MULTILEVEL COVARIANCE STRUCTURE ANALYSIS BY FITTING MULTIPLE SINGLE-LEVEL MODELS

Data in social and behavioral sciences are often hierarchically organized. Multilevel statistical procedures have been developed to analyze such data while taking into account the dependence of observations. When simultaneously evaluating models at all levels, a significant statistic provides no information on the level at which the model is misspecified. Model misspecification can exist at one or several levels simultaneously. When one level is misspecified, the other levels may be affected even when they are correctly specified. Motivated by these observations, we propose to separate a multilevel covariance structure into multiple single-level covariance structure models and to fit these single-level models as in conventional covariance structure analysis. A procedure for segregating the multilevel model into single-level models is developed. Five test statistics for evaluating a model at each level are provided. Standard error formulas for the separate estimators are also provided, and their efficiency is compared to simultaneous estimators. Empirical and Monte Carlo results demonstrate the advantages of the segregated procedure over the simultaneous procedure. Computer programs that will allow the developed procedure to be used in practice are also presented.     

Finite Normal Mixture SEM Analysis by Fitting Multiple Conventional SEM Models

This paper proposes a two-stage maximum likelihood (ML) approach to normal mixture structural equation modeling (SEM), and develops statistical inference that allows distributional misspecification. Saturated means and covariances are estimated at stage-1 together with a sandwich-type covariance matrix. These are used to evaluate structural models at stage-2. Techniques accumulated in the conventional SEM literature for model diagnosis and evaluation can be used to study the model structure for each component. Examples show that the two-stage ML approach leads to correct or nearly correct models even when the normal mixture assumptions are violated and initial models are misspecified. Compared to single-stage ML, two-stage ML avoids the confounding effect of model specification and the number of components, and is computationally more efficient. Monte-Carlo results indicate that two-stage ML loses only minimal efficiency under the condition where single-stage ML performs best. Monte-Carlo results also indicate that the commonly used model selection criterion BIC is more robust to distribution violations for the saturated model than that for a structural model at moderate sample sizes. The proposed two-stage ML approach is also extremely flexible in modeling different components with different models. Potential new developments in the mixture modeling literature can be easily adapted to study issues with normal mixture SEM.     

USING INSTRUMENTAL VARIABLE TESTS TO EVALUATE MODEL SPECIFICATION IN LATENT VARIABLE STRUCTURAL EQUATION MODELS

Structural equation modeling (SEM) with latent variables is a powerful tool for social and behavioral scientists, combining many of the strengths of psychometrics and econometrics into a single framework. The most common estimator for SEM is the full-information maximum likelihood (ML) estimator, but there is continuing interest in limited information estimators because of their distributional robustness and their greater resistance to structural specification errors. However, the literature discussing model fit for limited information estimators for latent variable models is sparse compared with that for full-information estimators. We address this shortcoming by providing several specification tests based on the 2SLS estimator for latent variable structural equation models developed by Bollen (1996) . We explain how these tests can be used not only to identify a misspecified model but to help diagnose the source of misspecification within a model. We present and discuss results from a Monte Carlo experiment designed to evaluate the finite sample properties of these tests. Our findings suggest that the 2SLS tests successfully identify most misspecified models, even those with modest misspecification, and that they provide researchers with information that can help diagnose the source of misspecification.     

The evaluation and selection of adequate causal models: a compensatory education example

Procedures for ascertaining relative model adequacy in latent variable structural relations models are discussed. Under diverse methods of estimation, this determination may be assessed using the chi square goodness of fit statistic, incremental fit indices for covariance structure models, and latent variable coefficients of determination. An example from evaluation research is taken (cf. Magidson, 1977; Bentler & Woodward, 1978). Numerical sensitivity of parameter estimates under alternative model specifications is demonstrated. Interpretive implications based on these procedures are discussed in terms of parameter sensitivity to alternative model specifications.     

