Structural Equation Modeling Research Paper

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In 1980, Peter Bentler (1980, p. 420) stated that structural equation modeling held ‘the greatest promise for furthering psychological science.’ Since then, there have been many important theoretical and practical advances in the field. So much so, in fact, that Muthen (2001) announced a ‘second generation’ of structural equation modeling. The purpose of this research paper is to provide a description of the ‘first generation’ of structural equation modeling and to outline briefly the most recent developments that constitute the ‘second generation’ of structural equation modeling. This research paper will not devote space to the comparison of specific software packages. For the most part, existing software packages differ in terms of the user interface. The competitive market in structural equation modeling software means that new developments available in one software package will soon be available in others.

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1. The First Generation: Definition And Brief History Of Structural Equation Modeling

Structural equation modeling can be defined as a class of methodologies that seeks to represent hypotheses about the means, variances, and covariances of observed data in terms of a smaller number of ‘structural’ parameters defined by a hypothesized underlying conceptual or theoretical model. Historically, structural equation modeling derived from the hybrid of two separate statistical traditions. The first tradition is factor analysis developed in the disciplines of psychology and psychometrics. The second tradition is simultaneous equation modeling developed mainly in econometrics, but having an early history in the field of genetics and introduced to the field of sociology under the name path analysis.

The subject of this research paper concerns the combination of the latent variable/factor analytic approach with simultaneous equation modeling. The combination of these methodologies into a coherent analytic framework was based on the work of Joreskog (1973), Keesling (1972) and Wiley (1973).




2. The General Model

The general structural equation model as outlined by Joreskog (1973) consists of two parts: (a) the structural part linking latent variables to each other via systems of simultaneous equations, and (b) the measurement part which links latent variables to observed variables via a restricted (confirmatory) factor model. The structural part of the model can be written as

Structural Equation Modeling Research Paper Formula 1

where η is a vector of endogenous (criterion) latent variables, ξ is a vector of exogenous (predictor) latent variables, B is a matrix of regression coefficients relating the latent endogenous variables to each other, Γ is a matrix of regression coefficients relating endogenous variables to exogenous variables, and ζ is a vector of disturbance terms.

The latent variables are linked to observable variables via measurement equations for the endogenous variables and exogenous variables. These equations are defined as

Structural Equation Modeling Research Paper Formula 2

and

Structural Equation Modeling Research Paper Formula 3

where Λy and Λx are matrices of factor loadings, respectively, and ε and δ are vectors of uniqueness, respectively. In addition, the general model specifies variances and covariances for ξ, ζ, ε, and δ, denoted Φ, Ψ, Θε, and Θδ, respectively.

Structural equation modeling is a methodology designed primarily to test substantive theories. As such, a theory might be sufficiently developed to suggest that certain constructs do not affect other constructs, that certain variables do not load on certain factors, and that certain disturbances and measurement errors do not covary. This implies that some of the elements of B, Γ, Λy, Λx, Φ, Ψ, Θε, and Θδ are fixed to zero by hypothesis. The remaining parameters are free to be estimated. The pattern of fixed and free parameters implies a specific structure for the covariance matrix of the observed variables. In this way, structural equation modeling can be seen as a special case of a more general covariance structure model defined as Σ = Σ(Ω), where Σ is the population covariance matrix and Σ(Ω) is a matrix valued function of the parameter vector Ω containing all of the parameters of the model.

3. Identification And Estimation Of Structural Equation Models

A prerequisite for the estimation of the parameters of the model is to establish whether the parameters are identified. We begin with a definition of identification by considering the problem from the covariance structure modeling perspective. The population covariance matrix Σ contains population variances and covariances that are assumed to follow a model characterized by a set of parameters contained in Ω. We know that the variances and covariances in Σ can be estimated by their sample counterparts in the sample covariance matrix S using straightforward equations for the calculation of sample variances and covariances. Thus, the parameters in Σ are identified. What needs to be established prior to estimation is whether the unknown parameters in Ω are identified from the elements of Σ.

We say that the elements in Ω are identified if they can be expressed uniquely in terms of the elements of the covariance matrix Σ. If all elements in Ω are identified then the model is identified. A variety of necessary and sufficient conditions for establishing the identification of the model exist and are described in Kaplan (2000).

