Diagonally Weighted Least Squares Research Articles

43 Comparison of Latent Structures for the Neuropsychiatric Inventory Questionnaire (NPI-Q)

Objective:Existing research has demonstrated that neuropsychiatric/behavioral-psychological symptoms of dementia (BPSD) frequently contribute to worse prognosis in patients with neurodegenerative conditions (e.g., increased functional dependence, worse quality of life, greater caregiver burden, faster disease progression). BPSD are most commonly measured via the Neuropsychiatric Inventory (NPI), or its briefer, informant-rated questionnaire (NPI-Q). Despite the NPI-Q’s common use in research and practice, there is disarray in the literature concerning the NPI-Q’s latent structure and reliability, possibly related to differences in methods between studies. Also, hierarchical factor models have not been considered, even though such models are gaining favor in the psychopathology literature. Therefore, we aimed to compare different factor structures from the current literature using confirmatory factor analyses (CFAs) to help determine the best latent model of the NPI-Q.Participants and Methods:This sample included 20,500 individuals (57% female; 80% White, 12% Black, 8% Hispanic), with a mean age of 71 (SD = 10.41) and 15 average years of education (SD = 3.43). Individuals were included if they had completed an NPI-Q during their first visit at one of 33 Alzheimer Disease Research Centers reporting to the National Alzheimer Coordinating Center (NACC). All CFA and reliability analyses were performed with lavaan and semTools R packages, using a diagonally weighted least squares (DWLS) estimator. Eight single-level models using full or modified versions of the NPI-Q were compared, and the top three were later tested in bifactor form.Results:CFAs revealed all factor models of the full NPI-Q demonstrated goodness of fit across multiple indices (SRMR = 0.039-0.052, RMSEA = 0.025-0.029, CFI = 0.973-0.983, TLI = 0.9670.977). Modified forms of the NPI-Q also demonstrated goodness of fit across multiple indices (SRMR = 0.025-0.052, RMSEA = 0.0180.031, CFI = 0.976-0.993, TLI = 0.968-0.989). Top factor models later tested in bifactor form all demonstrated consistently stronger goodness of fit regardless of whether they were a full form (SRMR = 0.023-0.035, RMSEA = 0.015-0.02, CFI = 0.992-0.995, TLI = 0.985-0.991) or a modified form (SRMR = 0.023-0.042, RMSEA = 0.015-0.024, CFI = 0.985-0.995, TLI = 0.9770.992). Siafarikas and colleagues’ (2018) 3-factor model demonstrated the best fit among the full-form models, whereas Sayegh and Knight’s (2014) 4-factor model had the best fit among all single-level models, as well as among the bifactor models.Conclusions:Although all factor models had adequate goodness of fit, the Sayegh & Knight 4-factor model had the strongest fit among both single-level and bifactor models. Furthermore, all bifactor models had consistently stronger fit than single-level models, suggesting that BPSD are best theoretically explained by a hierarchical, non-nested framework of general and specific contributors to symptoms. These findings also inform consistent use of NPI-Q subscales.

42 Cognitive Impairment Stage and Dementia Syndromes Explain Latent Structure Variability on the Neuropsychiatric Inventory Questionnaire (NPI-Q)