Three Likelihood-Based Methods For Mean and Covariance Structure Analysis With Nonnormal Missing Data

Survey and longitudinal studies in the social and behavioral sciences generally contain missing data. Mean and covariance structure models play an important role in analyzing such data. Two promising methods for dealing with missing data are a direct maximum-likelihood and a two-stage approach based on the unstructured mean and covariance estimates obtained by the EM-algorithm. Typical assumptions under these two methods are ignorable nonresponse and normality of data. However, data sets in social and behavioral sciences are seldom normal, and experience with these procedures indicates that normal theory based methods for nonnormal data very often lead to incorrect model evaluations. By dropping the normal distribution assumption, we develop more accurate procedures for model inference. Based on the theory of generalized estimating equations, a way to obtain consistent standard errors of the two-stage estimates is given. The asymptotic efficiencies of different estimators are compared under various assumptions. We also propose a minimum chi-square approach and show that the estimator obtained by this approach is asymptotically at least as efficient as the two likelihood-based estimators for either normal or nonnormal data. The major contribution of this paper is that for each estimator, we give a test statistic whose asymptotic distribution is chi-square as long as the underlying sampling distribution enjoys finite fourth-order moments. We also give a characterization for each of the two likelihood ratio test statistics when the underlying distribution is nonnormal. Modifications to the likelihood ratio statistics are also given. Our working assumption is that the missing data mechanism is missing completely at random. Examples and Monte Carlo studies indicate that, for commonly encountered nonnormal distributions, the procedures developed in this paper are quite reliable even for samples with missing data that are missing at random.     

NEW SPECIFICATIONS FOR EXPONENTIAL RANDOM GRAPH MODELS

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.     

A Methodological Comparison of Age‐Period‐Cohort Models: The Intrinsic Estimator and Conventional Generalized Linear Models

Age‐period‐cohort (APC) accounting models have long been objects of attention in statistical studies of human populations. It is well known that the identification problem created by the linear dependency of age, period, and cohort (Period = Age + Cohort or P = A + C) presents a major methodological challenge to APC analysis, a problem that has been widely addressed in demography, epidemiology, and statistics. This paper compares parameter estimates and model fit statistics produced by two solutions to the identification problem in age‐period‐cohort models—namely, the conventional demographic approach of constrained generalized linear models (Fienberg and Mason 1978, 1985; Mason and Smith 1985) and the intrinsic estimator method recently developed by Fu (2000; Knight and Fu 2000; Fu, Hall, and Rohan 2004). We report empirical analyses of applications of these two methods to population data on U.S. female mortality rates. Comparisons of parameter estimates suggest that both constrained generalized linear models and the intrinsic estimator method can yield similar estimates of age, period, and cohort effects, but estimates obtained by the intrinsic estimator are more direct and do not require prior information to select appropriate model identifying constraints. We also describe three statistical properties of the estimators: (1) finite‐time‐period bias, (2) relative statistical efficiency, and (3) consistency as the number of periods of observed data increases. These empirical analyses and theoretical results suggest that the intrinsic estimator may well provide a useful alternative to conventional methods for the APC analysis of demographic rates .     

Editorial note: referees

Abstract

This study investigates three methods to handle dependency among effect size estimates in meta-analysis arising from studies reporting multiple outcome measures taken on the same sample. The three-level approach is compared with the method of robust variance estimation, and with averaging effects within studies. A simulation study is performed, and the fixed and random effect estimates of the three methods are compared with each other. Both the robust variance estimation and three-level approach result in unbiased estimates of the fixed effects, corresponding standard errors and variances. Averaging effect sizes results in overestimated standard errors when the effect sizes within studies are truly independent. Although the robust variance and three-level approach are more complicated to use, they have the advantage that they do not require an estimate of the correlation between outcomes, and they still result in unbiased parameter estimates.     