Major advances in structural equation modeling have, arguably, been made in the area of parameter estimation. The problem of parameter estimation concerns obtaining estimates of the free parameters of the model, subject to the constraints imposed by the fixed parameters of the model. In other words, the goal is to apply a model to the sample covariance matrix S, which is an estimate of the population covariance matrix Σ that is assumed to follow a structural equation model. Estimates are obtained such that the discrepancy between the estimated covariance matrix Σ = Σ(Ω) and the sample covariance matrix S, is as small as possible. Therefore, parameter estimation requires choosing among a variety of fitting functions. Fitting functions have the properties that (a) the value of the function is greater than or equal to zero, (b) the value of the function equals zero if, and only if, the model fits the data perfectly, and (c) the function is continuous. We first consider estimators that assume multivariate normality of the data. We will discuss estimators for non-normal data in Sect. 5.

3.1 Normal Theory-Based Estimation

The most common estimation method that rests on the assumption of multivariate normality is the method of maximum likelihood. If the assumption of multivariate normality of the observed data holds, the estimates obtained from maximum likelihood estimation possess certain desirable properties. Specifically, maximum likelihood estimation will yield unbiased and efficient estimates as well as a large sample goodness-of-fit test.

Another fitting function that enjoys the same properties under the same assumption is the generalized least squares (GLS) estimator developed for the structural equation modeling context by Joreskog and Goldberger (1972).

4. Testing

Once the parameters of the model have been estimated, the model can be tested for global fit to the data. In addition, hypotheses concerning individual parameters can also be evaluated. We consider model testing in this section.

4.1 Testing Global Model Fit

A feature of maximum likelihood estimation is that one can explicitly test the hypothesis that the model fits data. The null hypothesis states that the specified model holds in the population. The corresponding alternative hypothesis is that the population covariance matrix is any symmetric (and positive definite) matrix. Under maximum likelihood estimation, the statistic for testing the null hypothesis that the model fits in the population is referred to as the likelihood ratio (LR) test. The large sample distribution of the likelihood ratio test is chi-squared with degrees of freedom given by the difference in the number of nonredundant elements in the population covariance matrix and the number of free parameters in the model.

4.2 Testing Hypotheses Regarding Individual Parameters

In addition to a global test of whether the model holds exactly in the population, one can also test hypotheses regarding the individual fixed and freed parameters in the model. Specifically, we can consider three alternative ways to evaluate the fixed and freed elements of the model vis-a-vis overall fit.

The first method rests on the difference between the likelihood ratio chi-squared statistics comparing a given model against a less restrictive model. Recall that the initial hypothesis, say H01, is tested against the alternative hypothesis Ha that Σ is a symmetric positive definite matrix. Consider a second hypothesis, say H02 that differs from H01 in that a single parameter that was restricted to zero is now relaxed. Note that the alternative hypothesis is the same in both cases. Therefore, the change in the chi-squared value between the two models can be used to test the improvement in fit due to the relaxation of the restriction. The distribution of the difference between the two chi-squared tests is distributed as chi-squared with degrees of freedom equaling the difference in degrees-of- freedom between the model under H01 and the less restrictive model under H02. In the case of a single restriction described here, the chi-squared difference test is evaluated with one degree of freedom.

There are other ways of assessing the change in the model without estimating a second nested model. The Lagrange multiplier (LM) test can be used to assess whether freeing a fixed parameter would result in a significant improvement in the overall fit of the model. The LM test is asymptotically distributed as chisquared with degrees of freedom equaling the difference between the degrees of freedom of the more restrictive model and the less restrictive model. Again, if one restriction is being evaluated, then the LM test will be evaluated with one degree of freedom. The LM test is also referred to as the modification index.

Finally, we can consider evaluating whether fixing a free parameter would result in a significant decrement in the fit of the model. This is referred to as the Wald test.

The Wald test is asymptotically distributed as chi- squared with degrees of freedom equaling the number imposed restrictions. When interest is in evaluating one restriction the Wald test is distributed as chi- squared with one degree of freedom. The square root of the Wald statistic in this case is a z statistic. The reason why the LM and Wald tests can be used as alternative approaches for testing model parameters is that they are asymptotically equivalent to each other and to the chi-squared difference test.

