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Over the last few decades, many robust statistics have been proposed in order to assess the fit of structural equation models. To date, however, no clear recommendations have emerged as to which te...

Classical accounts of maximum likelihood (ML) estimation of structural equation models for continuous outcomes involve normality assumptions: standard errors (SEs) are obtained using the expected information matrix and the goodness of fit of the model is tested using the likelihood ratio (LR) statistic. Satorra and Bentler (1994) introduced SEs and mean adjustments or mean and variance adjustments to the LR statistic (involving also the expected information matrix) that are robust to nonnormality. However, in recent years, SEs obtained using the observed information matrix and alternative test statistics have become available. We investigate what choice of SE and test statistic yields better results using an extensive simulation study. We found that robust SEs computed using the expected information matrix coupled with a mean- and variance-adjusted LR test statistic (i.e., MLMV) is the optimal choice, even with normally distributed data, as it yielded the best combination of accurate SEs and Type I errors.

Structural equation models are typically evaluated on the basis of goodness-of-fit indexes. Despite their popularity, agreeing what value these indexes should attain to confidently decide between the acceptance and rejection of a model has been greatly debated. A recently proposed approach by means of equivalence testing has been recommended as a superior way to evaluate the goodness of fit of models. The approach has also been proposed as providing a necessary vehicle that can be used to advance the inferential nature of structural equation modeling as a confirmatory tool. The purpose of this article is to introduce readers to key ideas in equivalence testing and illustrate its use for conducting model–data fit assessments. Two confirmatory factor analysis models in which a priori specified latent variable models with known structure and tested against data are used as examples. It is advocated that whenever the goodness of fit of a model is to be assessed researchers should always examine the resulting ...

A Monte Carlo simulation study was conducted to evaluate the sensitivities of the likelihood ratio test and five commonly used delta goodness-of-fit (ΔGOF) indices (i.e., ΔGamma, ΔMcDonald’s, ΔCFI, ΔRMSEA, and ΔSRMR) to detect a lack of metric invariance in a bifactor model. Experimental conditions included factor loading differences, location and number of noninvariant items, and sample size. The results indicated all ΔGOF indices held Type I error to a minimum and overall had adequate power for the study. For detecting the violation of metric invariance, only ΔGamma and ΔCFI, in addition to Δχ2, are recommended to use in the bifactor model with values of −.016 to −.023 and −.003 to −.004, respectively. Moreover, in the variance component analysis, the magnitude of the factor loading differences contributed the most variation to all ΔGOF indices, whereas sample size affected Δχ2 the most.

In 1959, Campbell and Fiske introduced the use of multitrait–multimethod (MTMM) matrices in psychology, and for the past 4 decades confirmatory factor analysis (CFA) has commonly been used to analyze MTMM data. However, researchers do not always fit CFA models when MTMM data are available; when CFA modeling is used, multiple models are available that have attendant strengths and weaknesses. In this article, we used a Monte Carlo simulation to investigate the drawbacks of either using CFA models that fail to match the data-generating model or completely ignore the MTMM structure of data when the research goal is to uncover associations between trait constructs and external variables. We then used data from the National Institute of Child Health and Human Development Study of Early Child Care and Youth Development to illustrate the substantive implications of fitting models that partially or completely ignore MTMM data structures. Results from analyses of both simulated and empirical data show noticeable bi...

This article proposes a model-based procedure, intended for personality measures, for exploiting the auxiliary information provided by the certainty with which individuals answer every item (response certainty). This information is used to (a) obtain more accurate estimates of individual trait levels, and (b) provide a more detailed assessment of the consistency with which the individual responds to the test. The basis model consists of 2 submodels: an item response theory submodel for the responses, and a linear-in-the-coefficients submodel that describes the response certainties. The latter is based on the distance-difficulty hypothesis, and is parameterized as a factor-analytic model. Procedures for (a) estimating the structural parameters, (b) assessing model–data fit, (c) estimating the individual parameters, and (d) assessing individual fit are discussed. The proposal was used in an empirical study. Model–data fit was acceptable and estimates were meaningful. Furthermore, the precision of the indivi...

