The effect of density on the level of bias in the network autocorrelation model

Researchers interested in the effects of social network ties on behavior are increasingly turning to the network autocorrelation model, which allows for the simultaneous computation of individual-level and network-level effects. Earlier research, however, had pointed to the possibility that the maximum likelihood estimates used to compute the network autocorrelation model yielded negatively biased parameter estimates. In this paper we use simulations to examine whether – and the conditions under which – a negative bias exists. We show that the network parameter estimate ρ is negatively biased under nearly all conditions, and that this bias becomes more severe at higher levels of both ρ and network density. We conclude by discussing the implications of these findings for researchers planning to use the network autocorrelation model.     

Structure and bias in the network autocorrelation model

In a recent paper (Mizruchi and Neuman, 2008), we showed that estimates of ρ in the network autocorrelation model exhibited a systematic negative bias and that the magnitude of this bias increased monotonically with increases in network density. We showed that this bias held regardless of the size of the network, the number of exogenous variables in the model, and whether the matrix W was normalized or in raw form. The networks in our simulations were random, however, which raises the question of the extent to which the negative bias holds in various structured networks. In this paper, we reproduce the simulations from our earlier paper on a series of networks drawn to represent well-known structures, including star, caveman, and small-world structures. Results from these simulations indicate that the pattern of negative bias in ρ continues to hold in all of these structures and that the negative bias continues to increase at increasing levels of density. Interestingly, the negative bias in ρ is especially pronounced at extremely low-density levels in the star network. We conclude by discussing the implications of these findings.     

Statistical power of the social network autocorrelation model

The network autocorrelation model has become an increasingly popular tool for conducting social network analysis. More and more researchers, however, have documented evidence of a systematic negative bias in the estimation of the network effect (ρ). In this paper, we take a different approach to the problem by investigating conditions under which, despite the underestimation bias, a network effect can still be detected by the network autocorrelation model. Using simulations, we find that moderately-sized network effects (e.g., ρ = .3) are still often detectable in modest-sized networks (i.e., 40 or more nodes). Analyses reveal that statistical power is primarily a nonlinear function of network effect size (ρ) and network size (N), although both of these factors can interact with network density and network structure to impair power under certain rare conditions. We conclude by discussing implications of these findings and guidelines for users of the autocorrelation model.     

Bayesian estimation of the network autocorrelation model

The network autocorrelation model has been extensively used by researchers interested modeling social influence effects in social networks. The most common inferential method in the model is classical maximum likelihood estimation. This approach, however, has known problems such as negative bias of the network autocorrelation parameter and poor coverage of confidence intervals. In this paper, we develop new Bayesian techniques for the network autocorrelation model that address the issues inherent to maximum likelihood estimation. A key ingredient of the Bayesian approach is the choice of the prior distribution. We derive two versions of Jeffreys prior, the Jeffreys rule prior and the Independence Jeffreys prior, which have not yet been developed for the network autocorrelation model. These priors can be used for Bayesian analyses of the model when prior information is completely unavailable. Moreover, we propose an informative as well as a weakly informative prior for the network autocorrelation parameter that are both based on an extensive literature review of empirical applications of the network autocorrelation model across many fields. Finally, we provide new and efficient Markov Chain Monte Carlo algorithms to sample from the resulting posterior distributions. Simulation results suggest that the considered Bayesian estimators outperform the maximum likelihood estimator with respect to bias and frequentist coverage of credible and confidence intervals.     

Modeling social influence through network autocorrelation: constructing the weight matrix

Many physical and social phenomena are embedded within networks of interdependencies, the so-called ‘context’ of these phenomena. In network analysis, this type of process is typically modeled as a network autocorrelation model. Parameter estimates and inferences based on autocorrelation models, hinge upon the chosen specification of weight matrix W, the elements of which represent the influence pattern present in the network. In this paper I discuss how social influence processes can be incorporated in the specification of W. Theories of social influence center around ‘communication’ and ‘comparison’; it is discussed how these can be operationalized in a network analysis context. Starting from that, a series of operationalizations of W is discussed. Finally, statistical tests are presented that allow an analyst to test various specifications against one another or pick the best fitting model from a set of models.     

