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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References

Networks of relationships between individuals influence individual and collective outcomes and are therefore of interest in social psychology, sociology, the health sciences, and other fields. We consider network panel data, a common form of longitudinal network data. In the framework of estimating functions, which includes the method of moments as well as the method of maximum likelihood, we propose score-type tests. The score-type tests share with other score-type tests, including the classic goodness-of-fit test of Pearson, the property that the score-type tests are based on comparing the observed value of a function of the data to values predicted by a model. The score-type tests are most useful in forward model selection and as tests of homogeneity assumptions, and possess substantial computational advantages. We derive one-step estimators which are useful as starting values of parameters in forward model selection and therefore complement the usefulness of the score-type tests. The finite-sample behaviour of the score-type tests is studied by Monte Carlo simulation and compared to t-type tests.


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References

Human beings are interdependent due to the need to produce food, clothing, shelter, and everything else considered to be essential or desirable. Interdependence is reflected in social relationships between individuals, and social relationships tend to influence the attitudes, actions, and social and economic well-being of individuals as well as collective outcomes: for example, friends may influence adolescents to initiate health-risky or criminal behaviour such as smoking, drinking alcohol, or taking drugs; co-workers in the workplace may influence the commitment of employees; and sexual relationships between individuals may spread diseases through populations. Social network analysis (Wasserman & Faust, 1994; Freeman, 2004) acknowledges that individual and collective outcomes depend on relationships between individuals, and studies the structure of social relationships between individuals and its impact on individual and collective outcomes. We consider networks of social relationships through time, acknowledging that social relationships change through time.

Longitudinal network data come frequently in the form of panel data. Holland and Leinhardt (1977) proposed to model network panel data as outcomes of continuous-time Markov processes, and Snijders (2001, 2006, 2009) introduced an attractive family of continuous-time Markov models with a wide range of possible applications.

To estimate continuous-time Markov models from network panel data, Snijders (2001) proposed the method of moments, Koskinen and Snijders (2007) proposed Bayesian methods, and Snijders, Koskinen, and Schweinberger (2010) proposed the method of maximum likelihood.

We are concerned with goodness of fit in the classic sense of the term (see Pearson, 1900), that is, with tests that compare the observed value of a function of the data to values predicted by a model. In the framework considered here, it is frequently desired to test the goodness of fit of a model with respect to functions of the data which are correlated as well as homogeneity assumptions with respect to time and individuals. In applications, estimating a model with correlated statistics and without homogeneity assumptions may be time-consuming and result in computational issues. Therefore, it is preferable to follow a forward model selection approach by estimating a model with restrictions on parameters (e.g., restrictions on parameters corresponding to correlated statistics and restrictions imposed by homogeneity assumptions) and testing whether the goodness of fit of the restricted model is unacceptable and the restrictions should be dropped. The literature (e.g., Snijders, 1996, 2001, 2003; Koskinen & Snijders, 2007; Snijders et al., 2010), to be discussed in more detail below, has not come up with goodness-of-fit tests which satisfy these requirements and can be computed in reasonable time.

In the framework of estimating functions, which includes the method of moments as well as the method of maximum likelihood, we propose score-type tests which allow us to test a wide range of restrictions on a model without estimating unrestricted models. The score-type tests share with other score-type tests, including the classic goodness-of-fit test of Pearson (1900)– the special case of the classic score test of Rao (1948) in the framework of multinomial distributions (see Bera & Bilias, 2001a,b) – and the classic score test of Rao (1948) in the framework of exponential family distributions, the property that the score-type tests have an appealing interpretation in terms of goodness of fit in the classic sense of the term. The score-type tests are most useful in forward model selection and as tests of homogeneity assumptions and possess substantial computational advantages. We derive one-step estimators which are useful as starting values of parameters in forward model selection and therefore complement the usefulness of the score-type tests.

The paper is structured as follows. Model specification and estimation are described in Section 2. Score-type tests are proposed in Section 3. In Section 4 the finite-sample behaviour of the score-type tests is studied by Monte Carlo simulation and compared to t-type tests. The usefulness of the score-type tests and the one-step estimators in forward model selection is demonstrated in Section 5 by an application to the cross-ownership network of more than 400 business firms in post-communist Slovenia.

2. Model specification and estimation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References

A binary, directed relation inline image (or digraph) on a finite set of nodes inline image is considered. The digraph is observed at discrete, ordered time points t1 < t2 < … < tM, and the observations are represented as binary matrices x (t1), x (t2), …, x (tM), where the element xij(tm) of the n×n matrix x (tm) is defined by

  • image(1)

in which inline image means that node i is related to node j; the diagonal elements xii(tm) are regarded as structural zeros.

2.1 .  Model specification

The observed digraph x (t1) is taken for granted, that is, the model conditions on x (t1). It is postulated that the observed digraphs x (t2), …, x (tM) conditional on x (t1) are generated by an unobserved continuous-time Markov process operating in the time interval [t1, tM]. Continuous-time Markov processes are described by, for example, Norris (1997) and Resnick (2002). We follow the application of continuous-time Markov processes to network panel data proposed by Snijders (2001, 2006, 2009). Throughout this paper, the case M= 2 is considered; the extension to the case M > 2 is straightforward due to the Markov property.