Determining an appropriate sample size for use in latent variable modeling techniques has presented ongoing challenges to researchers. In particular, small sample sizes are known to present concerns over sampling error for the variances and covariances on which model estimation is based, as well as for fit indexes and convergence failures. The literature on the topic has focused on conducting power analyses as well as identifying rules of thumb for deciding an appropriate sample size. Often the advice involves an assumption that sample size requirement is moderated by aspects of the model in question. In this study, an effort was undertaken to extend the findings of Gagne and Hancock (2006) on measurement model quality and solution propriety to a broader set of confirmatory factor analysis models. As well, we examined whether Herzog, Boomsma, and Reinecke's (2007) findings for the Swain correction to the χ2 statistic for large models would generalize to models in our study. Our findings suggest that Gagne...

Latent means methods such as multiple-indicator multiple-cause (MIMIC) and structured means modeling (SMM) allow researchers to determine whether or not a significant difference exists between groups' factor means. Strong invariance is typically recommended when interpreting latent mean differences. The extent of the impact of noninvariant intercepts on conclusions made when implementing both MIMIC and SMM methods was the main purpose of this study. The impact of intercept noninvariance on Type I error rates, power, and two model fit indices when using MIMIC and SMM approaches under various conditions were examined. Type I error and power were adversely affected by intercept noninvariance. Although the fit indices did not detect small misspecifications in the form of noninvariant intercepts, one did perform more optimally.

An interval estimation procedure for proportion of explained observed variance in latent curve analysis is discussed, which can be used as an aid in the process of choosing between linear and nonli...

Proper model specification is an issue for researchers, regardless of the estimation framework being utilized. Typically, indexes are used to compare the fit of one model to the fit of an alternate model. These indexes only provide an indication of relative fit and do not necessarily point toward proper model specification. There is a procedure in the Bayesian framework called posterior predictive checking that is designed theoretically to detect model misspecification for observed data. However, the performance of the posterior predictive check procedure has thus far not been directly examined under different conditions of mixture model misspecification. This article addresses this task and aims to provide additional insight into whether or not posterior predictive checks can detect model misspecification within the context of Bayesian growth mixture modeling. Results indicate that this procedure can only identify mixture model misspecification under very extreme cases of misspecification.

This study investigated the performance of fit indexes in selecting a covariance structure for longitudinal data. Data were simulated to follow a compound symmetry, first-order autoregressive, first-order moving average, or random-coefficients covariance structure. We examined the ability of the likelihood ratio test (LRT), root mean square error of approximation (RMSEA), comparative fit index (CFI), and Tucker–Lewis Index (TLI) to reject misspecified models with varying degrees of misspecification. With a sample size of 20, RMSEA, CFI, and TLI are high in both Type I and Type II error rates, whereas LRT has a high Type II error rate. With a sample size of 100, these indexes generally have satisfactory performance, but CFI and TLI are affected by a confounding effect of their baseline model. Akaike's Information Criterion (AIC) and Bayesian Information Criterion (BIC) have high success rates in identifying the true model when sample size is 100. A comparison with the mixed model approach indicates that se...

Because random assignment is not possible in observational studies, estimates of treatment effects might be biased due to selection on observable and unobservable variables. To strengthen causal inference in longitudinal observational studies of multiple treatments, we present 4 latent growth models for propensity score matched groups, and evaluate their performance with a Monte Carlo simulation study. We found that the 4 models performed similarly with respect to model fit, bias of parameter estimates, Type I error, and power to test the treatment effect. To demonstrate a multigroup latent growth model with dummy treatment indicators, we estimated the effect of students changing schools during elementary school years on their reading and mathematics achievement, using data from the Early Childhood Longitudinal Study Kindergarten Cohort.

This article shows how interfactor correlation is affected by error correlations. Theoretical and practical justifications for error correlations are given, and a new equivalence class of models is presented to explain the relationship between interfactor correlation and error correlations. The class allows simple, parsimonious modeling of error correlations via prespecifying reliabilities. Within the class, the correlation between latent factors can be as high as 1.0, and as low as the correlation between certain component scores. The models are indistinguishable in terms of parameter parsimony, identifiability, and fit statistics, implying that interfactor correlation is not identifiable within the class. The existence of the class is problematic for psychometric measurement, because estimates of interfactor correlation form the foundation of much of the literature.