THE EFFECT OF MINIMUM WAGES ON EMPLOYMENT: A FACTOR MODEL APPROACH

This paper uses factor model methods to resolve issues in the minimum wage‐employment debate. Factor model methods provide a more flexible way of addressing concerns related to unobserved heterogeneity that are robust to critiques from either side of the debate. The factor model estimators produce minimum wage‐employment elasticity estimates that are much smaller than the traditional ordinary least squares (OLS) results and are not statistically different from zero. These results hold for many specifications and datasets from the minimum wage‐employment literature. A simulation shows that unobserved common factors can explain the different estimates seen across methodologies in the literature. (JEL C23, J21, K31)     

Transformational leadership and follower proactivity in a volunteer workforce

This study explored how transformational leaders can enhance volunteers’ proactive behavior in an all‐volunteer nonprofit organization. Based on Parker, Bindl, and Strauss’s model of motivation, it was hypothesized that role breadth self‐efficacy, work values (self‐direction/stimulation and universalism/benevolence), and positive affect would mediate the transformational leadership—proactive behavior relationship. Data came from 141 volunteers in Brazilian chapters of an international not‐for‐profit organization. The model was tested using structural equation modeling, with mediation hypotheses tested by estimating the indirect effects using bias‐corrected intervals. Comparative fit index (.97) and standardized root mean square residual (.05) fit statistics indicate the model is plausible. These findings contribute to the understanding of the role that leaders play in increasing followers’ proactive behavior in volunteer organizations.     

SIMULTANEOUS EQUATION ECONOMETRICS: THE MISSING EXAMPLE

For introductory presentation of issues involving simultaneous equation systems, a natural vehicle consists of supply and demand relationships for a single good. One would expect to find in econometrics textbooks a supply-demand example featuring actual data in which structural estimation methods yield more satisfactory results than does ordinary least squares. But a search of 26 existing textbooks finds no example with actual data in which all crucial parameter estimates are of the proper sign and are statistically significant. The present article accordingly develops a simple but satisfying example, for broiler chickens, based on U.S. annual data from 1960 to 1999. (JEL C30 )     

Attitudes toward Immigrants,Immigration, and Multiculturalism in New Zealand: A Social Psychological Analysis1

The research examines attitudes toward immigrants and immigration policy based on a random sample of 2,020 New Zealand households. The analyses revealed that New Zealanders have positive attitudes toward immigrants and endorse multiculturalism to a greater extent than Australians and EU citizens. In addition, structural equation modeling produced an excellent fit of the data to a social psychological model commencing with multicultural ideology and intercultural contact as exogenous variables, leading, in turn, to diminished perceptions of threat, more positive attitudes toward immigrants, and, finally, support for New Zealand's policies on the number and sources of migrants.     

A Liang–Zeger Method for Modeling Dyadic Interdependence in the Analysis of Social Networks

This article introduces a modified Liang–Zeger method for the estimation of the variance–covariance matrix of parameter estimates for models of social network data that include variables to characterize dyadic nonindependence. While the pseudolikelihood method has been used recently to estimate parameters for such models, the issue of estimating their standard errors, or the variance–covariance matrix more generally, has been neglected. This article addresses the issue by proposing a method for such estimation and also presents an illustrative application of the method to empirical social network data.     

Multilevel mediation modeling in group-based intervention studies

This article proposes and evaluates a method to test for mediation in multilevel data sets formed when an intervention administered to intact groups is designed to produce change in individual mediator and outcome variables. Simulated data of this form were used to compare ordinary least squares (OLS) and two multilevel estimators of the mediated effect. OLS and multilevel standard error approximations were also evaluated and recommendations given for optimal estimator choice. These methods were applied to data from an existing substance use intervention to show the impact multilevel mediation modeling can have on the conclusions drawn from real-world evaluation studies.     

Addressing self-selection effects in evaluations of mutual help groups and professional mental health services: An introduction to two-stage sample selection models

Because random assignment to conditions is often neither possible nor desirable in longitudinal evaluations of mutual help organizations, the influence of self-selection effects must be assessed in order to accurately interpret outcome data. One approach to adjusting for self-selection effects is to control for covariates that predict outcome using statistical procedures such as analysis of covariance (ANCOVA), partial correlations, and hierarchical regression. This approach has considerable power, but is less useful when an evaluator is interested in directly modeling the process of entry into a program and incorporating information on the factors affecting self-selection into estimation of program effects. Two-stage sample selection models are designed to address such situations. These models rely on regression procedures in which program participation is modeled in an initial equation, which yields a sample selection correction factor. The correction factor is included with participation in a second equation that predicts outcome. This two-stage procedure allows the evaluator to interpret the observed effects of a professional service or a self-help group in the context of the magnitude and direction of selection effects. We compare and contrast the covariate control and sample selection models in a longitudinal study of the effects of participation in Alcoholics Anonymous on drinking behavior.     

