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In the social sciences, accurately identifying the dimensionality of measurement scales is crucial for understanding latent constructs such as anxiety, happiness, and self-efficacy. This study presents a rigorous comparison between Parallel Analysis (PA) and Exploratory Graph Analysis (EGA) for assessing the dimensionality of scales, particularly focusing on ordinal data. Through an extensive simulation study, we evaluated the effectiveness of these methods under various conditions, including varying sample size, number of factors and their association, patterns of loading magnitudes, and symmetrical or skewed item distributions with assumed underlying normality or non-normality. Results show that the performance of each method varies across different scenarios, depending on the context. EGA consistently outperforms PA in correctly identifying the number of factors, particularly in complex scenarios characterized by more than a single factor, high inter-factor correlations and low to medium primary loadings. However, for datasets with simpler and stronger factor structures, specifically those with a single factor, high primary loadings, low cross-loadings, and low to moderate interfactor correlations, PA is suggested as the method of choice. Skewed item distributions with assumed underlying normality or non-normality were found to noticeably impact the performance of both methods, particularly in complex scenarios. The results provide valuable insights for researchers utilizing these methods in scale development and validation, ensuring that measurement instruments accurately reflect theoretical constructs. [ABSTRACT FROM AUTHOR]

Psychological distance from climate change has emerged as an important construct in understanding sustainable behavior and attempts to mitigate and/or adapt to climate change. Yet, few measures exist to assess this construct and little is known about the properties of the existing measures. In this article, the author conducted two studies of a psychological distance measure developed by Wang and her colleagues. In Study 1, the author assessed the test–retest reliability of the measure over a two-week interval and found the scores to be acceptably stable over time. In Study 2, the author conducted two exploratory factor analyses, using different approaches to the correlation and factor extraction. Similar results were observed for each factor analysis: one factor was related to items that specified greater psychological distance from climate change; a second factor involved items that specified closeness to climate change; and a third involved the geographic/spatial distance from climate change. The author discussed the results and provided recommendations on ways that the measure may be used to research the construct of psychological distance from climate change. [ABSTRACT FROM AUTHOR]

In behavioral and social research, questionnaires are an important assessment tool, through which individuals can be categorized according to how they classify themselves in respect to a personal trait. One example is the Major Depression Inventory (MDI), which is widely used for the assessment of depression. It can also be used as a depression severity scale, with scores ranging from 0 to 50 constructed considering the same weight for each item in the MDI. However, the dependence among the items of the questionnaire suggests that a score with better properties could be obtained through factor models, which besides allowing to reduce the dimensionality of multivariate data, provides the estimation of common factors and factor loadings that often have an interesting theoretical interpretation. Additionally, auxiliary information could be available and, the effect of these variables in the latent factor could be estimated and provide interesting results. Thus, the main aim of this paper is to propose a factor model for ordered categorical data which incorporates auxiliary variables to explain the latent factors. The proposed model provides an alternative score to MDI based on the estimated latent factors that takes the uncertainty in the data and auxiliary information into account. [ABSTRACT FROM AUTHOR]

Although there has been significant research on the relationship between alcohol consumption and demographic and psychological influences, this does not consider the effect of social influence among older drinkers and if these effects differ between men and women. One aspect of social influence is social capital. The aim of this paper is to examine whether relational and cognitive social capital are associated with higher or lower risk of alcohol use among adults aged 50 years or older and to assess the extent to which this relationship differs between men and women. To investigate this, data were collected from a cross-sectional questionnaire survey of adults over the age of 50 in the United Kingdom who were recruited from general practitioners. The sample consisted of 9,984 individuals whose mean age was 63.87 years. From these data, we developed proxy measures of social capital and associate these with the respondent's level of alcohol consumption as measured on the Alcohol Use Disorders Identification Test (AUDIT-10) scale. In the sample, just over 20 per cent reported an increasing risk or dependency on alcohol. Using two expressions of social capital – relational (social relationships) and cognitive (knowledge acquisition and understanding) – we found that greater levels of both are associated with a reduced risk of higher drinking risk. Being female had no significant effect when combined with relational capital but did have a significant effect when combined with cognitive capital. It is argued that interventions to enhance social relations among older people and education to help understand alcohol risks would be helpful to protect older people from the damaging effects of excessive alcohol consumption. [ABSTRACT FROM AUTHOR]