Objective:Neuropsychiatric/behavioral-psychological symptoms of dementia (BPSD) frequently contribute to worse prognosis of patients with neurodegenerative conditions. BPSD are commonly measured via a brief, informant-rated version of the Neuropsychiatric Inventory (NPI), the NPI-Q. Previously (see our other submission to this conference), we established optimal latent structures by comparing different factor models in the literature using confirmatory factor analyses (CFAs). However, questions remain as to why so many different models were found in the literature. One possibility is sampling differences, including different proportions of individuals across cognitive stages (e.g., mild cognitive impairment, moderate dementia) or syndromes (e.g., Alzheimer’s amnestic syndrome, Dementia with Lewy Bodies). We tested this hypothesis by subjecting candidate models to measurement invariance (MI) analyses stratified by cognitive stage and syndrome.Participants and Methods:Individuals were included if they had completed an NPI-Q during their first visit at an Alzheimer Disease Research Center reporting to the National Alzheimer Coordinating Center (NACC). This resulted in 20,500 individuals (57% female; 80% White, 13% Black, 8% Hispanic), with a mean age of 71 (SD = 10.41) and 15 average years of education (SD = 3.43). Regarding staging, 75.9% of individuals did not meet criteria for all-cause dementia, whereas 24.1% individuals had all-cause dementia. Regarding syndromes, 35.6% had an Alzheimer’s presentation (“AD-type”) and 5.6% had either a behavioral variant frontotemporal dementia or Lewy-Body dementia presentation (“behavioral-type”). A 3-factor and 4-factor model were subject to MI across these groupings. We conducted MI analyses for equal forms, equal loadings, and equal intercepts using the lavaan R package with a diagonally weighted least squares (DWLS) estimator.Results:The 3-factor model demonstrated good fit among individuals experiencing (CFI = 0.965, TLI = 0.955) and not experiencing (CFI = 0.984, TLI = 0.979) dementia, as well as among AD-type (CFI = 0.983, TLI = 0.978) presentations, but had borderline poor fit for behavioral-type (CFI = 0.932, TLI = 0.912) presentations. The 4-factor model had better fit among those experiencing (CFI = 0.985, TLI = 0.977) and not experiencing (CFI = 0.995, TLI = 0.992) dementia. Additionally, the 4-factor model demonstrated good of fit for AD-type (CFI = 0.993, TLI = 0.989) and poorer fit for behavioral-type (CFI = 0.949, TLI = 0.922) syndromes. Chi-square differences suggested that equal loading and equal intercept hypotheses should be rejected for both 3- and 4-factor models, for both staging and syndromal groupings. However, relative fit indices suggested that the equal form, equal loading, and equal intercept hypotheses could be adequate for only the 4-factor model.Conclusions:The variability of factor structures in the BPSD literature appears, at least partially, explained by sampling variability among cognitive stages and dementia syndromes. The best models in the literature appear to have good fit in non-demented individuals and, among those who have dementia, in those with an AD syndrome. Only Sayegh & Knight’s 4-factor model had adequate (albeit, not optimal) fit among those with all-cause dementia and, more specifically, among those with a behavioral-type dementia syndrome. These findings inform BPSD theory and practical implementation of NPI-Q subscales.

Psychometric evaluation of the Social Support Questionnaire for Transactions in Turkish gender-based violence sample

AimThis study aims to investigate the reliability and predictive validity of the Social Support Questionnaire for Transactions (SSQT) scale. DesignA psychometric design using cross-sectional data. MethodThis cross-sectional descriptive study was conducted between February–June 2021 using snowball sampling through an online survey panel. 204 Turkish survivors of gender-based violence (GBV) participated in the research. ResultsThe mean age of participants was 35.66 ± 12.50. Exploratory and confirmatory factor analysis examined the construct validity of the SSQT scale. The principle axis factoring (PAF) estimation method was performed, including oblique rotation (Promax) for EFA, and the diagonally weighted least squares (DWLS) estimation method was used for CFA. Test-retest reliability coefficients (r) were moderate to excellent, ranging from 0.48 to 0.88. The analyzes supported the 5-factor solution, and the reliability was evaluated with Cronbach's Alpha coefficients for Social Friendship, Daily Emotional Support, Problem-Oriented Emotional Support, Daily Instrumental Support, and Problem-Focused Instrumental Support sub-dimensions, and the total score of SSQT showing 0.87, 0.84, 0.90, 0.73, 0.83, and 0.93, respectively. The relationships between sub-dimensions of SSQT showed moderate positive correlations ranging from 0.40 to 0.60. There were weak negative correlations between SSQT and PHQ-9, PHQ-15 and GAD-7, ranging from −0.20 to −0.34, which shows the external validity of the SSQT. ConclusionThe relationships between sub-dimensions of SSQT and PHQ-9, PHQ-15, GAD-7 are given in the results support using the SSQT scale as a research and clinical instrument for assessing women who had experienced GBV and have/have not received social support in Turkish culture.