Effects of missing data in social networks

We perform sensitivity analyses to assess the impact of missing data on the structural properties of social networks. The social network is conceived of as being generated by a bipartite graph, in which actors are linked together via multiple interaction contexts or affiliations. We discuss three principal missing data mechanisms: network boundary specification (non-inclusion of actors or affiliations), survey non-response, and censoring by vertex degree (fixed choice design), examining their impact on the scientific collaboration network from the Los Alamos E-print Archive as well as random bipartite graphs. The simulation results show that network boundary specification and fixed choice designs can dramatically alter estimates of network-level statistics. The observed clustering and assortativity coefficients are overestimated via omission of affiliations or fixed choice thereof, and underestimated via actor non-response, which results in inflated measurement error. We also find that social networks with multiple interaction contexts may have certain interesting properties due to the presence of overlapping cliques. In particular, assortativity by degree does not necessarily improve network robustness to random omission of nodes as predicted by current theory.     

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.     

Linked Ego Networks: Improving estimate reliability and validity with respondent-driven sampling

Respondent-driven sampling (RDS) is currently widely used for the study of HIV/AIDS-related high risk populations. However, recent studies have shown that traditional RDS methods are likely to generate large variances and may be severely biased since the assumptions behind RDS are seldom fully met in real life. To improve estimation in RDS studies, we propose a new method to generate estimates with ego network data, which is collected by asking respondents about the composition of their personal networks, such as “what proportion of your friends are married?”. By simulations on an extracted real-world social network of gay men as well as on artificial networks with varying structural properties, we show that the precision of estimates for population characteristics is greatly improved. The proposed estimator shows superior advantages over traditional RDS estimators, and most importantly, the method exhibits strong robustness to the recruitment preference of respondents and degree reporting error, which commonly happen in RDS practice and may generate large estimate biases and errors for traditional RDS estimators. The positive results henceforth encourage researchers to collect ego network data for variables of interests by RDS, for both hard-to-access populations and general populations when random sampling is not applicable.     

Assessing the Effect of Model Misspecifications on Parameter Estimates in Structural Equation Models

Model misspecifications may have a systematic effect on parameters, causing biases in their estimates. In the application of structural equation models, every interesting model is fallible. When simultaneously evaluating a model, it is of interest to study whether all parameters are affected by a misspecification. This paper provides three procedures for evaluating such an effect: (1) analyzing the path, (2) using a functional relationship, and (3) using a significance test. Analyzing the path is illustrated through a confirmatory factor model. This method is ad hoc but intuitive. A more rigorous approach is built upon the concept of orthogonality of two sets of parameters. When parameter a is orthogonal to parameter b, omitting parameter b will not affect the estimation of parameter a. The functional relationship of two sets of parameters is used to check their orthogonality. The distribution of the difference between estimates based on different models is obtained, which provides a Hausman–like way to check significant parameter differences that are due to biases. Examples illustrate that these procedures can provide valuable information on identifying parameter estimates that are systematically affected by a model misspecification.     

Negotiating Risk and Promoting Empowerment through Rap Music: Development of a Measure to Capture Risk and Empowerment Pathways to Change

The purpose of this study was to report on the development and construction of the Individual and Community Empowerment (ICE) inventory, a measure seeking to capture the specific pathways by which either risk-enhancing impacts or empowering impacts of rap music manifest. Data were analyzed via structural equation modeling from a convenience sample of 128 high school and college students. Results found that respondents elicited (1) empowering themes that related to them individually and to the broader community and (2) high-risk themes that may promote risky health behaviors. Implications about research and practice relevance of the ICE inventory are discussed.