In the social sciences, measurement scales often consist of ordinal items and are commonly analyzed using factor analysis. Either data are treated as continuous, or a discretization framework is imposed in order to take the ordinal scale properly into account. Correlational analysis is central in both approaches, and we review recent theory on correlations obtained from ordinal data. To ensure appropriate estimation, the item distributions prior to discretization should be (approximately) known, or the thresholds should be known to be equally spaced. We refer to such knowledge as substantive because it may not be extracted from the data, but must be rooted in expert knowledge about the data-generating process. An illustrative case is presented where absence of substantive knowledge of the item distributions inevitably leads the analyst to conclude that a truly two-dimensional case is perfectly one-dimensional. Additional studies probe the extent to which violation of the standard assumption of underlying normality leads to bias in correlations and factor models. As a remedy, we propose an adjusted polychoric estimator for ordinal factor analysis that takes substantive knowledge into account. Also, we demonstrate how to use the adjusted estimator in sensitivity analysis when the continuous item distributions are known only approximately. In psychology and other social sciences surveys are often used to collect data from individuals. These surveys contain items that are answered using limited discrete scales, for instance a five-point scale with values ranging from 1 = completely disagree to 5 = completely agree. The data collected is therefore only crudely measured. Complex statistical models of human traits like, e.g., openness or self-efficacy, are routinely estimated from such survey data. Due to the crudity of the measurements, these models rest on strong and partly unverifiable statistical assumptions. For correct modeling, these assumptions must hold, and expert knowledge which cannot be extracted from the survey data is needed. This article explains that standard statistical procedures may be unreliable when expert knowledge is not available, and proposes a way forward that incorporates expert knowledge. [ABSTRACT FROM AUTHOR]

In the social sciences, accurately identifying the dimensionality of measurement scales is crucial for understanding latent constructs such as anxiety, happiness, and self-efficacy. This study presents a rigorous comparison between Parallel Analysis (PA) and Exploratory Graph Analysis (EGA) for assessing the dimensionality of scales, particularly focusing on ordinal data. Through an extensive simulation study, we evaluated the effectiveness of these methods under various conditions, including varying sample size, number of factors and their association, patterns of loading magnitudes, and symmetrical or skewed item distributions with assumed underlying normality or non-normality. Results show that the performance of each method varies across different scenarios, depending on the context. EGA consistently outperforms PA in correctly identifying the number of factors, particularly in complex scenarios characterized by more than a single factor, high inter-factor correlations and low to medium primary loadings. However, for datasets with simpler and stronger factor structures, specifically those with a single factor, high primary loadings, low cross-loadings, and low to moderate interfactor correlations, PA is suggested as the method of choice. Skewed item distributions with assumed underlying normality or non-normality were found to noticeably impact the performance of both methods, particularly in complex scenarios. The results provide valuable insights for researchers utilizing these methods in scale development and validation, ensuring that measurement instruments accurately reflect theoretical constructs.

An iteratively reweighted least squares (IRLS) method is proposed for the estimation of polyserial and polychoric correlation coefficients in this article. It calculates the slopes in a series of weighted linear regression models fitting on conditional expected values. For polyserial correlation, conditional expectations of the latent predictor is derived from the observed ordinal categorical variable, and the regression coefficient is obtained with the weighted least squares method. In estimating polychoric correlation coefficient, conditional expectations of the response variable and the predictor are updated in turns. Standard errors of the estimators are obtained using the delta method based on data summaries instead of the whole data. Conditional univariate normal distribution is exploited and a single integral is numerically evaluated in the proposed algorithm, comparing to the double integral computed numerically based on the bivariate normal distribution in the traditional maximum likelihood (ML) approaches. This renders the new algorithm very fast in the estimation of both polyserial and polychoric correlation coefficients. Thorough simulation studies are conducted to compare the performances of the proposed method with the classical ML methods. Real data analyses illustrate the advantage of the new method in computing speed. for this article are available online. [ABSTRACT FROM AUTHOR]