This simulation study investigated the sensitivity of commonly used cutoff values for global-model-fit indexes, with regard to different degrees of violations of the assumption of uncorrelated errors in confirmatory factor analysis. It is shown that the global-model-fit indexes fell short in identifying weak to strong model misspecifications under both different degrees of correlated error terms, and various simulation conditions. On the basis of an example misspecification search, it is argued that global model testing must be supplemented by this procedure. Implications for the use of structural equation modeling are discussed.

Latent class models are becoming more popular in behavioral research. When models with a large number of latent classes relative to the number of manifest indicators are estimated, researchers must consider the possibility that the model is not identified. It is not enough to determine that the model has positive degrees of freedom. A well-known example is the latent class model with 4 binary indicators and 3 classes. This study shows that nonidentified latent class models can appear to be estimated normally when parameter estimates become fixed to a boundary of the parameter space. The consequences for the posterior probabilities of class membership given observed data are examined, along with the potential to bias predictions of distal outcomes contingent on class membership.

Bayesian approaches to modeling are receiving an increasing amount of attention in the areas of model construction and estimation in factor analysis, structural equation modeling (SEM), and related...

This article relates a still-popular motivation for using parceling to an unrecognized cost. The still-popular motivation is improvement in fit with respect to the item-solution. The cost is uncert...

Linear factor analysis (FA) models can be reliably tested using test statistics based on residual covariances. We show that the same statistics can be used to reliably test the fit of item response theory (IRT) models for ordinal data (under some conditions). Hence, the fit of an FA model and of an IRT model to the same data set can now be compared. When applied to a binary data set, our experience suggests that IRT and FA models yield similar fits. However, when the data are polytomous ordinal, IRT models yield a better fit because they involve a higher number of parameters. But when fit is assessed using the root mean square error of approximation (RMSEA), similar fits are obtained again. We explain why. These test statistics have little power to distinguish between FA and IRT models; they are unable to detect that linear FA is misspecified when applied to ordinal data generated under an IRT model.

Ordinal variables are common in many empirical investigations in the social and behavioral sciences. Researchers often apply the maximum likelihood method to fit structural equation models to ordin ...

When comparing latent variables among groups, it is important to first establish the equivalence or invariance of the measurement model across groups. Confirmatory factor analysis (CFA) is a commonly used methodological approach to examine measurement equivalence/invariance (ME/I). Within the CFA framework, the chi-square goodness-of-fit test and chi-square difference tests are used to evaluate ME/I, and these tests rely on the assumption of independence among groups. Limitations in the study design can hinder the practicality of the independence among groups assumption. This article illustrates, algebraically, the effects of violations independence on the chi-square goodness-of-fit and chi-square difference test statistics in a multigroup CFA.

Reliability can be estimated using structural equation modeling (SEM). Two potential problems with this approach are that estimates may be unstable with small sample sizes and biased with misspecified models. A Monte Carlo study was conducted to investigate the quality of SEM estimates of reliability by themselves and relative to coefficient alpha. The SEM approach showed minimal bias when the model was correctly specified if items were relatively well defined by their underlying factor(s). They tended to demonstrate somewhat greater bias when the model was misspecified, particularly underspecified. Overall, SEM estimates were more stable than anticipated. Researchers are more likely to obtain accurate estimates of reliability using SEM by conducting large-sample studies with well-constructed scales and critically assessing model fit.

Two general frameworks have been proposed for evaluating statistical power of tests of model fit in structural equation modeling (SEM). Under the Satorra–Saris (1985) approach, to evaluate the power of the test of fit of Model A, a Model B, within which A is nested, is specified as the alternative hypothesis and considered as the true model. We then determine the power of the test of fit of A when B is true. Under the MacCallum–Browne–Sugawara (1996) approach, power is evaluated with respect to the test of fit of Model A against an alternative hypothesis specifying a true degree of model misfit. We then determine the power of the test of fit of A when a specified degree of misfit is assumed to exist as the alternative hypothesis. In both approaches the phenomenon of isopower is present, which means that different alternative hypotheses (in the Satorra–Saris approach) or combinations of alternative hypotheses and other factors (in the MacCallum–Browne–Sugawara approach) yield the same level of power. We sh...