The model is specified by the generator of the Markov process, which corresponds to a W×W matrix Qθ indexed by a parameter vector θ, where W= 2n(n− 1) is the number of digraphs. The elements qθ(x, x) of the generator Qθ are the rates of moving from digraph x to digraph x. It is assumed that, if x deviates from x in more than one element xij, then qθ(x, x) = 0, implying that the process moves forward by changing no more than one element xij at a time. Let x be an arbitrary digraph, and let x be the digraph that is obtained from x by changing one and only one specified element (say, xij). Then qθ(x, x) can be rewritten as qθ(x, i, j) and decomposed as follows:

  • image(2)

where λθ(i, x) is called the rate function of node i, while

  • image(3)

is the conditional probability that i changes element xij, given that i changes some element xih, hi.

A constant rate function is given by

  • image(4)

where ρ > 0 is a parameter, while non-constant rate functions are given by

  • image(5)

where inline image is a vector-valued parameter and ai= (aik) is a vector-valued function of x and covariates.

A convenient, multinomial logit parametrization of μθ is given by

  • image(6)

where

  • image(7)

is called the objective function of node i, where inline image is a vector-valued parameter and si= (sik) is a vector-valued function of x and covariates.

Examples of specifications of rate functions and objective functions are given in Sections 4 and 5.

2.2 .  Model estimation

Model estimation is concerned with estimating the parameter vector θ, consisting of the parameters ρ, inline image, and inline image, from the observed data z, consisting of the observed digraphs x (t1), x (t2), and covariates. The observed data z are incomplete in the sense that the holding times as well as the changes of the digraph in the time interval [t1, t2] are missing. Three methods of estimation were proposed to estimate θ from the incomplete data z, the method of moments (Snijders, 2001), Bayesian methods (Koskinen & Snijders, 2007), and the method of maximum likelihood (Snijders et al., 2010). We focus on the method of moments and the method of maximum likelihood, which can be considered as special cases of the framework of estimating functions (see Godambe, 1960, 1991).

A moment estimator of the parameter vector θ (see Snijders, 2001) is a root of the estimating function

  • image(8)

where the vector of statistics u to estimate the parameter vector θ is chosen so that estimating function (8) is increasing in θ and sensitive to changes in θ (see Snijders, 2001). A natural choice is the statistic uk=∑ni, j= 1|xij(t2) −xij(t1)| to estimate the parameter ρ, the statistic inline image to estimate the parameter αk, and the statistic inline image to estimate the parameter βk (see Snijders, 2001).

The log-likelihood function Z(θ; z) of the parameter vector θ given the incomplete data z cannot be written in closed form. Snijders et al. (2010) exploited Fisher’s identity (see Fisher, 1925; Efron, 1977) to rewrite the analytically intractable incomplete-data score function ∂ℓZ(θ; z)/∂θ as the conditional expectation of the analytically tractable complete-data score function ∂ℓC(θ; c)/∂θ given X (t1) =x (t1), where c denotes the complete data, corresponding to the observed digraph x (t1), the holding times and the changes in the digraph in the time interval [t1, t2], the observed digraph x (t2), and covariates. A maximum likelihood estimator of θ (see Snijders et al., 2010) is therefore a root of the estimating function

  • image(9)

In terms of computing, finding the roots of the estimating functions (8) and (9) as a function of the parameter vector θ can be accomplished by iterative, stochastic approximation algorithms relying on Monte Carlo simulation in the case of (8) (see Snijders, 2001) and Markov chain Monte Carlo simulation in the case of (9) (see Snijders et al., 2010).

Based on a Monte Carlo simulation study, Snijders et al. (2010) concluded that, in terms of statistical efficiency, the moment estimator is inferior to the maximum likelihood estimator given small data sets, but almost as good as the maximum likelihood estimator given medium to large data sets. The asymptotic properties of the moment estimator and the maximum likelihood estimator are unknown (see Snijders et al., 2010). In terms of computational efficiency, the moment estimator can be computed 17.5 times faster than the maximum likelihood estimator (see Snijders et al., 2010). Therefore, if a medium to large data set is given, maximum likelihood estimation is hardly necessary on statistical grounds and hardly attractive on computational grounds.

3. Goodness-of-fit tests

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References

In Section 3.1 the existing goodness-of-fit tests in the considered family of models are reviewed. In Section 3.2 a new goodness-of-fit test is introduced.

To acknowledge that the data z depend on the number of nodes n, we index z and functions of z by n throughout this section.

3.1 .  Existing tests

We discuss the existing literature with the following desirable properties of goodness-of-fit tests in mind: whether the tests allow us to test the goodness of fit with respect to a wide range of functions of the data; whether the tests allow us to test homogeneity assumptions; and whether the tests can be computed in reasonable time.