Psychological distance from climate change has emerged as an important construct in understanding sustainable behavior and attempts to mitigate and/or adapt to climate change. Yet, few measures exist to assess this construct and little is known about the properties of the existing measures. In this article, the author conducted two studies of a psychological distance measure developed by Wang and her colleagues. In Study 1, the author assessed the test–retest reliability of the measure over a two-week interval and found the scores to be acceptably stable over time. In Study 2, the author conducted two exploratory factor analyses, using different approaches to the correlation and factor extraction. Similar results were observed for each factor analysis: one factor was related to items that specified greater psychological distance from climate change; a second factor involved items that specified closeness to climate change; and a third involved the geographic/spatial distance from climate change. The author discussed the results and provided recommendations on ways that the measure may be used to research the construct of psychological distance from climate change.

The polychoric correlation is a popular measure of association for ordinal data. It estimates a latent correlation, i.e., the correlation of a latent vector. This vector is assumed to be bivariate normal, an assumption that cannot always be justified. When bivariate normality does not hold, the polychoric correlation will not necessarily approximate the true latent correlation, even when the observed variables have many categories. We calculate the sets of possible values of the latent correlation when latent bivariate normality is not necessarily true, but at least the latent marginals are known. The resulting sets are called partial identification sets, and are shown to shrink to the true latent correlation as the number of categories increase. Moreover, we investigate partial identification under the additional assumption that the latent copula is symmetric, and calculate the partial identification set when one variable is ordinal and another is continuous. We show that little can be said about latent correlations, unless we have impractically many categories or we know a great deal about the distribution of the latent vector. An open-source R package is available for applying our results. [ABSTRACT FROM AUTHOR]

Background: Measures in nursing research frequently use Likert scales that yield ordinal data. Confirmatory factor analysis using Pearson correlations commonly applies to such data, although this violates ordinal scale assumptions. Objectives: The aim of this study was to illustrate the application of polychoric correlations and polychoric confirmatory factor analysis as a valid alternative statistical approach using data on family members' perceived support from nurses as an exemplar. Methods: A primary analysis of cross-sectional data from a sample of 800 participants using data collected with the Iceland-Family Perceived Support Questionnaire was conducted using polychoric versus Pearson correlations, analysis of variance, and confirmatory factor analysis. Results: A two-factor measurement model was compatible with data from family members in the Ugandan care settings. Two contextual factors (cognitive and emotional support) constituted the family support measurement model. A factor correlation indicated that the two factors reflected distinct but closely related aspects of family support. Polychoric correlation revealed 13.8% (range: 5.5%–25.2%) higher correlations compared to Pearson correlations. Moreover, the polychoric agreed with the data, whereas the Pearson confirmatory factor analysis did not fit based on multiple statistical criteria. Analyses indicated a difference in emotional and cognitive support perception across two family characteristics: education and relationship to the patient. Discussion: A polychoric correlation suggests stronger associations, and consequently, the approach can be more credible with an ordinal Likert scale than Pearson correlations. Hence, polychoric confirmatory factor analysis can address a larger proportion of variance. In nursing research, polychoric confirmatory factor analysis can confidently be utilized when conducting confirmatory factor analysis of ordinal variables in Likert scales. Furthermore, when a Pearson confirmatory factor analysis is used for ordinal Likert scales, the researcher should carefully evaluate the difference between the two approaches and justify their methodological choice. Even though we do not suggest dispensing with Pearson correlations entirely, we recommend using polychoric correlation for ordinal Likert scales. [ABSTRACT FROM AUTHOR]

The association between two ordinal variables can be expressed with a polychoric correlation coefficient. This coefficient is conventionally based on the assumption that responses to ordinal variables are generated by two underlying continuous latent variables with a bivariate normal distribution. When the underlying bivariate normality assumption is violated, the estimated polychoric correlation coefficient may be biased. In such a case, we may consider other distributions. In this paper, we aimed to provide an illustration of fitting various bivariate distributions to empirical ordinal data and examining how estimates of the polychoric correlation may vary under different distributional assumptions. Results suggested that the bivariate normal and skew-normal distributions rarely hold in the empirical datasets. In contrast, mixtures of bivariate normal distributions were often not rejected.