Reflecting on the “Robust” Standard Errors for Two-Stage SEM Estimation With Categorical Data: Mistakes and Correction

Estimating structural equation models with categorical data often proceeds in two stages. Stage 1 obtains polychoric correlations among categorical variables. Stage 2 fits the researcher’s model to the polychoric correlations, often with diagonally-weighted least squares (DWLS) or unweighted least squares (ULS). For DWLS/ULS model parameter estimates, the current literature contains two standard error (SE) methods: the classic SE and robust SE. Although the robust SE is generally believed to be satisfactory, in this paper we show it is reliable only given perfect models. When a model is misspecified, the robust SE can sometimes be egregiously biased, and the bias is not directly related to the model’s overall misfit. We also propose a new SE method for DWLS and ULS. The new SE is consistent regardless of the model fit, and substantially outperforms the robust SE in our simulation studies. The classic SE is the worst among the three SE methods.

An introduction to model implied instrumental variables using two stage least squares (MIIV-2SLS) in structural equation models (SEMs).

Structural equation models (SEMs) are widely used to handle multiequation systems that involve latent variables, multiple indicators, and measurement error. Maximum likelihood (ML) and diagonally weighted least squares (DWLS) dominate the estimation of SEMs with continuous or categorical endogenous variables, respectively. When a model is correctly specified, ML and DWLS function well. But, in the face of incorrect structures or nonconvergence, their performance can seriously deteriorate. Model implied instrumental variable, two stage least squares (MIIV-2SLS) estimates and tests individual equations, is more robust to misspecifications, and is noniterative, thus avoiding nonconvergence. This article is an overview and tutorial on MIIV-2SLS. It reviews the six major steps in using MIIV-2SLS: (a) model specification; (b) model identification; (c) latent to observed (L2O) variable transformation; (d) finding MIIVs; (e) using 2SLS; and (f) tests of overidentified equations. Each step is illustrated using a running empirical example from Reisenzein's (1986) randomized experiment on helping behavior. We also explain and illustrate the analytic conditions under which an equation estimated with MIIV-2SLS is robust to structural misspecifications. We include additional sections on MIIV approaches using a covariance matrix and mean vector as data input, conducting multilevel SEM, analyzing categorical endogenous variables, causal inference, and extensions and applications. Online supplemental material illustrates input code for all examples and simulations using the R package MIIVsem. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Shared mobility adoption from 2016 to 2018 in the Greater Toronto and Hamilton Area: Demographic or geographic diffusion?

Shared mobility services, such as on-demand ride-hailing, car sharing, and bike sharing, have significantly expanded the mobility tools available to urban residents. Understanding how these technologies are adopted over time is critical towards developing appropriate transportation policy measures. Most fundamentally: is adoption over time driven by demographic or geographic diffusion or a combination of the two? This study uses 2016 and 2018 survey data to explore the demographic and spatial predictors of changes in the adoption of on-demand ride-hailing, car sharing, and bike sharing in the Greater Toronto and Hamilton Area (GTHA) in Ontario, Canada.This study uses both descriptive statistics and inferential models to identify changes in the adoption process between 2016 and 2018. Trivariate ordered models estimated using diagonally weighted least squares (DWLS) suggest that spatial diffusion plays a role (accounting for 9–44% of changes in adoption between 2016 and 2018), that joint demographic-spatial patterns explain some variation in adoption over time (4–11%), but that demographic controls explain most changes in adoption over time (approximately 39–49%). Notably, adoption is concentrated (albeit somewhat less so over time) among younger cohorts, and it increases among women, among larger households, and among households without children. It is unclear how these changes in adoption will be impacted by the Covid-19 pandemic and future studies should explore the non-linear and complex nature of technology adoption, including the relative roles of service supply and demand changes.