In the method of moments framework, Snijders (1996) proposed t-type tests based on the test statistic

  • image(10)

where inline image is the moment estimator of the parameter θk and inline image is its standard error. The finite-sample distribution and the asymptotic distribution of (10) are unknown (see Snijders, 1996), though we will shed light on the finite-sample distribution by Monte Carlo simulation (see Section 4). In practice, Snijders (1996) suggested carrying out t-type tests under the assumption that the distribution of test statistic (10) under the null hypothesis H0: θk= 0 is approximately standard Gaussian. The t-type tests are not well suited to testing models with correlated statistics and testing homogeneity assumptions, because the t-type tests require the estimation of the unrestricted model, which may be time-consuming and result in computational issues.

A second test in the method of moments framework was proposed by Snijders (2003). Snijders (2003) studied the goodness of fit with respect to the (out-)degree distribution, which is a fundamental function of the data. The goodness-of-fit test of Snijders (2003) is, however, limited to the (out-)degree distribution and does not allow the goodness of fit to be tested with respect to other fundamental functions of the data.

In the maximum likelihood framework, Snijders et al. (2010) proposed likelihood ratio tests. The finite-sample distribution and the asymptotic distribution of the likelihood ratio test statistic are unknown (see Snijders et al., 2010). The likelihood ratio test is not well suited to testing homogeneity assumptions, because it requires an estimate of both the restricted model with homogeneity assumptions and the unrestricted model without homogeneity assumptions. If the unrestricted model includes a large number of parameters, such as in tests of homogeneity assumptions, the path-sampler which Snijders et al. (2010) used to approximate the likelihood ratio test statistic may give rise to bad approximations. Worse, as the likelihood ratio test requires both the restricted model and the unrestricted model to be estimated, the likelihood ratio test tends to consume 2 · 17.5 = 35 times as much computation time as the t-type tests of Snijders (1996)– see Section 2.2.

In both the method of moments and maximum likelihood framework, Monte Carlo simulations can be carried out conditional on moment estimators and maximum likelihood estimators, respectively. Monte Carlo simulations are useful for exploring goodness of fit with respect to a wide range of functions in an informal fashion, but are not well suited to testing models with correlated statistics and testing homogeneity assumptions.

In the Bayesian framework, Koskinen and Snijders (2007) used posterior predictions and cross-validation to study goodness of fit. In terms of computation time, however, the Bayesian framework is worse than the maximum likelihood framework.

In sum, a goodness-of-fit test in the classic sense, which can be computed in reasonable time and can be used in forward model selection to test the goodness of fit with respect to a wide range of functions of the data and as tests of homogeneity assumptions, is not available.

3.2 .  A new score-type test

In practice, computation time and computing in general are an important concern.

In the maximum likelihood framework, the score test of Rao (1948) is preferable to the Wald test and the likelihood ratio test in terms of computation time, because it requires the estimation of the model restricted by the null hypothesis, while the Wald test requires the more time-consuming estimation of the unrestricted model and the likelihood ratio test the even more time-consuming estimation of both the restricted model and the unrestricted model. In addition, if it is desired to test models with correlated statistics or test homogeneity assumptions, root-finding algorithms to estimate the unrestricted model may need a large number of iterations and struggle to find the root(s) of estimating function (9). Therefore, in situations where computation time and computing in general are an important concern, the score test of Rao (1948) is preferable to the Wald test and the likelihood ratio test. Under regularity conditions, the asymptotic behaviour of the three tests is equivalent (see Rao, 2002; Lehmann & Romano, 2005) and, in practice, it is therefore both legitimate and prudent to let practical considerations along these lines guide the choice of test.

However, given a medium to large data set, maximum likelihood estimation is hardly necessary on statistical grounds and hardly attractive on computational grounds (see Section 2.2), and therefore tests in the method of moments framework are preferable to tests in the maximum likelihood framework. Therefore, it is desirable to obtain a score-type test in the method of moments framework. A score-type test in the method of moments framework, which is the natural counterpart of the score test of Rao (1948) when nuisance parameters are estimated by maximum likelihood estimators and of the C(α) test of Neyman (1959) when nuisance parameters are estimated by consistent estimators, can be obtained by replacing the score function by regular estimating functions (Godambe, 1960, 1991) along the lines of Basawa (1985, 1991).

The section proceeds as follows. In Section 3.2.1, first the case is considered where the estimating function is given by the score function (9), and the score test of Rao (1948) and the C(α) test of Neyman (1959) in the family of models of interest are introduced; then the case is considered where the estimating function is given by (8), and a new score-type test in the family of models of interest is introduced. Remarks and extensions are given in Section 3.2.2.

3.2.1.  Basic score-type test

Let θ1 be a vector of nuisance parameters and θ2 be a vector of parameters of primary interest, and θ′= (θ1, θ2).

In the classical Neyman–Pearson tradition, goodness of fit can be studied by specifying hypotheses regarding the postulated family of probability distributions inline image, for instance the null hypothesis

  • image(11)

tested against

  • image(12)

where θ2,0 is a specified value (such as θ2,0=0), and θ1 is unspecified. Let θ0= (θ1, θ2,0) be the parameter vector under H0: θ2=θ2,0. Let gn= g n(zn, θ0) be an estimating function satisfying regularity conditions (see Godambe, 1960, 1991; Basawa, 1985, 1991). Partition g′n= (g′1n, g′2n) in accordance with θ0= (θ1, θ2,0).