Exploratory factor analysis (EFA) is a frequently used multivariate analysis technique in statistics. Jennrich and Sampson (1966) solved a significant EFA factor loading matrix rotation problem by deriving the direct Quartimin rotation. Jennrich was also the first to develop standard errors for rotated solutions, although these have still not made their way into most statistical software programs. This is perhaps because Jennrich's achievements were partly overshadowed by the subsequent development of confirmatory factor analysis (CFA) by Joreskog (1969). The strict requirement of zero cross-loadings in CFA, however, often does not fit the data well and has led to a tendency to rely on extensive model modification to find a well-fitting model. In such cases, searching for a well-fitting measurement model may be better carried out by EFA (Browne, 2001). Furthermore, misspecification of zero loadings usually leads to distorted factors with over-estimated factor correlations and subsequent distorted structur...

In research concerning model invariance across populations, researchers have discussed the limitations of the conventional χ2 difference test (Δχ2 test). There have been some research efforts in using goodness-of-fit indexes (i.e., δgoodness-of-fit indexes) for assessing multisample model invariance, and some specific recommendations have been made (Cheung & Rensvold, 2002). Because δgoodness-of-fit indexes were designed to assess model fit in terms of covariance structure, it is not clear how they will perform when mean structure invariance is the research focus. This study extends the previous work (Cheung & Rensvold, 2002), and evaluates how δgoodness-of-fit indexes perform in mean structure invariance analysis. By using a Monte Carlo simulation experiment, the performance of δgoodness-of-fit indexes in detecting population mean structure difference is evaluated. The findings suggest that, in general, δgoodness-of-fit indexes are so sensitive to model size that they are not generally useful in mean str...

Traditional estimators of fit measures based on the noncentral chi-square distribution (root mean square error of approximation [RMSEA], Steiger's , etc.) tend to overreject acceptable models when the sample size is small. To handle this problem, it is proposed to employ Bartlett's (1950), Yuan's (2005), or Swain's (1975) correction of the maximum likelihood chi-square statistic for the estimation of noncentrality-based fit measures. In a Monte Carlo study, it is shown that Swain's correction especially produces reliable estimates and confidence intervals for different degrees of model misspecification (RMSEA range: 0.000-0.096) and sample sizes (50, 75, 100, 150, 200). In the second part of the article, the study is extended to incremental fit indexes (Tucker-Lewis Index, Comparative Fit Index, etc.). For their small-sample robust estimation, use of Swain's correction is recommended only for the target model, not for the independence model. The Swain-corrected estimators only require a ratio of sample size to estimated parameters of about 2:1 (sometimes even less) and are thus strongly recommended for applied research. R software is provided for convenient use.

The purpose of this study is to investigate the effects of missing data techniques in longitudinal studies under diverse conditions. A Monte Carlo simulation examined the performance of 3 missing data methods in latent growth modeling: listwise deletion (LD), maximum likelihood estimation using the expectation and maximization algorithm with a nonnormality correction (robust ML), and the pairwise asymptotically distribution-free method (pairwise ADF). The effects of 3 independent variables (sample size, missing data mechanism, and distribution shape) were investigated on convergence rate, parameter and standard error estimation, and model fit. The results favored robust ML over LD and pairwise ADF in almost all respects. The exceptions included convergence rates under the most severe nonnormality in the missing not at random (MNAR) condition and recovery of standard error estimates across sample sizes. The results also indicate that nonnormality, small sample size, MNAR, and multicollinearity might advers...

We evaluated the statistical power of single-indicator latent growth curve models to detect individual differences in change (variances of latent slopes) as a function of sample size, number of longitudinal measurement occasions, and growth curve reliability. We recommend the 2 degree-of-freedom generalized test assessing loss of fit when both slope-related random effects, the slope variance and intercept-slope covariance, are fixed to 0. Statistical power to detect individual differences in change is low to moderate unless the residual error variance is low, sample size is large, and there are more than four measurement occasions. The generalized test has greater power than a specific test isolating the hypothesis of zero slope variance, except when the true slope variance is close to 0, and has uniformly superior power to a Wald test based on the estimated slope variance.

