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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.

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.

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 versionsof 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 dierential 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 appliedto study the dynamics of a friendship network among the students at an Australian high school. In particular, we model the relationshipbetween friendship and obesity, focusing on body mass index as a continuous co-evolving attribute.

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. (C) 2013 Elsevier Inc. All rights reserved.

With the development of an MCMC algorithm, Bayesian model selection for the p2 model for directed graphs has become possible. This paper presents an empirical exploration in using approximate Bayes factors for model selection. For a social network of Dutch secondary school pupils from different ethnic backgrounds it is investigated whether pupils report that they receive more emotional support from within their own ethnic group. Approximated Bayes factors seem to work, but considerable margins of error have to be reckoned with. © VVS, 2005.

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, Markov chain Monte Carlo (MCMC) algorithms have been developed that 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 statistics 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.

The p(2) 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. It was found that the IGLS estimates have a smaller variance but are severely biased, while the MCMC estimates have a larger variance with a small bias. For networks with 20 actors, mean squared errors are generally comparable or smaller for the IGLS estimates. For networks with 40 actors, mean squared errors are the smallest for the MCMC estimates. Coverage rates of confidence intervals are good for the MCMC estimates but not for the IGLS estimates.

This article reviews new specifications for exponential random graph models proposed by Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology] 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 near-degeneracy 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. (c) 2006 Elsevier B.V. All rights reserved.

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.

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. (C) 2011 Elsevier B.V. All rights reserved.

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.

A class of statistical models is proposed that aims to recover latent settings structures in social networks. Settings may be regarded as clusters of vertices. The measurement model is based on two assumptions. (1) The observed network is generated by hierarchically nested latent transitive structures, expressed by ultrametrics, and (2) the 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. Bayesian methods as well as maximum-likelihood methods are applied for statistical inference. Both approaches are implemented using Markov chain Monte Carlo methods.

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. 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 a way to achieve a fully multivariate analysis of repeated measures with incomplete data.

A number of estimation methods for the variance components in the Wing-Kristofferson 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 Wing-Kristofferson model is recognized as a moving average model.

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 distribution of the relation 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).

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.

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.

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) [Snijders, T.A.B., 1991. Enumeration and simulation methods for 0-1 matrices with given marginals. Psychometrika 56, 397-417.], for generating random graphs with fixed marginals, we suggest a measure of centrality that is logically but not necessarily empirically independent of degree. We examine the measure using data from Davis (1941) [Davis, A., Gardner, B., Gardner, M.R., 1941. Deep South. Univ. of Chicago Press, Chicago.] and Oliver (1993) [Oliver, A., 1993. New Biotechnology Firms: A Multilevel Analysis of Interorganizational Relations in an Emerging Industry. PhD dissertation, Univ. of California, Los Angeles.]. (C) 1998 Elsevier Science B.V.

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. (Less)

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, p2 = 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.

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 tha...

Exploring and Relating Model to Data in Practice Previous chapters concentrated on the formulation and specification of exponential random graph models (ERGMs) for different types of relational data. In Chapter 6, we saw that effects represented by configurations and corresponding parameters define a distribution of graphs where the probability of getting any particular graph depends on the configurations in the graph. Chapter 7 showed that configurations are sufficient information in the sense that the probability of a graph is completely determined by statistics that are the counts of relevant configurations. If we increase the strength of a parameter for a given configuration, graphs with more of that configuration become more likely in the resulting distribution. This simple fact is used in the three methods presented in this chapter: Simulation: For a given model by fixing parameter values, it is possible to examine the features of graphs in the distribution through simulation to gain insight into the outcomes of the model. Estimation: Empirically, for a given model and a given dataset, it is possible to estimate the parameter values that are most likely to have generated the observed graph, the “maximum likelihood estimates (MLEs).” Furthermore, it can be shown that the observed graph is central in the distribution of graphs determined by these estimates–but as we will see, because of the dependencies in the data, MLE requires simulation procedures. Heuristic goodness of fit (GOF): For a fitted model (i.e., with parameters estimated from data), it is then possible to simulate the distribution of graphs to see whether other features of the data (i.e., nonfitted effects) are central or extreme in the distribution. If a graph feature is not extreme, there is no evidence to suggest that it may not have arisen from processes implicit in this model, and hence we can say that the model can explain that particular feature of the data – in other words, that such a feature is well fitted by the model.

The concept of explained proportion of variance 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 the explanatory (independent) 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 the proportional reduction in mean squared prediction error 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.

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.

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. Crown Copyright (C) 2009 Published by Elsevier B.V. All rights reserved.

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.

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.

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, or null) at time I, and of some (same or other) type at time II, are proposed as possible test statistics.

Parameters are derived of distributions of three coefficients of similarity between pairs (dyads) of operational taxonomic units for multivariate binary data (presence/absence of attributes) under statistical independence. These are applied to test independence for dyadic data. Association among attributes within operational taxonomic units is allowed. It is also permissible for the two units in the dyad to be 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.

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 semi-partialing", 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. © 2007 The Psychometric Society.

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 (when ML is 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. (C) 2007 Elsevier BX All rights reserved.

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. (c) 2006 Elsevier B.V. All rights reserved.

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. (c) 2007 Elsevier B.V. All rights reserved.

In performing a multilevel analysis, what is a level? And when should a variable have a random slope? This depends on whether the units in the design should be regarded as being representative of a population and on whether the researcher wishes to draw conclusions primarily about the observed units or primarily about the population. Keywords: fixed effects; random effects; linear model; multilevel analysis; mixed model; population; dummy variables

Sample size determination in multilevel designs requires attention to the fact that statistical power depends on the total sample sizes for each level. It is usually desirable to have as many units as possible at the top level of the multilevel hierarchy. Some formulas are given to obtain insight in the design aspects that are most influential for standard errors and power. Keywords: power; statistical tests; design; multilevel analysis; sample size; multisite trial; cluster randomization

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. Preprint of: Snijders, Tom A.B. (2005). Models for Longitudinal Network Data. Chapter 11 (pp. 215–247) in P. Carrington, J. Scott, and S. Wasserman (Eds.), Models and methods in social network analysis. New York: Cambridge University Press.

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 usually decreases the asymptotic variance.

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.

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.

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.

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.

This paper explores some possibilities for variance reduction by the use of antithetic variates when estimating probabilities.

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.

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.

The problem is considered of testing symmetry of a bivariate distribution $\mathscr{L}(X, Y)$ against "asymmetry towards high $X$-values," subject to the restriction of invariance under the transformations $(x_i, y_i) \mapsto (g(x_i), g(y_i)) (1 \leq i \leq n)$ for increasing bijections $g$. This invariance restriction prohibits the common reduction to the differences $x_i - y_i$. 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.

for all j 6= l. The lengths of the vectors yj , β, and δj , respectively, are nj , r, and s. Like in all regression-type models, the explanatory variables X and Z are regarded as fixed variables, which can also be expressed by saying that the distributions of the random variables ǫ and δ are conditional on X and Z. The random variables ǫ and δ are also called the vectors of residuals at levels 1 and 2, respectively. The variables δ are also called random slopes. Level-two units are also called clusters. The standard and most frequently used specification of the covariance matrices is that level-one residuals are i.i.d., i.e.