Statistical modeling with sparsity has become an active research topic in the fields of statistics and machine learning. Because the true sparsity pattern of a model is generally unknown aforehand, it is often explored by a sparse estimation procedure, like least absolute shrinkage and selection operator (lasso). In this study, a penalized least squares (PLS) method for structural equation modeling (SEM) with ordinal data is developed. PLS describes data generation by an underlying response approach, and uses a least squares (LS) fitting function to construct a penalized estimation criterion. A numerical simulation was used to compare PLS with existing penalized likelihood (PL) in terms of averaged mean square error, absolute bias, and the correctness of the model. Based on these empirical findings, a hybrid PLS was also proposed to improve both PL and PLS. The hybrid PLS first chooses an optimal sparsity pattern by PL, then estimates model parameters by an unpenalized LS under the model selected by PL. We also extended PLS to cases of mixed type data and multi-group analysis. All proposed methods could be realized in the R package lslx. [ABSTRACT FROM AUTHOR]

To overcome the environmental side effects of over-utilizing fossil fuels and road traffic emissions, Alternative Fuel Vehicles (AFVs) are alternative power engines. However, as AFVs have not yet penetrated the automotive market in Iran, using the choice experiment model, this study considers the heterogeneous preference for Electric Vehicles (EVs) in four megacities of Iran. Therefore, 1000 respondents are interviewed and are asked to choose their preferred vehicle between hybrid electric vehicles (HEVs), battery electric vehicles (BEVs), and internal combustion engine vehicles. Furthermore, this study contributes to the literature by employing the Policoric correlation matrix, not Pearson, to investigate the influence of psychological factors besides the macro, micro, and product factors in adopting Evs. The latent class and mixed logit model reveal the monetary factors negatively affected consumers' willingness to pay for HEVs and BEVs. In contrast, the charging availability and non-monetary policies are positively connected to the acceptance rate of BEVs, While risk-aversion and normative-face perspectives reduce it. Furthermore, the class membership variable shows that the younger tend to purchase HEVs, and high-income groups are the most significant potential buyers of HEVs and BEVs. Therefore, the younger and high-income groups could be considered the target group in Iran's early penetrating EVs. [ABSTRACT FROM AUTHOR]

Factor analysis is a flexible technique for assessment of multivariate dependence and codependence. Besides being an exploratory tool used to reduce the dimensionality of multivariate data, it allows estimation of common factors that often have an interesting theoretical interpretation in real problems. However, standard factor analysis is only applicable when the variables are scaled, which is often inappropriate, for example, in data obtained from questionnaires in the field of psychology, where the variables are often categorical. In this framework, we propose a factor model for the analysis of multivariate ordered and non-ordered polychotomous data. The inference procedure is done under the Bayesian approach via Markov chain Monte Carlo methods. Two Monte Carlo simulation studies are presented to investigate the performance of this approach in terms of estimation bias, precision and assessment of the number of factors. We also illustrate the proposed method to analyze participants' responses to the Motivational State Questionnaire dataset, developed to study emotions in laboratory and field settings. [ABSTRACT FROM AUTHOR]

The severity of traffic barrier crashes has been modeled in the literature based on human, road, and traffic barrier factors. However, all these factors interact in a complicated way so the relationship between these factors still remains unclear. A structural equation model (SEM) can be used to capture the intricate relationships between the contributory factors and the latent or unobservable factors. This study was conducted using the SEM to model the complicated relationships between the confounding factors and traffic barrier crash severity. Due to possible differences across different road classifications, the SEM was applied separately to the highway and interstate systems. The SEM involves the measurement or confirmatory factor analysis (CFA) model and the structural model involving the interrelationships between the factors. This study evaluates traffic barrier crash severity in terms of numbers of death, injury, and severity of crashes. This study examined the nature and causes of severe traffic barrier crashes in Wyoming. The results indicated that road conditions, traffic barrier types, risky driving, and force are factors that need to be considered in addressing the severity of traffic barrier crashes. The methodology in this study also addresses categorical predictors and model selection techniques for specifying the measurement model and the structural model which make up the SEM. A careful sequential modeling framework was used to build the structural model characterizing the interrelationships among the latent variables as well as evaluation of the fitted SEM based on goodness of fit indices. [ABSTRACT FROM AUTHOR]