This study examined the efficacy of 4 different parceling methods for modeling categorical data with 2, 3, and 4 categories and with normal, moderately nonnormal, and severely nonnormal distributions. The parceling methods investigated were isolated parceling in which items were parceled with other items sharing the same source of variance, and distributed parceling in which items were parceled with items influenced by different factors. These parceling strategies were crossed with strategies in which items were either parceled with similarly distributed or differently distributed items, to create 4 different parceling methods. Overall, parceling together items influenced by different factors and with different distributions resulted in better model fit, but high levels of parameter estimate bias. Across all parceling methods, parameter estimate bias ranged from 20% to over 130%. Parceling strategies were contrasted with use of the WLSMV estimator for categorical, unparceled data. Results based on this es...

Univariate latent growth modeling (LGM) of composites of multiple items (e.g., item means or sums) has been frequently used to analyze the growth of latent constructs. This study evaluated whether LGM of composites yields unbiased parameter estimates, standard errors, chi-square statistics, and adequate fit indexes. Furthermore, LGM was compared with 2 alternatives: LGM with fixed error variances and the curve-of-factors model (McArdle, 1988), which is a multivariate latent growth model. It was found that the 2 univariate models only yield adequate results when the items are essentially tau-equivalent and there is strict factorial invariance. The curve-of-factors model was found to produce adequate results in all conditions, but it usually requires large sample sizes.

Two Monte Carlo studies were conducted to examine the sensitivity of goodness of fit indexes to lack of measurement invariance at 3 commonly tested levels: factor loadings, intercepts, and residual variances. Standardized root mean square residual (SRMR) appears to be more sensitive to lack of invariance in factor loadings than in intercepts or residual variances. Comparative fit index (CFI) and root mean square error of approximation (RMSEA) appear to be equally sensitive to all 3 types of lack of invariance. The most intriguing finding is that changes in fit statistics are affected by the interaction between the pattern of invariance and the proportion of invariant items: when the pattern of lack of invariance is uniform, the relation is nonmonotonic, whereas when the pattern of lack of invariance is mixed, the relation is monotonic. Unequal sample sizes affect changes across all 3 levels of invariance: Changes are bigger when sample sizes are equal rather than when they are unequal. Cutoff points for t...

The goals of this article are twofold: (a) briefly highlight the merits of residual centering for representing interaction and powered terms in standard regression contexts (e.g., Lance, 1988), and (b) extend the residual centering procedure to represent latent variable interactions. The proposed method for representing latent variable interactions has potential advantages over extant procedures. First, the latent variable interaction is derived from the observed covariation pattern among all possible indicators of the interaction. Second, no constraints on particular estimated parameters need to be placed. Third, no recalculations of parameters are required. Fourth, model estimates are stable and interpretable. In our view, the orthogonalizing approach is technically and conceptually straightforward, can be estimated using any structural equation modeling software package, and has direct practical interpretation of parameter estimates. Its behavior in terms of model fit and estimated standard errors is v...

Two-level data sets are frequently encountered in social and behavioral science research. They arise when observations are drawn from a known hierarchical structure, as when individuals are randomly drawn from groups that are randomly drawn from a target population. When the covariance structures in the group level and the individual level are the same, standard structural equation modeling software such as LISREL can be used easily to analyze 2-level data sets. However, if the data set consists of variables that have group-level characteristics, LISREL cannot be used in a straightforward manner to implement the analysis of structural equation models that have different structures at the group and individual level. This article illustrates how 2-level structural equation models with additional group-level variables, and hence different structures at the group and individual levels, can be analyzed by LISREL with appropriate settings. A small-scale simulation study was conducted to examine the performance ...

The usefulness of assessing individual fit in latent growth curve models was examined. The study used simulated data based on an unconditional and a conditional latent growth curve model with a linear component and a small quadratic component and a linear model was fit to the data. Then the overall fit of linear and quadratic models to these data was examined. The quadratic models fit the data well but the linear models did not, even though the quadratic component was very small. Then fit was examined at the individual level. Based on individual log likelihood values, a linear model fit most of the individual trajectories well. The article then demonstrates the use of individual fit indexes in empirical data and offers suggestions for assessing individual fit in applied research.

