Kenneth A. Bollen

Testing structural equation models

Introduction - Kenneth A Bollen and J Scott Long Multifaceted Conceptions of Fit in Structural Equation Models - J S Tanaka Monte Carlo Evaluations of Goodness-of-Fit Indices for Structural Equation Models - David W Gerbing and James C Anderson Some Specification Tests for the Linear Regression Model - J Scott Long and Pravin K Trivedi Bootstrapping Goodness-of-Fit Measures in Structural Equation Models - Kenneth A Bollen and Robert A Stine Alternative Ways of Assessing Model Fit - Michael W Browne and Robert Cudeck Bayesian Model Selection in Structural Equation Models - Adrian E Raftery Power Evaluations in Structural Equation Models - Willem E Saris and Albert Satorra Goodness-of-Fit with Categorical and Other Nonnormal Variables - Bengt O Muthen Some New Covariance Structure Model Improvement Statistics - P M Bentler and Chih-Ping Chou Nonpositive Definite Matrices in Structural Modeling - Werner Wothke Testing Structural Equation Models - Karl G Joreskog

A New Incremental Fit Index for General Structural Equation Models

Assessing overall model fit is an important problem in general structural equation models. One of the most widely used fit measures is Bentler and Bonetts (1980) normed index. This article has three purposes: (1) to propose a new incremental fit measure that provides an adjustment to the normed index for sample size and degrees of freedom, (2) to explain the relation between this new fit measure and the other ones, and (3) to illustrate its properties with an empirical example and a Monte Carlo simulation. The simulation suggests that the mean of the sampling distribution of the new fit measure stays at about one for different sample sizes whereas that for the normed fit index increases with N. In addition, the standard deviation of the new measure is relatively low compared to some other measures (e.g., Tucker and Lewiss (1973) and Bentler and Bonetts (1980) nonnormed index). The empirical example suggests that the new fit measure is relatively stable for the same model in different samples. In sum, it appears that the new incremental measure is a useful complement to the existing fit measures.

Bootstrapping Goodness-of-Fit Measures in Structural Equation Models

Assessing overall fit is a topic of keen interest to structural equation modelers, yet measuring goodness of fit has been hampered by several factors. First, the assumptions that underlie the chi-square tests of model fit often are violated. Second, many fit measures (e.g., Bentler and Bonetts [1980] normed fit index) have unknown statistical distributions so that hypothesis testing, confidence intervals, or comparisons of significant differences in these fit indices are not possible. Finally, modelers have little knowledge about the distribution and behavior of the fit measures for misspecified models or for nonnested models. Given this situation, bootstrapping techniques would appear to be an ideal means to tackle these problems. Indeed, Bentlers (1989) EQS 3.0 and Jöreskog and Sörboms (forthcoming) LISREL 8 have bootstrap resampling options to bootstrap fit indices. In this article the authors (a) demonstrate that the usual bootstrapping methods will fail when applied to the original data, (b) explain why this occurs, and, (c) propose a modified bootstrap method for the chi-square test statistic for model fit. They include simulated and empirical examples to illustrate their results.

DIRECT AND INDIRECT EFFECTS: CLASSICAL AND BOOTSTRAP ESTIMATES OF VARIABILITY

The decomposition of effects in structural equation models has been of considerable interest to social scientists. Finite-sample or asymptotic results for the sampling distribution of estimators of direct effects are widely available. Statistical inferences about indirect effects have relied exclusively on asymptotic methods which assume that the limiting distribution of the estimator is normal, with a standard error derived from the delta method. We examine bootstrap procedures as another way to generate standard errors and confidence intervals and to estimate the sampling distributions of estimators of direct and indirect effects. We illustrate the classical and the bootstrap methods with three empirical examples. We find that in a moderately large sample, the bootstrap distribution of an estimator is close to that assumed with the

Overall Fit in Covariance Structure Models: Two Types of Sample Size Effects

A controversial area in covariance structure models is the assessment of overall model fit. Researchers have expressed concern over the influence of sample size on measures of fit. Many contradictory claims have been made regarding which fit statistics are affected by N. Part of the confusion is due to there being two types of sample size effects that are confounded. The first is whether N directly enters the calculation of a fit measure. The second is whether the means of the sampling distributions of a fit index are associated with sample size. I explain these types of sample size effects and illustrate them with the major structural equation fit indices. In addition, I examine the current debate on sample size influences in light of this distinction. Structural equation models, including confirmatory factor analyses, are becoming increasingly popular in psychology. Key to these procedures is the hypothesis that the population covariance matrix of observed variables is a function of the unknown free parameters of a model. Many measures of overall model fit have been proposed to assess the degree to which this hypothesis holds (e.g., Rentier B Bollen, 1986;Hoelter, 1983; Joreskog & Sorbom, 1986; Tucker & Lewis, 1973).