Partial association refers to the relationship between variables Y 1 , Y 2 , ... , Y K while adjusting for a set of covariates X = { X 1 , ... , X p } . To assess such an association when Yk's are recorded on ordinal scales, a classical approach is to use partial correlation between the latent continuous variables. This so-called polychoric correlation is inadequate, as it requires multivariate normality and it only reflects a linear association. We propose a new framework for studying ordinal-ordinal partial association by using Liu-Zhang's surrogate residuals. We justify that conditional on X , Yk, and Yl are independent if and only if their corresponding surrogate residual variables are independent. Based on this result, we develop a general measure ϕ to quantify association strength. As opposed to polychoric correlation, ϕ does not rely on normality or models with the probit link, but instead it broadly applies to models with any link functions. It can capture a nonlinear or even nonmonotonic association. Moreover, the measure ϕ gives rise to a general procedure for testing the hypothesis of partial independence. Our framework also permits visualization tools, such as partial regression plots and three-dimensional P-P plots, to examine the association structure, which is otherwise unfeasible for ordinal data. We stress that the whole set of tools (measures, p-values, and graphics) is developed within a single unified framework, which allows a coherent inference. The analyses of the National Election Study (K = 5) and Big Five Personality Traits (K = 50) demonstrate that our framework leads to a much fuller assessment of partial association and yields deeper insights for domain researchers. for this article are available online. [ABSTRACT FROM AUTHOR]

The measurement of many human traits, states, and disorders begins with a set of items on a questionnaire. The response format for these questions is often simply binary (e.g., yes/no) or ordered (e.g., high, medium or low). During data analysis, these items are frequently summed or used to estimate factor scores. In clinical applications, such assessments are often non-normally distributed in the general population because many respondents are unaffected, and therefore asymptomatic. As a result, in many cases these measures violate the statistical assumptions required for subsequent analyses. To reduce the influence of the non-normality and quasi-continuous assessment, variables are frequently recoded into binary (affected–unaffected) or ordinal (mild–moderate–severe) diagnoses. Ordinal data therefore present challenges at multiple levels of analysis. Categorizing continuous variables into ordered categories typically results in a loss of statistical power, which represents an incentive to the data analyst to assume that the data are normally distributed, even when they are not. Despite prior zeitgeists suggesting that, e.g., variables with more than 10 ordered categories may be regarded as continuous and analyzed as if they were, we show via simulation studies that this is not generally the case. In particular, using Pearson product-moment correlations instead of maximum likelihood estimates of polychoric correlations biases the estimated correlations towards zero. This bias is especially severe when a plurality of the observations fall into a single observed category, such as a score of zero. By contrast, estimating the ordinal correlation by maximum likelihood yields no estimation bias, although standard errors are (appropriately) larger. We also illustrate how odds ratios depend critically on the proportion or prevalence of affected individuals in the population, and therefore are sub-optimal for studies where comparisons of association metrics are needed. Finally, we extend these analyses to the classical twin model and demonstrate that treating binary data as continuous will underestimate genetic and common environmental variance components, and overestimate unique environment (residual) variance. These biases increase as prevalence declines. While modeling ordinal data appropriately may be more computationally intensive and time consuming, failing to do so will likely yield biased correlations and biased parameter estimates from modeling them. [ABSTRACT FROM AUTHOR]