Effects of incomplete data on fit indexes remain relatively unexplored. We evaluate a wide set of fit indexes (χ2, root mean squared error of appproximation, Normed Fit Index [NFI], Tucker-Lewis Index, comparative fit index, gamma-hat, and McDonald's Centrality Index) varying conditions of sample size (100-1,000 in increments of 50), factor loadings (.4 or. 8), factor covariances (.4 or. 8), type of missing data (missing completely at random or missing at random), and extent of missing data (0-95% on 3 of 9 indicators in increments of 5%) for correct and 2 misspecified (measurement or structural) models. Incremental and absolute fit indexes indicate better fit with higher proportions of missing data. Effects of missing data on the NFI were more varied, indicating poorer model fit as missing data increased for the correct model, and indicating better or poorer fit as an interaction of all the other factors for misspecified models. Recommendations are made for researchers and software developers.

In previous research (Hu & Bentler, 1998, 1999), 2 conclusions were drawn: standardized root mean squared residual (SRMR) was the most sensitive to misspecified factor covariances, and a group of other fit indexes were most sensitive to misspecified factor loadings. Based on these findings, a 2-index strategy-that is, SRMR coupled with another index-was proposed in model fit assessment to detect potential misspecification in both the structural and measurement model parameters. Based on our reasoning and empirical work presented in this article, we conclude that SRMR is not necessarily most sensitive to misspecified factor covariances (structural model misspecification), the group of indexes (TLI, BL89, RNI, CFI, Gamma hat, Mc, or RMSEA) are not necessarily more sensitive to misspecified factor loadings (measurement model misspecification), and the rationale for the 2-index presentation strategy appears to have questionable validity.

This article describes a SAS macro to assess model fit of structural equation models by employing a test of the model-implied vanishing tetrads. Use of this test has been limited in the past, in part due to the lack of software that fully automates the test in a user-friendly way. The current SAS macro provides a straightforward method for researchers to use the vanishing tetrads implied by models to assess the fit of (a) structural equation models containing continuous endogenous variables; (b) structural equation models containing continuous endogenous variables nested for vanishing tetrads; and (c) structural equation models containing dichotomous, ordinal, or censored endogenous variables. Besides providing an alternative assessment of model fit to the usual likelihood-ratio test (LRT), the vanishing tetrads test occasionally provides a statistical assessment of competing models nested for vanishing tetrads but not nested for the LRT. The macro permits formal comparisons between tetrad-nested structur...

There has been relatively little systematic investigation of the effect of the number of variables on measures of model fit in structural equation modeling. There is conflicting evidence as to whether measures of fit tend to improve or decline as more variables are added to the model. We consider 3 different types of specification error: minor factors, 2-factor models, and method errors. Using a formal method based on the noncentrality parameter (NCP), we find that root mean squared error of approximation (RMSEA) tends to improve regardless of the type of specification error and that the comparative fit index (CFI) and Tucker-Lewis Index (TLI), generally, though not always, tend to worsen as the number of variables in the model increases. The formal method that we develop can be used to investigate other measures of fit and other types of misspecification.

This study focused on misspecifications in composing parcels to represent a latent construct. Two measurement design factors, item reliability and intercorrelations among parcels, defined 12 true unidimensional parcel models. Deviations from the true model were examined via a 2-facet measurement model in which items and parcels represented the 2 facets. Unidimensionality was examined using a set of criteria developed for the 2-facet measurement model. Many misspecified parcel models produced admissible factor loadings despite poor overall fit and unacceptable residual covariances. The factor loadings alone may not be sufficient for evaluating latent factor representation. The findings suggest that parcels' unidimensionality should be analyzed before the measurement model to which they belong is entered into a comprehensive structural model. The 2-facet measurement model provides a relevant assessment of unidimensionality of the parcel compositions.