A unified model-implied instrumental variable approach for structural equation modeling with mixed variables

The model-implied instrumental variable (MIIV) estimator is an equation-by-equation estimator of structural equation models that is more robust to structural misspecifications than full information estimators. Previous studies have concentrated on endogenous variables that are all continuous (MIIV-2SLS) or all ordinal . We develop a unified MIIV approach that applies to a mixture of binary, ordinal, censored, or continuous endogenous observed variables. We include estimates of factor loadings, regression coefficients, variances, and covariances along with their asymptotic standard errors. In addition, we create new goodness of fit tests of the model and overidentification tests of single equations. Our simulation study shows that the proposed MIIV approach is more robust to structural misspecifications than diagonally weighted least squares (DWLS) and that both the goodness of fit model tests and the overidentification equations tests can detect structural misspecifications. We also find that the bias in asymptotic standard errors for the MIIV estimators of factor loadings and regression coefficients are often lower than the DWLS ones, though the differences are small in large samples. Our analysis shows that scaling indicators with low reliability can adversely affect the MIIV estimators. Also, using a small subset of MIIVs reduces small sample bias of coefficient estimates, but can lower the power of overidentification tests of equations.

Statistical estimation of structural equation models with a mixture of continuous and categorical observed variables.

In the social and behavioral sciences, observed variables of mixed scale types (i.e., both continuous and categorical observed variables) have long been included in structural equation models. However, little is known about the impact of mixed continuous and categorical observed variables on the performance of existing estimation methods. This study compares two popular estimation methods with robust corrections, robust maximum likelihood (MLR) and diagonally weighted least squares (DWLS), when mixed continuous and categorical observed data are analyzed, evaluating the behavior of DWLS and MLR estimates in both measurement and full structural equation models. Monte Carlo simulation was carried out to examine the performance of DWLS and MLR in estimating model parameters, standard errors, and chi-square statistics. Two population models, a correlated three-factor measurement model and a five-factor structural equation model, were tested in combination with 36 other experimental conditions characterized by the number of observed variables' categories (2, 3, 4, 5, 6, and 7), categorical observed distribution shape (symmetry and slight asymmetry), and sample size (200, 500, and 1000). Data generation and analysis were performed with Mplus 8. Results reveal that (1) DWLS yields more accurate factor loading estimates for categorical observed variables than MLR, whereas DWLS and MLR produce comparable factor loading estimates for continuous observed variables; (2) inter-factor correlations and structural paths are estimated equally well by DWLS and MLR in nearly all conditions; (3) robust standard errors of parameter estimates obtained by MLR are slightly more accurate than those produced by DWLS in almost every condition, but the superiority of MLR over DWLS is not clearly evident once a medium or large sample is used (i.e., n = 500 or 1000); and (4) DWLS is systematically superior to MLR in controlling Type I error rates, but this superiority is attenuated with increasing sample size. The article concludes with a general discussion of the findings and some recommendations for practice and future research.

The Role of Perceived Support and Perceived Prejudice in the Health of LGBT Soldiers

Very little is known about factors that underlie the behavioral health status of LBGT compared with heterosexual soldiers. To address this knowledge gap, the current study explored the potential mechanistic roles of perceived prejudice and support for the LGBT community as they related to observed differences in behavioral health symptoms between LGBT and heterosexual soldiers. Between May 10 and 12, 2016, a sample of active-duty soldiers (N = 759) completed a battery of study measures while attending an academic training institute. Latent variable models using diagonally weighted least squares (DWLS) estimation were used to test for the direct and indirect effects of LGBT identity on behavioral health symptoms when treating perceived prejudice and support as simultaneous mediators. There is no evidence for a direct effect of LGBT identity on behavioral health symptoms after accounting for the role of perceived prejudice and support. There are indirect effects of LGBT status on these behavioral health outcomes manifested through shared associations with perceived support for and prejudice against the LBGT community. Even though “Don’t Ask, Don’t Tell” is no longer an explicit guideline, there are critical differences in the experience of LGBT and heterosexual soldiers, which may explain differences in mental and behavioral health. Specifically, perceived prejudice and perceived support appear to play a mechanistic role in those differences. The culture of the military following the repeal of Don’t Ask Don’t Tell continues to impact both heterosexual and LGBT Soldiers. Understanding the mechanistic role culture plays in the behavioral health of LGBT Soldiers may be one means of addressing their behavioral health needs.