Weight matrices for social influence analysis: An investigation of measurement errors and their effect on model identification and estimation quality

Weight matrices, such as used in network autocorrelation models, are useful to investigate social influence processes. The objective of this paper is to investigate a key topic that has received relatively little attention in previous research, namely the issues that arise when observational limitations lead to measurement errors in these weight matrices. Measurement errors are investigated from two perspectives: when relevant ties are omitted, and when irrelevant ties are erroneously included as part of the matrix. The paper first shows analytically that these two situations result in biased estimates. Next, a simulation experiment provides evidence of the effect of erroneously coding the weight matrix on model performance and the ability of a network autocorrelation test to identify social influence effects. The results suggest that depending on the level of autocorrelation and the topology attributes of the underlying matrix, there is a window of opportunity to identify and model social influence processes even in situations where the ties in a matrix cannot be accurately observed.     

A comparison of estimators for the network autocorrelation model based on observed social networks

Network autocorrelation models (NAMs) are widely used to study a response variable of interest among subjects embedded within a network. Although the NAM is highly useful for studying such networked observational units, several simulation studies have raised concerns about point estimation. Specifically, these studies have consistently demonstrated a negative bias of maximum likelihood estimators (MLEs) of the network effect parameter. However, in order to gain a practical understanding of point estimation in the NAM, these findings need to be expanded in three important ways. First, these simulation studies are based on relatively simple network generative models rather than observed networks, thereby leaving as an open question how realistic network topologies may affect point estimation in practice. Second, although there has been strong work done in developing two-stage least squares estimators as well as Bayesian estimators, only the MLE has received extensive attention in the literature, thus leaving practitioners in question as to best practices. Third, the performance of these estimators need to be compared using both bias and variance, as well as the coverage rate of each estimator's corresponding confidence or credible interval. In this paper we describe a simulation study which aims to overcome these shortcomings in the following way. We first fit real social networks using the exponential random graph model and used the Bayesian predictive posterior distribution to generate networks with realistic topologies. We then compared the performance of the three different estimators mentioned above.     

Imputation of attributes in networked data using Bayesian autocorrelation regression models

Misspecification in network autocorrelation models poses a challenge for parameter estimation, which is amplified by missing data. Model misspecification has been a focus of recent work in the statistics literature and new robust procedures have been developed, in particular cutting feedback. This paper shows how this helps in a misspecified network autocorrelation model. Where model misspecification is mild and the traits are fully observed, Bayesian imputation is routine. In settings with high missingness, Bayesian inference can fail, but a closely related cut model is robust. We illustrate this on a data set of graduate students using a Facebook-like messaging app.     

Bias decomposition and estimator performance in respondent-driven sampling

Respondent-Driven Sampling (RDS) is a method of network sampling that is used to sample hard-to-reach populations. The resultant sample is non-random, but different weighting methods can account for the over-sampling of (1) high-degree individuals and (2) homophilous groups that recruit members more effectively. While accounting for degree-bias is almost universally agreed upon, accounting for recruitment-bias has been debated as it can further increase estimate variance without substantially reducing bias. Simulation-based research has examined which weighting procedures perform best given underlying population network structures, group recruitment differences, and sampling processes. Yet, in the field, analysts do not have a priori knowledge of the network they are sampling. We show that the RDS sample data itself can determine whether a degree-based estimator is sufficient. Formulas derived from the decomposition of a ‘dual-component’ estimator can approximate the ‘recruitment component’ (RC) and ‘degree component’ (DC) of a sample’s bias. Simulations show that RC and DC values can predict the performance of different classes of estimators. Samples with extreme ‘RC’ values, a consequence of network homophily and differential recruitment, are better served by a classical estimator. The use of sample data to improve estimator selection is a promising innovation for RDS, as the population network features that should guide estimator selection are typically unknown.     

I can’t smile without you: Spousal correlation in life satisfaction

This paper tests whether one partner’s happiness significantly influences the happiness of the other partner. Using 10 waves of the British Household Panel Survey, it utilizes a panel-based GMM methodology to estimate a dynamic model of life satisfaction. The use of the GMM-system estimator corrects for correlated effects of partner’s life satisfaction and solves the problem of measurement error bias. The results show that, for both genders, there is a positive and statistically significant spillover effect of life satisfaction that runs from one partner to the other partner in a couple. The positive bias on the estimated spillover effect coming from assortative mating and shared social environment at cross-section is almost offset by the negative bias coming from systematic measurement errors in the way people report their life satisfaction. Moreover, consistent with the spillover effect model, couple dissolution at t + 1 is negatively correlated with partners’ life satisfaction at t.     