Conventional wisdom in missing data research dictates adding variables to the missing data model when those variables are predictive of (a) missingness and (b) the variables containing missingness. However, it has recently been shown that adding variables that are correlated with variables containing missingness, whether or not they are related to missingness, can substantially improve estimation (bias and efficiency). Including large numbers of these "auxiliary" variables is straightforward for researchers who use multiple imputation. However, what is the researcher to do if 1 of the FIML/SEM procedures is the analysis of choice? This article suggests 2 models for SEM analysis with missing data, and presents simulation results to show that both models provide estimation that is clearly as good as analysis with the EM algorithm, and by extension, multiple imputation. One of these models, the saturated correlates model, also provides good estimates of model fit.

This study investigated the impact of categorization on confirmatory factor analysis (CFA) parameter estimates, standard errors, and 5 ad hoc fit indexes. Models were generated that represented empirical research situations in terms of model size, sample sizes, and loading values. CFA results obtained from analysis of normally distributed, continuous data were compared to results obtained from 5-category Likert-type data with normal distributions. The ordered categorical data were analyzed using the estimators: Weighted Least Squares (WLS; with polychoric correlation [PC] input) and Maximum Likelihood (ML; with Pearson Product-Moment [PPM] input). ML-PPM-based parameter estimates reported moderate levels of negative bias for all conditions, WLS-PC-based standard errors showed high amounts of bias, especially with a small sample size and moderate loading values. With nonnormally distributed, ordered categorical data, ML-PPM-based parameter estimates, standard errors, and factor intercorrelation showed high...

Measurement invariance is usually tested using Multigroup Confirmatory Factor Analysis, which examines the change in the goodness-of-fit index (GFI) when cross-group constraints are imposed on a measurement model. Although many studies have examined the properties of GFI as indicators of overall model fit for single-group data, there have been none to date that examine how GFIs change when between-group constraints are added to a measurement model. The lack of a consensus about what constitutes significant GFI differences places limits on measurement invariance testing. We examine 20 GFIs based on the minimum fit function. A simulation under the two-group situation was used to examine changes in the GFIs (ΔGFIs) when invariance constraints were added. Based on the results, we recommend using Δcomparative fit index, ΔGamma hat, and ΔMcDonald's Noncentrality Index to evaluate measurement invariance. These three ΔGFIs are independent of both model complexity and sample size, and are not correlated with the o...

Two simulation studies were conducted to investigate the effects of the practice of item parceling. In Study 1, unidimensional sets of normally and nonnormally distributed item-level data were categorized into 2-, 3-, and 4-item parcels. Analyses revealed that the use of item parcels resulted in better fitting solutions, as measured by the root mean squared error of approximation (RMSEA), comparative fit index (CFI), and chi-square test, when items had a unidimensional structure. Parceled solutions also resulted in less bias in estimates of structural parameters under these conditions than did solutions based on the individual items. In Study 2 the issue of whether the use of item parceling could mask a known multidimensional factor structure among a set of items was investigated. Results indicated that certain types of item parceling can obfuscate a multidimensional factor structure in such a way that acceptable values of fit indexes are found for a misspecified solution. In addition, parceling under the...

A Monte Carlo simulation examined the performance of 4 missing data methods in structural equation models: full information maximum likelihood (FIML), listwise deletion, pairwise deletion, and similar response pattern imputation. The effects of 3 independent variables were examined (factor loading magnitude, sample size, and missing data rate) on 4 outcome measures: convergence failures, parameter estimate bias, parameter estimate efficiency, and model goodness of fit. Results indicated that FIML estimation was superior across all conditions of the design. Under ignorable missing data conditions (missing completely at random and missing at random), FIML estimates were unbiased and more efficient than the other methods. In addition, FIML yielded the lowest proportion of convergence failures and provided near-optimal Type 1 error rates across both simulations.

This research was designed to investigate how much more suitable moving average (MA) and autoregressive-moving average (ARMA) models are for longitudinal panel data in which measurement errors correlate than AR, quasi-simplex, and 1-factor models. The conclusions include (a) when testing for a stochastic process hypothesized to occur in a longitudinal data set, testing for other processes is necessary, because incorrect models often fit other processes well enough to be deceiving; (b) when measurement error correlations are flagged to be relatively high in panel data, the fit and propriety of an MA or ARMA model should be considered and compared to the fit and propriety of other models; (c) when an MA model is fit to AR data, measurement error correlations may nonetheless be deceptively high, though fortunately MA model fit indexes are almost always lower than those for an AR model; and (d) the assumption that longitudinal panel data always contain measurement error correlations is patently false. In summ...