Improving Fit Indices in Structural Equation Modeling with Categorical Data

Current computations of commonly used fit indices in structural equation modeling (SEM), such as RMSEA and CFI, indicate much better fit when the data are categorical than if the same data had not been categorized. As a result, researchers may be led to accept poorly fitting models with greater frequency when data are categorical. In this article, I first explain why the current computations of categorical fit indices lead to this problematic behavior. I then propose and evaluate alternative ways to compute fit indices with categorical data. The proposed computations approximate what the fit index values would have been had the data not been categorized. The developments in this article are for the DWLS (diagonally weighted least squares) estimator, a popular limited information categorical estimation method. I report on the results of a simulation comparing existing and newly proposed categorical fit indices. The results confirmed the theoretical expectation that the new indices better match the corresponding values with continuous data. The new fit indices performed well across all studied conditions, with the exception of binary data at the smallest studied sample size (N = 200), when all categorical fit indices performed poorly.

Correct Point Estimator and Confidence Interval for RMSEA Given Categorical Data

RMSEA estimation given nonnormal continuous data is usually based on the mean-adjusted () or mean-variance-adjusted () chi-square statistic, but a plain application of these statistics has poor performance. Savalei and colleagues gave a better way (the BSL method) to infer RMSEA using or . However, the BSL method is applicable to continuous data only. For categorical data, currently RMSEA inference is still based on a plain application of or , but such practice is already problematic under continuous data. In this paper, we first show that it is more meaningful to define RMSEA under unweighted least squares (ULS) than under weighted least squares (WLS) or diagonally weighted least squares (DWLS). Then, we propose a correct point estimator and confidence interval for RMSEA given categorical data and ULS. Simulation results show our methods perform well while all the traditional methods break down.

The Effect of Estimation Methods on SEM Fit Indices.

We examined the effect of estimation methods, maximum likelihood (ML), unweighted least squares (ULS), and diagonally weighted least squares (DWLS), on three population SEM (structural equation modeling) fit indices: the root mean square error of approximation (RMSEA), the comparative fit index (CFI), and the standardized root mean square residual (SRMR). We considered different types and levels of misspecification in factor analysis models: misspecified dimensionality, omitting cross-loadings, and ignoring residual correlations. Estimation methods had substantial impacts on the RMSEA and CFI so that different cutoff values need to be employed for different estimators. In contrast, SRMR is robust to the method used to estimate the model parameters. The same criterion can be applied at the population level when using the SRMR to evaluate model fit, regardless of the choice of estimation method.

The Influence of Number of Categories and Threshold Values on Fit Indices in Structural Equation Modeling with Ordered Categorical Data

This study examines the unscaled and scaled root mean square error of approximation (RMSEA), comparative fit index (CFI), and Tucker–Lewis index (TLI) of diagonally weighted least squares (DWLS) and unweighted least squares (ULS) estimators in structural equation modeling with ordered categorical data. We show that the number of categories and threshold values for categorization can unappealingly impact the DWLS unscaled and scaled fit indices, as well as the ULS scaled fit indices in the population, given that analysis models are misspecified and that the threshold structure is saturated. Consequently, a severely misspecified model may be considered acceptable, depending on how the underlying continuous variables are categorized. The corresponding CFI and TLI are less dependent on the categorization than RMSEA but are less sensitive to model misspecification in general. In contrast, the number of categories and threshold values do not impact the ULS unscaled fit indices in the population.

RMSEA, CFI, and TLI in structural equation modeling with ordered categorical data: The story they tell depends on the estimation methods.

In structural equation modeling, application of the root mean square error of approximation (RMSEA), comparative fit index (CFI), and Tucker-Lewis index (TLI) highly relies on the conventional cutoff values developed under normal-theory maximum likelihood (ML) with continuous data. For ordered categorical data, unweighted least squares (ULS) and diagonally weighted least squares (DWLS) based on polychoric correlation matrices have been recommended in previous studies. Although no clear suggestions exist regarding the application of these fit indices when analyzing ordered categorical variables, practitioners are still tempted to adopt the conventional cutoff rules. The purpose of our research was to answer the question: Given a population polychoric correlation matrix and a hypothesized model, if ML results in a specific RMSEA value (e.g., .08), what is the RMSEA value when ULS or DWLS is applied? CFI and TLI were investigated in the same fashion. Both simulated and empirical polychoric correlation matrices with various degrees of model misspecification were employed to address the above question. The results showed that DWLS and ULS lead to smaller RMSEA and larger CFI and TLI values than does ML for all manipulated conditions, regardless of whether or not the indices are scaled. Applying the conventional cutoffs to DWLS and ULS, therefore, has a pronounced tendency not to discover model-data misfit. Discussions regarding the use of RMSEA, CFI, and TLI for ordered categorical data are given.