THE HARDER THE TASK,THE HIGHER THE SCORE: FINDINGS OF A DIFFICULTY BIAS

Studies have found that going first or last in a sequential order contest leads to a biased outcome, commonly called order bias (or primacy and recency). Studies have also found that judges have a tendency to reward contestants they recognize with additional points, called reference bias. Controlling for known biases, we test for a new type of bias we refer to as “difficulty bias,” which reveals that athletes attempting more difficult routines receive higher execution scores, even when difficulty and execution are judged separately. Despite some identification challenges, we add to the literature by finding strong evidence of a difficulty bias in gymnastics. We also provide generalizations beyond athletics. (JEL L10, L83, D81, J70, Z1)     

A comparative study of social network models: Network evolution models and nodal attribute models

This paper reviews, classifies and compares recent models for social networks that have mainly been published within the physics-oriented complex networks literature. The models fall into two categories: those in which the addition of new links is dependent on the (typically local) network structure (network evolution models, NEMs), and those in which links are generated based only on nodal attributes (nodal attribute models, NAMs). An exponential random graph model (ERGM) with structural dependencies is included for comparison. We fit models from each of these categories to two empirical acquaintance networks with respect to basic network properties. We compare higher order structures in the resulting networks with those in the data, with the aim of determining which models produce the most realistic network structure with respect to degree distributions, assortativity, clustering spectra, geodesic path distributions, and community structure (subgroups with dense internal connections). We find that the nodal attribute models successfully produce assortative networks and very clear community structure. However, they generate unrealistic clustering spectra and peaked degree distributions that do not match empirical data on large social networks. On the other hand, many of the network evolution models produce degree distributions and clustering spectra that agree more closely with data. They also generate assortative networks and community structure, although often not to the same extent as in the data. The ERGM model, which turned out to be near-degenerate in the parameter region best fitting our data, produces the weakest community structure.     

Cultural convergence 10 years later: A reexamination of intergroup bias among journalists in the digital media age

This study revisits Filak’s 2004 research of print and broadcast journalists to assess whether changes in the field have diminished the levels of intergroup bias for these groups. The findings here demonstrate that print and broadcast journalists (n = 191) remained biased against each other, even in the face of obvious outside threats and outgroup benefits. In addition, the journalists were more likely to view convergence efforts negatively when these efforts were perceived to be the work of outgroup members. In comparing the data gathered here to that in the original study, dislike and distrust of each other remain consistent. Finally, the influx of digital media, although viewed as valuable by all participants, has had little impact regarding the levels of bias the journalists espoused.     

Network autocorrelation models with egocentric data

Network autocorrelation models have been widely used for decades to model the joint distribution of the attributes of a network's actors. This class of models can estimate both the effect of individual characteristics as well as the network effect, or social influence, on some actor attribute of interest. Collecting data on the entire network, however, is very often infeasible or impossible if the network boundary is unknown or difficult to define. Obtaining egocentric network data overcomes these obstacles, but as of yet there has been no clear way to model this type of data and still appropriately capture the network effect on the actor attributes in a way that is compatible with a joint distribution on the full network data. This paper adapts the class of network autocorrelation models to handle egocentric data. The proposed methods thus incorporate the complex dependence structure of the data induced by the network rather than simply using ad hoc measures of the egos’ networks to model the mean structure, and can estimate the network effect on the actor attribute of interest. The vast quantities of unknown information about the network can be succinctly represented in such a way that only depends on the number of alters in the egocentric network data and not on the total number of actors in the network. Estimation is done within a Bayesian framework. A simulation study is performed to evaluate the estimation performance, and an egocentric data set is analyzed where the aim is to determine if there is a network effect on environmental mastery, an important aspect of psychological well-being.     

Analytic strategies for longitudinal networks with missing data

Missing data are often problematic when analyzing complete longitudinal social network data. We review approaches for accommodating missing data when analyzing longitudinal network data with stochastic actor-based models. One common practice is to restrict analyses to participants observed at most or all time points, to achieve model convergence. We propose and evaluate an alternative, more inclusive approach to sub-setting and analyzing longitudinal network data, using data from a school friendship network observed at four waves (N = 694). Compared to standard practices, our approach retained more information from partially observed participants, generated a more representative analytic sample, and led to less biased model estimates for this case study. The implications and potential applications for longitudinal network analysis are discussed.     