3.2.1.1. Estimating function: score function

Let gn be the score function (9) and inline image its variance–covariance matrix, partitioned into submatrices inline image, inline image, inline image, and inline image in accordance with θ0= (θ1, θ2,0). The score test of Rao (1948) and the C(α) test of Neyman (1959) can be motivated as follows (see Bera & Bilias, 2001a,b).

Score test without nuisance parameters

In the absence of nuisance parameters, θ0 reduces to θ2,0. If θ2,0 were translated by inline image, the local change in the log-likelihood function due to the local change in θ2,0 would be given approximately by

  • image(13)

Under H0: θ2=θ2,0, (13) has mean 0 and variance inline image. A test of H0: θ2=θ2,0 could be based on the test statistic

  • image(14)

Test statistic (14) is based on the linear function inline image of the score function g2n and raises the question: which linear function of the score function g2n is optimal in the sense that, under H1: θ2θ2,0, test statistic (14) is as large as possible? By the Cauchy–Schwarz inequality,

  • image(15)

where the maximum on the right-hand side of (15) is attained at inline image. The right-hand side of (15) is the score test statistic of Rao (1948).

Score test with nuisance parameters

In the presence of nuisance parameters, the score test statistic of Rao (1948) is given by

  • image(16)

where inline image is the restricted maximum likelihood estimator of θ0 under H0: θ2=θ2,0, obtained by maximizing the log-likelihood function with respect to θ0 subject to the constraint H0: θ2=θ2,0.

C(α) test with nuisance parameters

The C(α) test is designed to test hypotheses in the presence of nuisance parameters, where nuisance parameters are replaced by consistent estimators. If θ1 and θ2,0 were translated by inline image and inline image, respectively, the local change of the log-likelihood function due to the local changes in θ1 and θ2,0 would approximately be given by

  • image(17)

If θ0 were estimated by the restricted maximum likelihood estimator under H0: θ2=θ2,0, then inline image would vanish. If, however, θ0 were replaced by a consistent estimator inline image under H0: θ2=θ2,0, then inline image would not, in general, vanish. Neyman (1959) showed that, under regularity conditions, the impact of replacing θ0 by a consistent estimator inline image under H0: θ2=θ2,0 on the test can be eliminated by basing tests on

  • image(18)

where

  • image(19)

Under H0: θ2=θ2,0, (18) has mean 0 and variance inline image, where Cn is the variance–covariance matrix of en. A test of H0: θ2=θ2,0 could be based on the test statistic

  • image(20)

An argument along the lines of the score test shows that the optimal choice of inline image is given by inline image, giving rise to the C(α) test statistic of Neyman (1959):

  • image(21)
3.2.1.2. Estimating function: non-score function

Let gn be given by (8). By the definition of gn in (8), the conditional expectation of gn given X (t1) =x (t1) is given by

  • image(22)

for all θ. Let inline image be the L×L variance–covariance matrix of gn, and denote the limit of inline image as inline image, if it exists, by inline image, where wn are suitable norming constants. The derivative of gn with respect to θ is given by

  • image(23)

Denote the limit of inline image as inline image, if it exists, by inline image. Partition inline image and inline image in accordance with θ′= (θ1, θ2):

  • image(24)

Suppose that

  • image(25)

where inline image denotes convergence in distribution, NL refers to the L-variate Gaussian distribution, and inline image is non-singular.

To make the impact of replacing the unknown nuisance parameter vector θ1 by an estimator (such as a moment estimator, see Section 2.2) on the test as small as possible, Neyman’s (1959) elimination method can be exploited, as suggested by Basawa (1985, 1991) in the case of regular estimating functions. Let

  • image(26)

where inline image is non-singular. Since both g2n and inline image have zero expectation by (22) and are asymptotically Gaussian distributed by (25), one obtains

  • image(27)

where the variance–covariance matrix inline image is given by

  • image(28)

where inline image and R is the number of coordinates of θ2. Thus, under H0: θ2=θ2,0,

  • image(29)

is asymptotically central chi-square distributed with R degrees of freedom.

The entities inline image and inline image can be replaced by inline image and inline image, respectively, without changing the asymptotic distribution of (29). The parameter vector θ0 can be replaced by a restricted moment estimator inline image under H0: θ2=θ2,0 (see Section 2.2). The test statistic, obtained by replacing inline image, inline image, and θ0 in (29) by inline image, inline image, and inline image, respectively, is given by

  • image(30)

The entities gn, inline image, and inline image are not available in closed form, but can be estimated by Monte Carlo methods. Monte Carlo estimation of gn and inline image, which are expectations, is straightforward (Hammersley & Handscomb, 1964). Monte Carlo estimators of inline image, corresponding to derivatives, can be found in Schweinberger and Snijders (2007). These Monte Carlo estimators are simulation-consistent in the sense that the Monte Carlo estimators converge in probability to the desired entities as the number of Monte Carlo simulations increases without bound. Therefore, the Monte Carlo estimator of test statistic Cn is simulation-consistent.

