially complete if there are only two possible outcomes.
Key words: Empirical Bayes classification, complete class, monotone
procedures.
Snijders, T.A.B., A test for randomness in behaviour.
Statistica Neerlandica, 29 (1975), 39-48.
Abstract.
In biological analysis of behaviour, transition matrices occur of which
the diagonal entries are essentially zero. For such transition matrices, a
model of randomness is constructed, with a test for the hypothesis that
this model holds.
Key words: Markov chain, ethology, transition analysis.
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