4.3 Alternative Methods Of Model Fit

Since the early 1980s, attention has focused on the development of alternative indices that provide relatively different perspectives on the fit of structural equation models. The development of these indices has been motivated, in part, by the known sensitivity of the likelihood ratio chi-squared statistic to large sample sizes. Other classes of indices have been motivated by a need to rethink the notion of testing exact fit in the population—an idea that is deemed by some to be unrealistic in most practical situations. Finally, another class of alternative indices has been developed that focuses on the cross-validation adequacy of the model.

In this section we will divide our discussion of alternative fit indices into three categories: (a) measures based on comparative fit to a baseline model, including those that add a penalty function for model complexity, (b) measures based on population errors of approximation, and (c) cross-validation measures.

4.3.1 Measures Based On Comparative fit To A Baseline Model. Arguably, the most active work in the area of alternative fit indices has been the development of what can be broadly referred to as measures of comparative fit. The basic idea behind these indices is that the fit of the model is compared to the fit of some baseline model that usually specifies complete independence among the observed variables. The fit of the baseline model will usually be fairly large and thus the issue is whether one’s model of interest is an improvement relative to the baseline model. The index is typically scaled to lie between zero and one, with one representing perfect fit relative to this baseline model. The sheer number of comparative fit indices precludes a detailed discussion of each one. We will consider a subset of indices here that serve to illustrate the basic ideas.

The quintessential example of a comparative fit index is the normed-fit index (NFI) proposed by Bentler and Bonett (1980). This index can be written as

Structural Equation Modeling Research Paper Formula 4

where χb+е is the chi-squared for the model of complete independence (the so-called baseline model) and χt2 is the chi-squared for the target model of interest. This index, which ranges from zero to one, provides a measure of the extent to which the target model is an improvement over the baseline model.

The NFI assumes a true null hypothesis and therefore a central chi-squared distribution of the test statistic. However, an argument could be made that the null hypothesis is never exactly true and that the distribution of the test statistic can be better approximated by a non-central chi-squared with noncentrality parameter λ. An estimate of the noncentrality parameter can be obtained as the difference between the statistic and its associated degrees of freedom. Thus, for models that are not extremely misspecified, an index developed by McDonald and Marsh (1990) and referred to as the relative noncentrality index (RNI) can be defined as

Structural Equation Modeling Research Paper Formula 5

The RNI can lie outside the 0–1 range. To remedy this, Bentler (1990) adjusted the RNI so that it would lie in the range of 0–1. This adjusted version is referred to as the comparative fit index (CFI).

Finally, there are classes of comparative fit indices that adjust existing fit indexes for the number of parameters that are estimated. These are so called parsimony-based comparative fit indices. The rationale behind these indices is that a model can be made to fit the data by simply estimating more and more parameters. Indeed, a model that is just identified fits the data perfectly. Therefore, an appeal to parsimony would require that these indices be adjusted for the number of parameters that are estimated.

It is important to note that use of comparative indices has not been without controversy. In particular, Sobel and Borhnstedt (1985) argued early on that these indices are designed to compare one’s hypothesized model against a scientifically questionable baseline hypothesis that the observed variables are completely uncorrelated with each other. Yet, as Sobel and Borhnstedt (1985) argue, one would never seriously entertain such a hypothesis, and that perhaps these indices should be compared with a different baseline hypothesis. Unfortunately, the conventional practice of structural equation modeling as represented in scholarly journals suggests that these indices have always been compared with the baseline model of complete independence.

4.3.2 Measures Based On Errors Of Approximation. It was noted earlier that the likelihood ratio chi-squared test assesses an exact null hypothesis that the model fits perfectly in the population. In reality, it is fairly unlikely that the model will fit the data perfectly. Not only is it unreasonable to expect our models to hold perfectly, but trivial misspecifications can have detrimental effects on the likelihood ratio test when applied to large samples. Therefore, a more sensible approach is to assess whether the model fits approximately well in the population. The difficulty arises when trying to quantify what is meant by ‘approximately.’

To motivate this work it is useful to differentiate among different types of discrepancies. First, there is the discrepancy due to approximation, which measures the lack of fit of the model to the population covariance matrix. Second, there is the discrepancy due to estimation, which measures the difference between the model fit to the sample covariance matrix and the model fit to the population covariance matrix. Finally, there is the discrepancy due to overall error that measures the difference between the elements of the population covariance matrix and the model fit to the sample covariance matrix.