3.2.2.  Remarks and extensions

Observe that, to test restrictions on the parameter vector θ2, θ2 need not be estimated, saving computation time and avoiding computational issues which may arise in the estimation of unrestricted models with correlated statistics or without homogeneity assumptions.

If θ2 (and therefore bn and inline image) is a scalar, then (30) can be used both in its quadratic form, as presented above, and in its corresponding linear form,

  • image(31)

The linear form is convenient when one-sided one-parameter tests are desired. The minus sign in (31) facilitates the interpretation in the sense that, if u2 denotes the statistic corresponding to the parameter θ2 and its conditional expectations given X (t1) =x (t1) are increasing functions of θ2, then, by the definition of gn in (8), θ2θ2,0 > 0 is associated with positive values of (31). By (27), the asymptotic distribution of (31) under H0: θ2=θ2,0 is standard Gaussian.

Furthermore, tests with R > 1 degrees of freedom can be complemented with one-degree-of-freedom tests, testing the restrictions one by one; two-sided one-parameter tests can be based on the test statistic Cn, while one-sided one-parameter tests can be based on the test statistic Dn. It is convenient to compute the one-parameter test statistics by using the simulations under the null hypothesis of the multi-parameter test, requiring no additional, time-consuming simulations. If the null hypothesis of the multi-parameter test is true, such one-parameter test statistics are computed correctly. Otherwise, they are computed incorrectly, but one can take them as an informal indication of where the model deviates from the null hypothesis of the multi-parameter test, without requiring additional, time-consuming simulations.

Observe that test statistics Cn and Dn have an appealing interpretation in terms of goodness of fit in the classic sense, because both are based on

  • image(32)

where u2 is the vector of statistics corresponding to the parameter vector θ2. In other words, the test statistics are based on the ‘distance’ between the expected value of the function u2 of the data – evaluated under H0: θ2=θ2,0– and the observed value of u2.

Since the one- and multi-parameter tests do not require the estimation of the unrestricted model, the tests are most useful in forward model selection and as tests of homogeneity assumptions with respect to time and nodes. To complement the usefulness of the tests, we derive one-step estimators which are useful as starting values of parameters in forward model selection. Suppose that tests indicate empirical evidence against the model restricted by H0: θ2=θ2,0 and it is desired to estimate the unrestricted model. If gn is differentiable at inline image, then, by definition (Magnus & Neudecker, 1988, p. 82),

  • image(33)

where

  • image(34)

Thus, in the limit, solving gn= g n(zn, θ) =0 is the same as solving

  • image(35)

suggesting the one-step estimator

  • image(36)

where inline image is non-singular. The one-step estimator θ is an approximation of the unrestricted estimator inline image. The one-step estimator θ is useful as a starting value of the parameter vector θ in the root-finding algorithm which is used to find the root(s) inline image of the estimating function gn= g n(zn, θ). If either gn is approximately linear as a function of θ or inline image is sufficiently close to inline image, the linear approximation of gn around inline image can be expected to result in good one-step estimators θ and therefore good starting values of θ. Otherwise, the one-step estimator θ is at least an improvement on inline image.

Finally, concerning the asymptotic Gaussian distribution of the estimating function gn (see (25)), note that the choice of statistics u of the estimating function gn is arbitrary as long as gn is increasing in θ and sensitive to changes in θ. The use of test statistics Cn and Dn is admissible for all choices of gn for which gn is asymptotically Gaussian distributed or at least approximately Gaussian distributed. In most applications, verifying the asymptotic distribution of gn is hard. Indeed, hardly anything is known about asymptotics of estimators and tests in the field of social networks, leaving aside simplistic models without dependence; for example, as noted in Section 3.1, the distribution of the t-type test is unknown.

4. Monte Carlo simulation study

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References

This section studies the finite-sample behaviour of the score-type test (30) and the t-type test (10) by Monte Carlo simulation. For convenience, the two tests will be referred to as the ‘score test’ and ‘t-test’, respectively. While it would be desirable to compare the likelihood ratio test of Snijders et al. (2010) to the score test and t-test, the fact that its computation tends to take 35 times as much time as the score test and the t-test (see Section 3.1) implies that such a comparison is not feasible.

Data are generated by simulating the Markov process with n= 30 (‘small data set’) and n= 60 nodes (‘moderate data set’) in the time interval [0, 2], starting with a real-world digraph at time point t1= 0 and ‘observing’ the random digraph at time points t2= 1 and t3= 2. In period m= 1, 2, the rate function is given by

  • image(37)

and the objective function is given by

  • image(38)

where

  • • 
    inline image is the number of arcs,
  • • 
    inline image is the number of reciprocated arcs,
  • • 
    inline image: is the number of transitive triples,
  • • 
    inline image is the number of indirect connections,
  • • 
    inline image is the number of arcs weighted by covariate cih,
  • • 
    inline image is the number of arcs weighted by covariate dh.