(2000). Doing the Four-Step, Right-2-3, Wrong-2-3: A Brief Reply to Mulaik and Millsap; Bollen; Bentler; and Herting and Costner. Structural Equation Modeling: A Multidisciplinary Journal: Vol. 7, No. 1, pp. 111-123.

It has been proposed that a clear separation of measurement from structural reasons for model failure can be obtained via a procedure testing 4 nested models: (a) a factor model, (b) a confirmatory factor model, (c) the anticipated structural equation model, and (d) possibly, a more constrained model. Advocates of the 4-step procedure contend that these nested models provide a trustworthy way of determining whether one's model is failing as a result of structural (conceptual) inadequacy, or as a result of measurement misspecification. We argue that measurement and structural issues can not be unambiguously separated by the 4 steps, and that the seeming separation is incomplete at best and illusory at worst. The prime difficulty is that the 4-step procedure is incapable of determining whether the proposed model contains the proper number of factors. As long as the number of factors is in doubt, measurement and structural assessments remain dubious and entwined. The assessment of model fit raises additional...

Two Lagrange multiplier (LM) methods may be used in specification searches for adding parameters to models: one based on univariate LM tests and respecification of the model (LM‐respecified method) and the other based on a partitioning of multivariate LM tests (LM‐incremental method). These methods may result in extraneous parameters being included in models due to either sampling error or the model being misspecified. A 2‐stage specification search may be used to reduce errors due to misspecification. In the 1st stage, parameters are added to models based on LM tests to maximize fit. Second, parameters added in the 1st stage are deleted if they are no longer necessary to maintain model fit. Illustrations are presented to demonstrate that errors due to misspecification occur with the LM‐respecified method and are even more likely with the LM‐incremental approach. These illustrations also show how the deletion stage can help eliminate some of these errors.

A Monte Carlo simulation study was conducted to investigate the effects on structural equation modeling (SEM) fit indexes of sample size, estimation method, and model specification. Based on a balanced experimental design, samples were generated from a prespecified population covariance matrix and fitted to structural equation models with different degrees of model misspecification. Ten SEM fit indexes were studied. Two primary conclusions were suggested: (a) some fit indexes appear to be noncomparable in terms of the information they provide about model fit for misspecified models and (b) estimation method strongly influenced almost all the fit indexes examined, especially for misspecified models. These 2 issues do not seem to have drawn enough attention from SEM practitioners. Future research should study not only different models vis‐a‐vis model complexity, but a wider range of model specification conditions, including correctly specified models and models specified incorrectly to varying degrees.

The effects of levels of aggregation on measures of goodness of fit and higher order parameter estimates obtained from confirmatory factor analysis (CFA) were investigated. For a higher order model of academic self‐concept, 3 levels of aggregation were considered—disaggregated, partially disaggregated, and partially aggregated. In the disaggregated model, measured variables represented individual items. In the partially disaggregated model, testlets (groups of 4 items) represented measured variables. In the partially aggregated model, subscale scores represented measured variables. Three indexes of fit were employed: the Tucker‐Lewis Index (TLI), the Comparative Fit Index (CFI), and chi‐square. Solutions for the disaggregated models consistently evidenced poor fit. TLI and CFI values for partially disaggregated and partially aggregated solutions were satisfactory. Standardized parameter estimates were similar across all solutions. Implications of these findings are discussed with consideration of other re...

Marsh and I continue to disagree on issues relating to the signs of relationships, the importance of scaling, the value of existing rules of thumb, and the relative usefulness of M0 and M= in the potential development of new inferential approaches to fit assessment. I introduce an alternative model, M=, to clarify the distinction between substantive and baseline models. However, I partially endorse Marsh's compromise proposal. I also offer tentative heuristic values for interpreting fit indices that involve M=.