Measures of approximate fit are concerned with the discrepancy due to approximation. Based on the work of Steiger and Lind (1980) it is possible to assess approximate fit of a model in the population. The method of Steiger and Lind (1980) utilizes the root mean square error of approximation (RMSEA) for measuring approximate fit. In utilizing the RMSEA for assessing approximate fit, a formal hypothesis testing framework is employed. Typically an RMSEA value ≤ 0.05 is considered a ‘close fit.’ This value serves to define a formal null hypothesis of close fit. In addition, a 90% confidence interval around the RMSEA can be formed, enabling an assessment of the precision of the estimate (Steiger and Lind 1980). Practical guidelines recommended suggest that values of the RMSEA between 0.05 and 0.08 are indicative of fair fit, while values between 0.08 and 0.10 are indicative of mediocre fit.

4.3.3 Measures That Assess Cross-Validation Adequacy. Another important consideration in evaluating a structural model is whether the model is capable of cross-validating well in a future sample of the same size, from the same population, and sampled in the same fashion. In some cases, an investigator may have a sufficiently large sample to allow it to be randomly split in half with the model estimated and modified on the calibration sample and then cross-validated on the validation sample. When this is possible, then the final fitted model from the calibration sample is applied to the validation sample covariance matrix with parameter estimates fixed to the estimated values obtained from the calibration sample.

In many instances, investigators may not have samples large enough to allow separation into calibration and validation samples. However, the cross-validation adequacy of the model remains a desirable piece of information in model evaluation. Two comparable methods for assessing cross-validation adequacy are the Akaike information criterion (AIC) and the expected cross-validation index (ECVI). In both cases, it is assumed that the investigator is in a position to choose among a set of competing models. The model with the lowest AIC or ECVI value will be the model that will have the best cross-validation capability.

5. Statistical Assumptions Underlying Structural Equation Modeling

As with all statistical methodologies, structural equation modeling requires that certain underlying assumptions be satisfied in order to ensure accurate inferences. The major assumptions associated with structural equation modeling include: multivariate normality, no systematic missing data, sufficiently large sample size, and correct model specification.

5.1 Sampling Assumptions

For the purposes of proper estimation and inference, a very important question concerns the sampling mechanism. In the absence of explicit information to the contrary, estimation methods such as maximum likelihood assume that data are generated according to simple random sampling. Perhaps more often than not, however, structural equation models are applied to data that have been obtained through something other than simple random sampling. For example, when data are obtained through multistage sampling, the usual assumption of independent observations is violated resulting in predictable biases. Structural equation modeling expanded to consider multistage sampling will be discussed in Sect. 6.

5.2 Non-Normality

A basic assumption underlying the standard use of structural equation modeling is that observations are drawn from a continuous and multivariate normal population. This assumption is particularly important for maximum likelihood estimation because the maximum likelihood estimator is derived directly from the expression for the multivariate normal distribution. If the data follow a continuous and multivariate normal distribution, then maximum likelihood attains optimal asymptotic properties, viz., that the estimates are normal, unbiased, and efficient.

The effects of non-normality on estimates, standard errors, and tests of model fit are well known. The extant literature suggests that non-normality does not affect parameter estimates. In contrast, standard errors appear to be underestimated relative to the empirical standard deviation of the estimates. With regard to goodness-of-fit, research indicates that non-normality can lead to substantial overestimation of likelihood ratio chi-squared statistic, and this overestimation appears to be related to the number of degrees of freedom of the model.

5.2.1 Estimators For Non-Normal Data. The estimation methods discussed in Sect. 3 assumed multivariate normal data. In the case where the data are not multivariate normal, two general estimators exist that differ depending on whether the observed variables are continuous or categorical.

One of the major breakthroughs in structural equation modeling has been the development of estimation methods that are not based on the assumption of continuous multivariate normal data. Browne (1984) developed an asymptotic distribution free (ADF) estimator based on weighted least-squares theory, in which the weight matrix takes on a special form. Specifically, when the data are not multivariate normal, the weight matrix can be expanded to incorporate information about the skewness and kurtosis of the data. Simulation studies have shown reasonably good performance of the ADF estimation compared with ML and GLS.