The values of the dyadic covariate cih are generated by independent draws from the Poisson(1) distribution, and the values of the node-bound covariate dh are generated by independent draws from the Bernoulli(1/2) distribution.

In practice, at least two main purposes of goodness-of-fit tests can be distinguished, and hence the simulation study consists of two main parts: (1) testing parameters which capture structural features of the data such as third-order dependence among arcs (‘clustering’), which gives rise to the consideration of correlated statistics such as si3 and si4; and (2) testing the impact of covariates on the digraph evolution. Tests of homogeneity assumptions with respect to time, an important application of the score test, have been studied elsewhere (see Lospinoso, Schweinberger, Snijders, & Ripley, 2011). The basic data-generating model corresponds to parameters ρ12= 4, β1=−1 (corresponding to si1), and β2= 1 (corresponding to si2), which are common to all models used for generating data. The two parts of the simulation study test hypotheses involving the parameters (1) β3 and β4, corresponding to si3 and si4, respectively, and (2) β5 and β6, corresponding to si5 and si6, respectively. All tests are two-sided.

4.1 .  Testing effects capturing third-order dependence

The basic model is Pθ,  θ= (ρ1, ρ2, β1, β2, β3, β4)′= (4, 4, − 1, 1, β3, β4)′; three values of β3 (0, 0.2, and 0.4) and β4 (0, −0.3, and −0.6) are considered, including all combinations, giving nine models. The selected values of the parameters β3 and β4 are in the range of estimates encountered in empirical applications and generate data that resemble real-world networks in terms of degree and other fundamental functions of the data. For each model, 500 data sets are generated; the reason for limiting the number of the data sets to 500 is that estimating nine models in restricted as well as unrestricted form from 500 data sets with n= 30 as well as n= 60 takes weeks (on one conventional PC). For each model and each data set, the score test is evaluated in one estimation run (where the four unrestricted parameters are estimated) and the t-test in another estimation run (where all six parameters are estimated); the parameters are estimated by the conditional method of moments of Snijders (2001).

Table 1 shows the empirical rejection probabilities for the hypotheses mentioned and the nine data-generating models using the nominal significance level .05. Observe that if H0 is true and the test is of size .05, then the Binomial(500, .05) distribution of the number of rejections implies that the empirical rejection probability should be roughly between .03 and .07. Under H0: β34= 0, the empirical rejection probabilities of the one- and two-parameter score tests are between .03 and .07 and the score tests are therefore neither conservative nor liberal. The t-tests, however, seem to be conservative.

Table 1. Monte Carlo results, model Pθ,  θ= (ρ1, ρ2, β1, β2, β3, β4)′= (4, 4, − 1, 1, β3, β4)′, n= 30 and n= 60: empirical rejection probabilities for tests with nominal significance level .05
True model H 0:Score test β34= 0Score test β3= 0Score test β4= 0 t-test β3= 0 t-test β4= 0
P θ , β3= 0, β4= 0 n= 30.024.050.034.032.014
n= 60.036.054.044.048.024
P θ , β3= 0, β4=−0.3 n= 30.492.078.596.018.458
n= 60.972.140.988.052.972
P θ , β3= 0, β4=−0.6 n= 30.892.180.940.014.862
n= 601.000.4721.000.0301.000
P θ , β3= 0.2, β4= 0 n= 30.446.540.136.422.026
n= 60.744.812.128.788.030
P θ , β3= 0.2, β4=−0.3 n= 30.816.512.800.220.612
n= 601.000.832.998.506.990
P θ , β3= 0.2, β4=−0.6 n= 30.982.658.990.162.944
n= 601.000.9461.000.3161.000
P θ , β3= 0.4, β4= 0 n= 30.984.990.664.980.026
n= 601.0001.000.6501.000.038
P θ , β3= 0.4, β4=−0.3 n= 30.992.978.976.870.786
n= 601.0001.0001.000.986.990
P θ , β3= 0.4, β4=−0.6 n= 301.000.9521.000.574.990
n= 601.0001.0001.000.9001.000

Figure 1 shows histograms of the distributions of the one- and two-parameter score test statistics and the t-test statistics under H0: β34= 0. The distributions of the score test statistics seem to agree with the chi-square reference distributions, while the distributions of the t-test statistics do not seem to agree with the Gaussian reference distribution suggested by Snijders (1996).

image

Figure 1. Monte Carlo results, model Pθ,  θ= (ρ1, ρ2, β1, β2, β3, β4)′= (4, 4, − 1, 1, 0, 0)′, n= 30 and n= 60: distribution of test statistics. The curves represent the reference distributions under H0; c and t refer to score test (30) and t-test (10), respectively.

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The two-parameter score test behaves reasonably under H0: β34= 0 and seems to have, for all practical purposes, enough power to detect departures from H0.