Examining Chi-Square Test Statistics Under Conditions of Large Model Size and Ordinal Data

This study examined the effect of model size on the chi-square test statistics obtained from ordinal factor analysis models. The performance of six robust chi-square test statistics were compared across various conditions, including number of observed variables (p), number of factors, sample size, model (mis)specification, number of categories, and threshold distribution. Results showed that the unweighted least squares (ULS) robust chi-square statistics generally outperform the diagonally weighted least squares (DWLS) robust chi-square statistics. The ULSM estimator performed the best overall. However, when fitting ordinal factor analysis models with a large number of observed variables and small sample size, the ULSM-based chi-square tests may yield empirical variances that are noticeably larger than the theoretical values and inflated Type I error rates. On the other hand, when the number of observed variables is very large, the mean- and variance-corrected chi-square test statistics (e.g., based on ULSMV and WLSMV) could produce empirical variances conspicuously smaller than the theoretical values and Type I error rates lower than the nominal level, and demonstrate lower power rates to reject misspecified models. Recommendations for applied researchers and future empirical studies involving large models are provided.

More efficient parameter estimates for factor analysis of ordinal variables by ridge generalized least squares.

Data in psychology are often collected using Likert-type scales, and it has been shown that factor analysis of Likert-type data is better performed on the polychoric correlation matrix than on the product-moment covariance matrix, especially when the distributions of the observed variables are skewed. In theory, factor analysis of the polychoric correlation matrix is best conducted using generalized least squares with an asymptotically correct weight matrix (AGLS). However, simulation studies showed that both least squares (LS) and diagonally weighted least squares (DWLS) perform better than AGLS, and thus LS or DWLS is routinely used in practice. In either LS or DWLS, the associations among the polychoric correlation coefficients are completely ignored. To mend such a gap between statistical theory and empirical work, this paper proposes new methods, called ridge GLS, for factor analysis of ordinal data. Monte Carlo results show that, for a wide range of sample sizes, ridge GLS methods yield uniformly more accurate parameter estimates than existing methods (LS, DWLS, AGLS). A real-data example indicates that estimates by ridge GLS are 9-20% more efficient than those by existing methods. Rescaled and adjusted test statistics as well as sandwich-type standard errors following the ridge GLS methods also perform reasonably well.

GW-SEM: A Statistical Package to Conduct Genome-Wide Structural Equation Modeling.

Improving the accuracy of phenotyping through the use of advanced psychometric tools will increase the power to find significant associations with genetic variants and expand the range of possible hypotheses that can be tested on a genome-wide scale. Multivariate methods, such as structural equation modeling (SEM), are valuable in the phenotypic analysis of psychiatric and substance use phenotypes, but these methods have not been integrated into standard genome-wide association analyses because fitting a SEM at each single nucleotide polymorphism (SNP) along the genome was hitherto considered to be too computationally demanding. By developing a method that can efficiently fit SEMs, it is possible to expand the set of models that can be tested. This is particularly necessary in psychiatric and behavioral genetics, where the statistical methods are often handicapped by phenotypes with large components of stochastic variance. Due to the enormous amount of data that genome-wide scans produce, the statistical methods used to analyze the data are relatively elementary and do not directly correspond with the rich theoretical development, and lack the potential to test more complex hypotheses about the measurement of, and interaction between, comorbid traits. In this paper, we present a method to test the association of a SNP with multiple phenotypes or a latent construct on a genome-wide basis using a diagonally weighted least squares (DWLS) estimator for four common SEMs: a one-factor model, a one-factor residuals model, a two-factor model, and a latent growth model. We demonstrate that the DWLS parameters and p-values strongly correspond with the more traditional full information maximum likelihood parameters and p-values. We also present the timing of simulations and power analyses and a comparison with and existing multivariate GWAS software package.