A problem with the ADF estimator is the computational difficulties encountered for models of moderate size. Specifically, with p variables there are u 1/2p( p + 1) elements in the sample covariance matrix S. The weight matrix is of order u × u. It can be seen that the size of the weight matrix grows rapidly with the number of variables. So, if a model was estimated with 20 variables, the weight matrix would contain 22,155 distinct elements. Moreover, ADF estimation required that the sample size (for each group if relevant) exceed p + 1/2p( p + 1) to ensure that the weight matrix is non-singular. These constraints have limited the utility of the ADF estimator in applied settings.

Another difficulty with the ADF estimator is that social and behavioral science data are rarely continuous. Rather, we tend to encounter categorical data and often, we find mixtures of scale types within a specific analysis. When this is the case, the assumptions of continuity as well as multivariate normality are violated. Muthen (1984) made a major contribution to the estimation of structural equation models under these more realistic conditions by providing a unified approach for estimating models containing mixtures of measurement scales.

The basic idea of Muthen’s (1984) method is that underlying each of the categorical variables is a latent continuous and normally distributed variable. The statistics for estimating the model are correlations among these latent variables. For example, the Pearson product–moment correlations between observed dichotomous variables are called phi coefficients. Phi coefficients can underestimate the ‘true’ or latent correlations between the variables. Thus, when phi coefficients are used in structural equation models, the estimates may be attenuated. In contrast, the latent correlations between dichotomous variables are referred to as tetrachoric correlations, and their use would result in disattenuated correlation coefficients. Other underlying correlations can also be defined.

The procedure developed by Muthen (1984) handles mixtures of scale types by calculating the appropriate correlations and the appropriate weight matrix corresponding to those correlations. These are entered into a weighted least-squares analysis yielding correct goodness-of-fit tests, estimates, and standard errors. Unfortunately, as with the ADF estimator, the method of Muthen is computationally intensive. However, research has shown how such estimation methods can be developed that are not as computationally intensive.

5.3 Missing Data

Generally, statistical procedures such as structural equation modeling assume that each unit of analysis has complete data. However, for many reasons, units may be missing values on one or more of the variables under investigation. Numerous ad hoc approaches exist for handling missing data, including list-wise and pairwise deletion. However, these approaches assume that the missing data are unrelated to the variables that have missing data, or any other variable in the model, i.e., missing completely at random (MCAR)— a heroic assumption at best. More sophisticated approaches for dealing with missing data derive from the seminal ideas of Little and Rubin (1987).

Utilizing the Rubin and Little framework, a major breakthrough in model-based approaches to missing data in the structural equation modeling context was made simultaneously by Allison (1987) and Muthen et al. (1987). They showed how structural equation modeling could be used when missing data are missingat-random (MAR), an assumption that is more relaxed than missing-completely-at-random.

The approach advocated by Muthen et al. (1987) was compared with standard listwise deletion and pairwise deletion approaches (which assume MCAR) in an extensive simulation study. The general findings were that Muthen et al.’s approach was superior to listwise and pairwise deletion approaches for handling missing data even under conditions where it was not appropriate to ignore the mechanism that generated the missing data.

The major problem associated with Muthen et al.’s approach is that it is restricted to modeling under a relatively small number of distinct missing data patterns. Small numbers for distinct missing data patterns will be rare in social and behavioral science applications. However, it has proved possible to apply MAR-based approaches (Little and Rubin 1987) to modeling missing data within standard structural equation modeling software in a way that mitigates the problems with Muthen et al.’s approach. Specifically, imputation approaches have become available for maximum likelihood estimation of structural model parameters under incomplete data assuming MAR. Simulation studies have demonstrated the effectiveness of these methods in comparison to listwise deletion and pairwise deletion under MCAR and MAR.

5.4 Specification Error

Structural equation modeling assumes that the model is properly specified. Specification error is defined as the omission of relevant variables in any equation of the system of equations defined by the structural equation model. This includes the measurement model equations as well as the structural model equations. As with the problems of non-normality and missing data, the question of concern is the extent to which omitted variables affect inferences.

The mid-1980s saw a proliferation of studies on the problem of specification error in structural equation models. On the basis of work by Kaplan and others (summarized in Kaplan 2000) the general finding is that specification errors in the form of omitted variables can result in substantial parameter estimate bias. In the context of sampling variability, specification errors have been found to be relatively robust to small specification errors. However, tests of free parameters in the model are affected in such a way that specification error in one parameter can propagate to affect the power of the test in another parameter in the model. Sample size also interacts with the size and type of the specification error.