The two one-parameter score tests have considerably more power than the two corresponding t-tests. Observe, however, that the two-parameter score test statistic and the two corresponding one-parameter score test statistics are computed under H0: β34= 0, and thus when data are generated according to models with either β3 or β4 or both non-zero, then the computation of (one of the two) one-parameter score test statistics is incorrect. The correct computation of the one-parameter score test statistics would require additional, time-consuming estimation runs, but in practice these incorrect one-parameter test statistics can be taken as an informal indication of where the model deviates from H0: β34= 0, without requiring additional, time-consuming estimation runs (see Section 3.2.2). In other words, in practice one can follow a forward model selection approach by restricting multiple parameters and, provided the multi-parameter score test indicates that there is empirical evidence against the restricted model, one can drop the restriction on the parameter for which the p-value of the one-parameter score test is lowest, estimate the model without the restriction, and test whether the restrictions on the other parameters should be dropped as well.

4.2 .  Testing covariate-related effects

The basic model is Pθ,  θ= (ρ1, ρ2, β1, β2, β3, β5, β6)′= (4, 4, − 1, 1, 0.2, β5, β6)′, which includes transitivity parameter β3; three values of β5 (0, 0.1, and 0.2) and β6 (0, 0.2, and 0.4) are considered, including all combinations, giving nine models.

Table 2 shows the empirical rejection probabilities for the hypotheses mentioned and the nine data-generating models, using the nominal significance level .05. The table indicates that the one- and two-parameter score tests are conservative, while the t-tests are neither conservative nor liberal.

Table 2. Monte Carlo results, model Pθ,  θ= (ρ1, ρ2, β1, β2, β3, β5, β6)′= (4, 4, − 1, 1, 0.2, β5,β6)′, n= 30 and n= 60: empirical rejection probabilities for tests with nominal significance level .05
True model H 0:Score test β56= 0Score test β5= 0Score test β6= 0 t-test β5= 0 t-test β6= 0
P θ , β5= 0, β6= 0 n= 30.024.024.030.044.068
n= 60.012.014.032.058.062
P θ , β5= 0, β6= 0.2 n= 30.152.034.220.068.326
n= 60.336.030.512.066.626
P θ , β5= 0, β6= 0.4 n= 30.588.020.696.070.808
n= 60.942.022.974.056.982
P θ , β5= 0.1, β6= 0 n= 30.182.250.044.326.078
n= 60.394.526.038.640.056
P θ , β5= 0.1, β6= 0.2 n= 30.344.276.190.374.292
n= 60.660.530.462.632.596
P θ , β5= 0.1, β6= 0.4 n= 30.734.262.728.378.822
n= 60.980.500.982.616.992
P θ , β5= 0.2, β6= 0 n= 30.690.780.034.846.058
n= 60.968.988.046.994.084
P θ , β5= 0.2, β6= 0.2 n= 30.790.792.232.858.318
n= 60.996.992.482.998.600
P θ , β5= 0.2, β6= 0.4 n= 30.932.812.712.886.818
n= 601.000.984.984.994.990

Figure 2 shows histograms of the distributions of the one- and two-parameter score test statistics and the t-test statistics under H0: β56= 0. The agreement of the distributions of the score test statistics with the chi-square reference distributions seems to be acceptable, while the agreement of the distributions of the t-test statistics with the Gaussian reference distribution suggested by Snijders (1996) is, once again, less than acceptable.

image

Figure 2. Monte Carlo results, model Pθ,  θ= (ρ1, ρ2, β1, β2, β3, β5, β6)′= (4, 4, − 1, 1, 0.2, 0, 0)′, n= 30 and n= 60: distribution of test statistics. The curves represent the reference distributions under H0; c and t refer to score test (30) and t-test (10), respectively.

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Table 2 indicates that the one-parameter score test has less power than the t-test, in particular if n= 30 and the departure from H0 is small.

5. Application

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References

Pahor (2003) and Pahor, Prasnikar, and Ferligoj (2004) studied the directed cross-ownerships among 413 business firms in post-communist Slovenia observed at five time points, where directed cross-ownership inline image means that firm A holds stock market shares in firm B. We reanalyse the data in the light of the score-type test (30).

The baseline model corresponds to the rate function

  • image(39)

where m refers to period m= 1, 2, 3, 4, and the objective function

  • image(40)

where

  • • 
    inline image is the number of arcs,
  • • 
    inline image is the number of reciprocated arcs,
  • • 
    inline image is the interaction of arcs and covariate dh1,
  • • 
    inline image is the interaction of arcs and covariate dh2,

in which dh1 indicates whether or not firm h was quoted on the stock exchange, while dh2 refers to the size of firm h. The parameter vector θ of the baseline model Pθ corresponds to inline image, where inline image.

Pahor (2003) suspected that the data may exhibit third-order dependence, leading to the transitivity parameter β5 and the indirect connections parameter β6 as candidates to be tested, which correspond to statistics ∑nh, l= 1xihxhlxil and ∑nh= 1(1 −xih)max lxilxlh, respectively. According to Table 3, the two-parameter score test of H0: β56= 0 clearly indicates that parameters capturing third-order dependence are required to improve the goodness of fit of the model, and the one-parameter score tests suggest including β5 but not β6. A one-sided test of H0: β5= 0 can be carried out by using the linear form of the score test (see (31)), giving 7.86: under the assumption that the linear form under H0: β5= 0 is approximately standard Gaussian distributed, it seems that β5 is positive, which is supported by the one-step estimate of β5 (see (36)) given by 0.996. As a result, β5 is henceforth included in the model, while β6 is not.