Abstract Background Core vestibular symptoms are vague, hard for patients to describe, and difficult for examiners to quantify. Reliable and validated patient-reported outcome measures (PROMs) have obtained acceptance and popularity in the specialty of vestibular disorders. In Kurdish, there is a critical shortage of such measures. The aim of this survey was to assess the psychometric properties of a central Kurdish version (VSS − SF − CK) of the Vertigo Symptom Scale−Short Form (VSS − SF). Methods The study utilized a regulated process of cross-cultural adaptation to produce the VSS − SF − CK. We examined its psychometric properties by using a cross-sectional survey. Owing to a non-normal distribution, both principal axis factoring and polychoric correlation were used to examine the structure. The internal consistency of the scales was evaluated using Cronbach’s alpha coefficient (α) and composite reliability. The discriminant validity was evaluated using the heterotrait–monotrait ratio of correlations (HTMT.85) and the Fornell-Larcker criterion. To assess convergent validity, the instrument was correlated with two comparators. Results The participants (n = 195) were composed of 165 patients with vestibular symptoms (mean − age 45 ± 15.8, range 61 years; 56.4% women) and 30 healthy participants (mean − age 35 ± 18.6; range 52 years; 60% women). Based on the scree plot, along with other criteria such as Horn’s parallel analysis and minimum average partial, two factors were extracted: vestibular (VSS − V) and autonomic-anxiety (VSS − AA). Both constructs showed a robust structure in terms of adequate loadings and weak cross-loadings. The scales’ αs were 0.81, 0.81, and 0.87 for VSS-V, VSS-AA, and the total scale (VSS − T), respectively. Discriminant validity was established with a value of 0.71 for HTMT (

Least-squares exploratory factor analysis based on tetrachoric/polychoric correlations is a robust, defensible and widely used approach for performing item analysis, especially in the first stages of scale development. A relatively common problem in this scenario, however, is that the inter-item correlation matrix fails to be positive definite. This paper, which is largely intended for practitioners, aims to provide a didactic discussion about the causes, consequences and remedies of this problem. The discussion is more applied than statistical and based on the factor analysis model, and the problem is linked to that of improper solutions. Solutions for preventing the problem from occurring, and the smoothing corrections available at present are described and discussed. A new smoothing algorithm is also proposed. [ABSTRACT FROM AUTHOR]

Ordinal data in social science statistics are often modeled as discretizations of a multivariate normal vector. In contrast to the continuous case, where SEM estimation is also consistent under non-normality, violation of underlying normality in ordinal SEM may lead to inconsistent estimation. In this article, we illustrate how underlying non-normality induces bias in polychoric estimates and their standard errors. This bias is strongly affected by how we discretize. It is therefore important to consider tests of underlying multivariate normality. In this study we propose a parametric bootstrap test for this purpose. Its performance relative to the test of Maydeu-Olivares is evaluated in a Monte Carlo study. At realistic sample sizes, the bootstrap exhibited substantively better Type I error control and power than the Maydeu-Olivares test in ordinal data with ten dimensions or higher. R code for the bootstrap test is provided. [ABSTRACT FROM AUTHOR]

Previous influential simulation studies investigate the effect of underlying non-normality in ordinal data using the Vale–Maurelli (VM) simulation method. We show that discretized data stemming from the VM method with a prescribed target covariance matrix are usually numerically equal to data stemming from discretizing a multivariate normal vector. This normal vector has, however, a different covariance matrix than the target. It follows that these simulation studies have in fact studied data stemming from normal data with a possibly misspecified covariance structure. This observation affects the interpretation of previous simulation studies. [ABSTRACT FROM AUTHOR]

Human capital (HC) plays an important role in modern agriculture. The difference in efficiency of assets explains only about a half of the economic performance of agricultural farms, while the other half relies on HC. Although education and training are the main components of HC, it may also be viewed from the perspective of behavioral theories that were taken under consideration in this study. The role of HC in sustainable farming has not been sufficiently explained when it comes to contractual governance (CG). In this study, the meaning of contractual governance was extended and the eco-contractual governance (ECG) concept was proposed, which stands for CG induced by agri-environmental contracts. The main objective of the article is to confirm the latent concepts of HC and ECG and to verify their correlation in view of the standards imposed by the agricultural policy. To achieve this goal, a structural equation model was developed and simultaneous confirmatory factor analysis with ordinal variables was carried out based on the sample of 674 small farms in Poland. The analysis has confirmed a relatively strong correlation between HC and ECG. It was revealed that training plays a crucial role in this relationship, while economic dependence on agricultural policy weakens the effectiveness of both HC and ECG.