Issues in the Comparative Measurement of Political Democracy

The empirical study of the causes and consequences of political democracy has been the subject of considerable research. Yet cumulative development of this research is hampered by the controversial aspects and limitations of the existing indices of political democracy. These issues concern the validity of the indicators, the unknown reliability, and the limited sample and temporal coverage of these indices. After a discussion of these issues, a revised index of political democracy that overcomes some of these limitations is presented. The indicators of the revised index are analyzed by means of confirmatory factor analysis and the reliability of the measure is discussed. The index is generally better than existing measures in reliability, sample size, and temporal coverage; but the remaining limitations of the index are reviewed. The moderate to high correlations of this index with other democracy indices support its external validity. Yet, the differences in empirical results possible by using different indices are demonstrated. Finally, there are two appendices. The first provides the technical details of the confirmatory factor analysis. The second appendix lists the values of the political democracy index.

Regression Diagnostics An Expository Treatment of Outliers and Influential Cases

Gauging the robustness of regression estimates is especially important in small-sample analyses. Here, we examine recent developments in the detection and analysis of outliers and influential cases in multivariate studies. Specifically, we review five diagnostic procedures: partial regression plots, the “hat” matrix, studentized residuals, DFITSi, and DFBETASij. The main part of the article presents two empirical applications (drawn from recent cross-national studies) that show (a) how the diagnostic procedures can be incorporated into the research process, and (b) what we can learn from them. These applications serve to underscore the point that the diagnostics cannot be employed mechanically. Instead, once a case is diagnosed as influential, remedial action requires a firm substantive grounding. Although case deletion may be warranted in some circumstances, it is an extreme remedy of last resort that should not be routinely followed. The more fruitful approach is to ask why a given case is influential. As our applications indicate, the diagnostics can be helpful in isolating such problems as sample composition, specification error, and errors in measurement.

POLITICAL DEMOCRACY AND THE SIZE DISTRIBUTION OF INCOME

This paper reviews the major theoretical arguments that link political democracy with economic inequality. It then shows that previous empirical analyses of the linkage have produced inconclusive results because they have suffered from specification, measurement, and sample-composition problems. A nonrecursive model is proposed that overcomes many of these limitations. Using 2SLS and a new weighted 2SLS procedure, we find no evidence of direct effects of political democracy on income inequality, or vice versa. However, economic development influences both variables, while world-system position, Protestantism, and British colonial experience affects political democracy. Population age-structure and systematic measurement error induce changes in income inequality. The paper concludes with a discussion both of the flaws in prior arguments linking democracy with inequality, and of the political processes that undermine any such linkage. In democracies the poor have more sovereign power than the men of property; for they are more numerous and the decisions of the majority prevail. Aristotle, The Politics, 1962:237 Where one set of people possesses a great deal and the other nothing, the result is either extreme democracy [mob rule] or unmixed oligarchy or a tyranny due to the excesses of the other two. Aristotle, The Politics, 1962:173 Debate over the linkage between political democracy and social equality has a venerable history. At least since Aristotle, many have held that by reducing inequalities in the distribution of political power, democracy helps to reduce inequalities of wealth and status. Paralleling this view, it also has been argued that extreme inequalities in wealth undermine democratic political structures. At the same time, it has been suggested that democracy and inequality have no meaningful bearing on each other. This paper offers a fresh approach to the democracy-inequality linkage. We begin with a survey of the pertinent substantive arguments and the assumptions on which they are based. This is followed by a review of representative empirical studies which shows that those studies have generated inconclusive results. Further, there is no readily apparent way to resolve these differences, because, as we indicate below, the empirical studies have employed a variety of samples, measurement procedures, and model specifications. Finally, we report our own empirical analysis, which seeks to correct some of the major specification and measurement problems of previous work. This analysis is based on a more comprehensive data set than has been available in the past, and centers on 60 Western and Third World countries.

Political Democracy: Conceptual and Measurement Traps

The movement toward democratic political systems in many nations in the 1980s has renewed interest in measurement of political democracy. This paper calls attention to the problems that surround both the definition and measurement of political democracy. The main conceptual problems are the failure to develop an adequate theoretical definition of this concept, the confounding of the concept with others, and treating democracy as a binary rather than a continuous concept. Four problems of measurement are: invalid indicators, subjective indicators, ordinal or dichotomous measures, and the failure to test reliability or validity. The paper offers several suggestions to improve measurement as well as a warning about the danger of repeating past errors.

Improper solutions in structural equation models: Causes, consequences, and strategies

In this article, the authors examine the most common type of improper solutions: zero or negative error variances. They address the causes of, consequences of, and strategies to handle these issues. Several hypotheses are evaluated using Monte Carlo simulation models, including two structural equation models with several misspecifications of each model. Results suggested several unique findings. First, increasing numbers of omitted paths in the measurement model were associated with decreasing numbers of improper solutions. Second, bias in the parameter estimates was higher in samples with improper solutions than in samples including only proper solutions. Third, investigations of the consequences of using constrained estimates in the presence of improper solutions indicated that inequality constraints helped some samples achieve convergence. Finally, the use of confidence intervals as well as four other proposed tests yielded similar results when testing whether the error variance was greater than or equal to zero.