Table 3. Pahor data: testing restricted models
Nuisance parametersTestScore testd.f. p-value
  1. The parameter vector inline image is given by inline image.

inline image H 0: β56= 061.952<.0001
inline image H 0: β5= 061.821<.0001
inline image H 0: β6= 0<0.011 .9905
inline image, β5 H 0: β789= 0127.593<.0001
inline image, β5 H 0: β7= 043.381<.0001
inline image, β5 H 0: β8= 070.341<.0001
inline image, β5 H 0: β9= 014.041 .0002
inline image, β5, β7, β8, β9 H 0: β101112= 020.193 .0002
inline image, β5, β7, β8, β9 H 0: β10= 010.371 .0013
inline image, β5, β7, β8, β9 H 0: β11= 04.601 .0319
inline image, β5, β7, β8, β9 H 0: β12= 013.831 .0002

Pahor (2003) argued that firms tend to hold shares in other firms close to them with respect to region (cih1), to industry branch (cih2), and to other firms having the same owner (cih3). Three parameters are added, β7, β8, and β9, corresponding to statistics ∑nh= 1xih cihl, l= 1, 2, 3, respectively. The three-parameter score test of H0: β789= 0 and the three corresponding one-parameter tests, shown in Table 3, suggest adding all three covariates to the model, which is done.

It is interesting to test whether the values of the parameters are constant across time intervals. A basic parameter for which such homogeneity tests frequently make sense is the ‘out-degree’ parameter β1, corresponding to statistic si1. A homogeneity test for β1 is conducted by testing H0: β101112= 0, where β10, β11, and β12 correspond to statistics e(m)isi1, m= 2, 3, 4, respectively, in which e(m)i is a period-dependent dummy variable with value 1 in the time interval [tm, tm+ 1] for all i and 0 otherwise. According to Table 3, the three-parameter score test of H0: β101112= 0 indicates that there is empirical evidence against H0, and the three corresponding one-parameter tests suggest that it is sensible to add β10, β11, and β12 to the model. The one-sided one-parameter score test statistics (see (31)) are −3.22, −2.15, and 3.72, respectively, suggesting that the values of β10 and β11 are negative while β12 is positive. Table 4 gives the one-step estimate (see (36)) of the parameter inline image based on the moment estimate of θ under the model restricted by H0: β101112= 0, and shows in addition the unrestricted moment estimate of θ, including standard errors and t-tests. The one-step estimates roughly agree with the unrestricted moment estimates. The t-tests by and large agree with the score tests (regarding β5, β7, β8, β9, and β10), but slightly disagree about β11 and clearly disagree about β12.

Table 4. Pahor data, model Pθ, inline image: estimates, standard errors, and t-tests
 One-step estimate θMoment estimate inline images.e.inline image t-test p-value
  1. To save space, the nuisance parameters inline image, β2, β3, and β4 are omitted. The one-step estimate θ is based on the moment estimate of θ under the model restricted by H0: β101112= 0; the moment estimate inline image is the unrestricted moment estimate of θ.

β1−2.65−2.670.097−27.54<.0001
β50.810.810.1027.91<.0001
β70.570.580.0767.60<.0001
β81.011.010.1069.57<.0001
β90.890.890.2583.45.0006
β10−0.31−0.290.134−2.12.0337
β11−0.31−0.280.158−1.79.0735
β120.120.130.1171.14.2544

6. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References

In the framework of continuous-time Markov models for network panel data, we proposed score-type tests which are goodness-of-fit tests in the classical sense of Pearson (1900), are most useful in forward model selection and as tests of homogeneity assumptions, and possess substantial computational advantages.

A Monte Carlo simulation study shed light on the finite-sample behaviour of the score-type tests as well as the t-type tests of Snijders (1996). The score-type tests turned out to be most useful as tests of third-order dependence and therefore as tests of the assertion of Faust (2007) that most of the third-order dependence found in data sets can be explained by degree distributions and second-order dependence. The t-type tests seem to be less suited to testing third-order dependence and more suited to testing covariates, though the null distributions of the t-type tests do not seem to be Gaussian as assumed by Snijders (1996).

Last, tests of homogeneity assumptions with respect to time, an important application of the score-type tests proposed here, were studied elsewhere (see Lospinoso et al., 2011).

The proposed test statistic is implemented in the R package RSiena (Ripley, Snijders, & Lopez, 2011).

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References

Support is acknowledged from the Netherlands Organization for Scientific Research (NWO grant 401-01-550). The author is grateful to Marko Pahor for sharing data with the author and to Tom A. B. Snijders for stimulating discussions and valuable comments.

References

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  2. Abstract
  3. 1. Introduction
  4. 2. Model specification and estimation
  5. 3. Goodness-of-fit tests
  6. 4. Monte Carlo simulation study
  7. 5. Application
  8. 6. Discussion
  9. Acknowledgements
  10. References
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