The performance of ML, DWLS, and ULS estimation with robust corrections in structural equation models with ordinal variables.

Three estimation methods with robust corrections-maximum likelihood (ML) using the sample covariance matrix, unweighted least squares (ULS) using a polychoric correlation matrix, and diagonally weighted least squares (DWLS) using a polychoric correlation matrix-have been proposed in the literature, and are considered to be superior to normal theory-based maximum likelihood when observed variables in latent variable models are ordinal. A Monte Carlo simulation study was carried out to compare the performance of ML, DWLS, and ULS in estimating model parameters, and their robust corrections to standard errors, and chi-square statistics in a structural equation model with ordinal observed variables. Eighty-four conditions, characterized by different ordinal observed distribution shapes, numbers of response categories, and sample sizes were investigated. Results reveal that (a) DWLS and ULS yield more accurate factor loading estimates than ML across all conditions; (b) DWLS and ULS produce more accurate interfactor correlation estimates than ML in almost every condition; (c) structural coefficient estimates from DWLS and ULS outperform ML estimates in nearly all asymmetric data conditions; (d) robust standard errors of parameter estimates obtained with robust ML are more accurate than those produced by DWLS and ULS across most conditions; and (e) regarding robust chi-square statistics, robust ML is inferior to DWLS and ULS in controlling for Type I error in almost every condition, unless a large sample is used (N = 1,000). Finally, implications of the findings are discussed, as are the limitations of this study as well as potential directions for future research. (PsycINFO Database Record

A Simulation Study of Polychoric Instrumental Variable Estimation in Structural Equation Models

Data collected from questionnaires are often in ordinal scale. Unweighted least squares (ULS), diagonally weighted least squares (DWLS) and normal-theory maximum likelihood (ML) are commonly used methods to fit structural equation models. Consistency of these estimators demands no structural misspecification. In this article, we conduct a simulation study to compare the equation-by-equation polychoric instrumental variable (PIV) estimation with ULS, DWLS, and ML. Accuracy of PIV for the correctly specified model and robustness of PIV for misspecified models are investigated through a confirmatory factor analysis (CFA) model and a structural equation model with ordinal indicators. The effects of sample size and nonnormality of the underlying continuous variables are also examined. The simulation results show that PIV produces robust factor loading estimates in the CFA model and in structural equation models. PIV also produces robust path coefficient estimates in the model where valid instruments are used. However, robustness highly depends on the validity of instruments.

Taking correlations in GPS least squares adjustments into account with a diagonal covariance matrix

Based on the results of Luati and Proietti (Ann Inst Stat Math 63:673–686, 2011) on an equivalence for a certain class of polynomial regressions between the diagonally weighted least squares (DWLS) and the generalized least squares (GLS) estimator, an alternative way to take correlations into account thanks to a diagonal covariance matrix is presented. The equivalent covariance matrix is much easier to compute than a diagonalization of the covariance matrix via eigenvalue decomposition which also implies a change of the least squares equations. This condensed matrix, for use in the least squares adjustment, can be seen as a diagonal or reduced version of the original matrix, its elements being simply the sums of the rows elements of the weighting matrix. The least squares results obtained with the equivalent diagonal matrices and those given by the fully populated covariance matrix are mathematically strictly equivalent for the mean estimator in terms of estimate and its a priori cofactor matrix. It is shown that this equivalence can be empirically extended to further classes of design matrices such as those used in GPS positioning (single point positioning, precise point positioning or relative positioning with double differences). Applying this new model to simulated time series of correlated observations, a significant reduction of the coordinate differences compared with the solutions computed with the commonly used diagonal elevation-dependent model was reached for the GPS relative positioning with double differences, single point positioning as well as precise point positioning cases. The estimate differences between the equivalent and classical model with fully populated covariance matrix were below the mm for all simulated GPS cases and below the sub-mm for the relative positioning with double differences. These results were confirmed by analyzing real data. Consequently, the equivalent diagonal covariance matrices, compared with the often used elevation-dependent diagonal covariance matrix is appropriate to take correlations in GPS least squares adjustment into account, yielding more accurate cofactor matrices of the unknown.