6. Modern Developments Constituting The ‘Second Generation’

Structural equation modeling is, arguably, one of the most popular statistical methodologies available to quantitative social scientists. The popularity of structural equation modeling has led to the creation of a scholarly journal devoted specifically to structural equation modeling as well as the existence of SEMNET, a very popular and active electronic discussion list that focuses on structural equation modeling and related issues. In addition, there are a large number of software programs that allow for sophisticated and highly flexible modeling.

In addition to ‘first-generation’ applications of structural equation modeling, later developments have allowed traditionally different approaches to statistical modeling to be specified as special cases of structural equation modeling. Space constraints make it impossible to highlight all of the modern developments in structural equation modeling. However, the more important modern developments have resulted from specifying the general model in a way that allows a ‘structural modeling’ approach to other types of analytic strategies. The most recent example of this development is the use of structural equation modeling to estimate multilevel data as well as specifying structural equation models to estimate growth parameters from longitudinal data.

6.1 Multilevel Structural Equation Modeling

Attempts have been made to integrate multilevel modeling with structural equation modeling so as to provide a general methodology that can account for issues of measurement error and simultaneity as well as multistage sampling.

Multilevel structural equation modeling assumes that the levels of the within-group endogenous and exogenous variables vary over between-group units. Moreover, it is possible to specify a model which is assumed to hold at the between-group level and that explains between-group variation of the within-group variables. An overview and application of multilevel structural equation modeling can be found in Kaplan (2000).

6.2 Growth Cur E Modeling From The Structural Equation Modeling Perspective

Growth curve modeling has been advocated for many years by numerous researchers for the study of interindividual differences in change. The specification of growth models can be viewed as falling within the class of multilevel linear models (Bryk and Raudenbush 1992), where level 1 represents intraindividual differences in initial status and growth and level 2 models individual initial status and growth parameters as a function of interindividual differences.

Research has also shown how growth curve models could also be incorporated into the structural equation modeling framework utilizing existing structural equation modeling software. The advantages of the structural equation modeling framework to growth curve modeling is its tremendous flexibility in specifying models of substantive interest. Developments in this area have allowed the merging of growth curve models and latent class models for the purpose of isolating unique populations defined by unique patterns of growth.

7. Pros And Cons Of Structural Equation Modeling

Structural equation modeling is, without question, one of the most popular methodologies in the quantitative social sciences. Its popularity can be attributed to the sophistication of the underlying statistical theory, the potential for addressing important substantive questions, and the availability and simplicity of software dedicated to structural equation modeling. However, despite the popularity of the method, it can be argued that ‘first-generation’ use of the method is embedded in a conventional practice that precludes further statistical as well as substantive advances.

The primary problem with the ‘first-generation’ practice of structural equation modeling lies in attempting to attain a ‘well-fitting’ model. That is, in conventional practice, if a model does not fit from the standpoint of one statistical criterion (e.g., the likelihood ratio chi-squared test), then other conceptually contradictory measures are usually reported (e.g., the NNFI). In addition, it is not uncommon to find numerous modifications made to an ill-fitting model to bring it in line with the data, usually supplemented by post hoc justification for how the modification fit into the original theoretical framework.

Perhaps the problem lies in an obsession with null hypothesis testing—certainly an issue that has received considerable attention. Or perhaps the practice of ‘first-generation’ structural equation modeling is embedded in the view that only a well-fitting model is worthy of being interpreted. Regardless, it has been argued by Kaplan (2000) that this conventional practice precludes learning valuable information about the phenomena under study that could other- wise be attained if the focus was on the predictive ability of a model. Such a focus on the predictive ability of the model combined with a change of view toward strict hypothesis testing might lead to further substantive and statistical developments. Opportunities for substantive development and model improvement emerge when the model does not yield accurate or admissible predictions derived from theory. Opportunities for statistical developments emerge when new methods are developed for engaging in prediction studies and evaluating predictive performance.

That is not to say that there are no bright spots in the field of structural equation modeling. Indeed, developments in multilevel structural equation modeling, growth curve modeling, and latent class applications suggest a promising future with respect to statistical and substantive developments. However, these ‘second-generation’ methodologies will have to be combined with a ‘second-generation’ epistemology so as to realize the true potential of structural equation modeling in the array of quantitative social sciences.

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