Advances in biased net theory: definitions,derivations, and estimations

Random and biased net theory, introduced by Rapoport and others in the 1950s, is one of the earliest approaches to the formal modeling of social networks. In this theory, intended as a theory of large-scale networks, ties between nodes derive both from random and non-random events of connection. The non-random connections are postulated to arise through “bias” events that incorporate known or suspected systematic tendencies in tie formation, such as, mutuality or reciprocity, transitivity or closure in triads, and homophily—the overrepresentation of ties between persons who share important socio-demographic attributes like race/ethnicity or level of educational attainment. A key problem for biased net theory has been analytical intractability of the models. Formal derivations require approximation assumptions and model parameters have been difficult to estimate. The accuracy of the derived formulas and the estimated parameters has been difficult to assess. In this paper, we attempt to address long-standing issues in biased net models stemming from their analytical intractability. We first reformulate and clarify the definitions of basic biases. Second, we derive from first principles the triad distribution in a biased net, using two different analytical strategies to check our derivations. Third, we set out a pseudo-likelihood method for parameter estimation of key bias parameters and then check the accuracy of this relatively simple but approximate scheme against the results obtained from the triad distribution derivation.     

Lead exposure is a risk for worsening bone mineral density in middle-aged male workers

Abstract

Objective: Lead exposure linked to osteoporosis in women. However, there is no direct evidence whether lead exposure has effects on bone metabolism in middle-aged male subjects. Therefore, the present study investigated the relationship between bone mineral densitometry measurements, bone markers, endocrine hormones and blood lead levels.

Material and methods: The present study included lead exposure patients (n: 30) and control subjects (n: 32). We recorded information on patient demographics and risk factors of osteoporosis. Blood lead levels were evaluated using Varian AA 240Z atomic absorption spectrophotometry. Bone mineral density measurements were measured using dual-energy X-ray absorptiometry.

Results: Each lumbar T and Z scores in the lead exposure group were lower than the control group. There were no significant differences in femur neck and femur total T and Z scores between two groups. Blood lead levels were also negatively correlated with lumbar 2-4 T score, total lumbar T score, lumbar 2-4 Z score and total lumbar Z score. Urinary hydroxyproline and urinary deoxypyridinoline levels in the lead exposure group were significantly higher compared to controls. Blood lead levels were strong, positively correlated with urinary deoxypyridinoline. Endocrine hormone levels and 1,25-dihydroxy-vitamin D3 levels were comparable between lead exposure and control group.

Conclusion: Lead exposure in male workers is an important factor for deterioration in bone mineral density. We should be screening blood lead levels and history of lead exposure in male osteoporosis.     

Structural effects of network sampling coverage I: Nodes missing at random

Network measures assume a census of a well-bounded population. This level of coverage is rarely achieved in practice, however, and we have only limited information on the robustness of network measures to incomplete coverage. This paper examines the effect of node-level missingness on 4 classes of network measures: centrality, centralization, topology and homophily across a diverse sample of 12 empirical networks. We use a Monte Carlo simulation process to generate data with known levels of missingness and compare the resulting network scores to their known starting values. As with past studies (0035 and 0135), we find that measurement bias generally increases with more missing data. The exact rate and nature of this increase, however, varies systematically across network measures. For example, betweenness and Bonacich centralization are quite sensitive to missing data while closeness and in-degree are robust. Similarly, while the tau statistic and distance are difficult to capture with missing data, transitivity shows little bias even with very high levels of missingness. The results are also clearly dependent on the features of the network. Larger, more centralized networks are generally more robust to missing data, but this is especially true for centrality and centralization measures. More cohesive networks are robust to missing data when measuring topological features but not when measuring centralization. Overall, the results suggest that missing data may have quite large or quite small effects on network measurement, depending on the type of network and the question being posed.     

A meta-analytic review of the effect of guilt on compliance

Meta-analytic procedures were used to estimate the effect of experienced guilt on compliance. Examination of 47 effect sizes indicated that inducing guilt is an effective means by which to increase compliance, ρ = .26. Moreover, despite coding for numerous substantive and methodological moderators, there was no evidence of moderation in these data. Instead, correcting for measurement error in the independent variable and restriction in range in the dependent explained all variance in effect sizes, yielding a corrected effect size of ρ′ = .35.