In this article, operational details of the R package OrdNor that is designed for the concurrent generation of correlated ordinal and normal data are described, and examples of some important functions are given. The package provides needed tools that have been lacking for generating multivariate data with a mixture of ordinal and normal components.

The polychoric instrumental variable (PIV) approach is a recently proposed method to fit a confirmatory factor analysis model with ordinal data. In this paper, we first examine the small‐sample properties of the specification tests for testing the validity of instrumental variables (IVs). Second, we investigate the effects of using different numbers of IVs. Our results show that specification tests derived for continuous data are extremely oversized at all sample sizes when applied to ordinal variables. Possible modifications for ordinal data are proposed in the present study. Simulation results show that the modified specification tests with all available IVs are able to detect model misspecification. In terms of estimation accuracy, the PIV approach where the IVs outnumber the endogenous variables by one produces a lower bias but a higher variation than the PIV approach with more IVs for correctly specified factor loadings at small samples. [ABSTRACT FROM AUTHOR]

This research concerns the estimation of polychoric correlations in the context of fitting structural equation models to observed ordinal variables by multistage estimation. The first main contribution of this research is to propose and evaluate a Monte Carlo estimator for the asymptotic covariance matrix (ACM) of the polychoric correlation estimates. In multistage estimation, the ACM plays a prominent role, as overall test statistics, derived fit indices, and parameter standard errors all depend on this quantity. The ACM, however, must itself be estimated. Established approaches to estimating the ACM use a sample-based version, which can yield poor estimates with small samples. A simulation study demonstrates that the proposed Monte Carlo estimator can be more efficient than its sample-based counterpart. This leads to better calibration for established test statistics, in particular with small samples. The second main contribution of this research is a further exploration of the consequences of violating the normality assumption for the underlying response variables. We show the consequences depend on the type of nonnormality, and the number and location of thresholds. The simulation study also demonstrates that overall test statistics have little power to detect the studied forms of nonnormality, regardless of the ACM estimator. [ABSTRACT FROM PUBLISHER]

This article introduces a new consistent variance-based estimator called ordinal consistent partial least squares (OrdPLSc). OrdPLSc completes the family of variance-based estimators consisting of PLS, PLSc, and OrdPLS and permits to estimate structural equation models of composites and common factors if some or all indicators are measured on an ordinal categorical scale. A Monte Carlo simulation (N =500) with different population models shows that OrdPLSc provides almost unbiased estimates. If all constructs are modeled as common factors, OrdPLSc yields estimates close to those of its covariance-based counterpart, WLSMV, but is less efficient. If some constructs are modeled as composites, OrdPLSc is virtually without competition. [ABSTRACT FROM AUTHOR]

The statistical multivariate analysis of Likert response scales, given their widespread use, is a controversial issue in the scientific community, mainly from the specification of the problem of measurement. This work aims to study various conditions of these ordinal scales affect the calculation of the product-moment and tetrachoric-polychoric correlation coefficients. For this purpose, a simulation study was applied in which 90 databases with 10 items each were generated. In the estimation of the databases, the following variables were controlled: number of response categories, symmetrical or asymmetric distributions of data, sample size and level of relationship between items. Thus, 90 matrices (10x10) were obtained which included the difference between the product-moment and tetrachoric-polychoric correlations. The graphical and variance analysis show how the product-moment correlation coefficient significantly underestimates the relationship between variables mainly when the number of response categories of the ordinal scale is small and the relationship between the variables is large. On the other hand, the statistical estimation of both coefficients is very similar when the starting relationship between pairs of variables is small and/or when the number of response options of the variables is greater than 5. The study concludes by making a recommendation to the applied researcher on the most appropriate correlation coefficient depending on the type of data available. Finally, the results are discussed from the previous studies, which reach some similar conclusions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]