Evaluating estimation methods for ordinal data in structural equation modeling

This study examined the performance of two alternative estimation approaches in structural equation modeling for ordinal data under different levels of model misspecification, score skewness, sample size, and model size. Both approaches involve analyzing a polychoric correlation matrix as well as adjusting standard error estimates and model chi-squared, but one estimates model parameters with maximum likelihood and the other with robust weighted least-squared. Relative bias in parameter estimates and standard error estimates, Type I error rate, and empirical power of the model test, where appropriate, were evaluated through Monte Carlo simulations. These alternative approaches generally provided unbiased parameter estimates when the model was correctly specified. They also provided unbiased standard error estimates and adequate Type I error control in general unless sample size was small and the measured variables were moderately skewed. Differences between the methods in convergence problems and the evaluation criteria, especially under small sample and skewed variable conditions, were discussed.     

Selection of influential variables in ordinal data with preponderance of zeros

Presence of excess zero in ordinal data is pervasive in areas like medical and social sciences. Unfortunately, analysis of such kind of data has so far hardly been looked into, perhaps for the reason that the underlying model that fits such data, is not a generalized linear model. Obviously some methodological developments and intensive computations are required. The current investigation is concerned with the selection of variables in such models. In many occasions where the number of predictors is quite large and some of them are not useful, the maximum likelihood approach is not the automatic choice. As, apart from the messy calculations involved, this approach fails to provide efficient estimates of the underlying parameters. The proposed penalized approach includes ?₁ penalty (LASSO) and the mixture of ?₁ and ?₂ penalties (elastic net). We propose a coordinate descent algorithm to fit a wide class of ordinal regression models and select useful variables appearing in both the ordinal regression and the logistic regression based mixing component. A rigorous discussion on the selection of predictors has been made through a simulation study. The proposed method is illustrated by analyzing the severity of driver injury from Michigan upper peninsula road accidents.     

The effect of estimation method and sample size in multilevel structural equation modeling

Multilevel structural equation modeling (multilevel SEM) has become an established method to analyze multilevel multivariate data. The first useful estimation method was the pseudobalanced method. This method is approximate because it assumes that all groups have the same size, and ignores unbalance when it exists. In addition, full information maximum likelihood (ML) estimation is now available, which is often combined with robust chi‐squares and standard errors to accommodate unmodeled heterogeneity (MLR). In addition, diagonally weighted least squares (DWLS) methods have become available as estimation methods. This article compares the pseudobalanced estimation method, ML(R), and two DWLS methods by simulating a multilevel factor model with unbalanced data. The simulations included different sample sizes at the individual and group levels and different intraclass correlation (ICC). The within‐group part of the model posed no problems. In the between part of the model, the different ICC sizes had no effect. There is a clear interaction effect between number of groups and estimation method. ML reaches unbiasedness fastest, then the two DWLS methods, then MLR, and then the pseudobalanced method (which needs more than 200 groups). We conclude that both ML(R) and DWLS are genuine improvements on the pseudobalanced approximation. With small sample sizes, the robust methods are not recommended.     

Estimating intercoder reliability: a structural equation modeling approach

Intercoder reliability is usually estimated with a summary index, and yet the limitations concerning the indexing approach have been well noted. This study critically reviewed all the existing major modeling approaches to estimating intercoder reliability, and empirically tested and further compared these approaches. It was found that latent variable modeling, also called the second-generation SEM, generally perform better than log-linear modeling, and is able to explain the paradox haunting some indices, and to spot the sources of disagreement among coders. Implications were discussed at last.     

Structural equation modeling and the latent linearity hypothesis in social and behavioral research

The issue of sensitivity of the structural equation modeling (SEM) methodology to violations of the underlying hypothesis of linear latent relationships is the focus of this paper. The identity of overall goodness-of-fit indices of an initially considered linear latent pattern model and of an equivalent model not making this assumption exemplifies the lack of routinely available global means within the methodology to evaluate the linearity assumption. It is next focused on the sensitivity of SEM to violations of presumed linearity for a general, nonlinear pattern of true relationship. The results of a simulation study are then presented which demonstrate that latent correlations and percentage explained variance as well as parameter standard errors and model residuals can provide critical information about violation of latent linearity, and should therefore also be focused on when examining departures from linear relationships at the latent level in applications of the SEM methodology in social and behavioral research.     

The nonrandom selection of don't knows in binary and ordinal responses: Corrections with the bivariate probit model with sample selection

On social surveysdon't knows are a common answer to attitudinal questions, which often have binary or ordinal response categories.Don't knows can be nonrandomly selected according to certain demographic or socioeconomic characteristics of the respondent. To model the sample selection and correct for its bias, this paper discusses two types of bivariate models —binary-probit and the ordinal probit model with sample selection. The difference between parameter estimates and predicted probabilities from the analysis modelling the sample selection bias ofdon't knows and those from the analysis not modellingdon't knows is emphasized. Two empirical examples using the 1989 General Social Survey data demonstrate the necessity to correct for the bias in the nonrandom selection ofdon't knows for binary and ordinal attitudinal response variables. A replication of the analyses using the 1990 and 1991 General Social Survey data helps demonstrate the reliability of the sample selection bias ofdon't knows.     

Estimating structural equation models with non-normal variables by using transformations

We discuss structural equation models for non-normal variables. In this situation the maximum likelihood and the generalized least-squares estimates of the model parameters can give incorrect estimates of the standard errors and the associated goodness-of-fit chi-squared statistics. If the sample size is not large, for instance smaller than about 1000, asymptotic distribution-free estimation methods are also not applicable. This paper assumes that the observed variables are transformed to normally distributed variables. The non-normally distributed variables are transformed with a Box–Cox function. Estimation of the model parameters and the transformation parameters is done by the maximum likelihood method. Furthermore, the test statistics (i.e. standard deviations) of these parameters are derived. This makes it possible to show the importance of the transformations. Finally, an empirical example is presented.     

Semi-parametric estimation of simultaneous equations with limited dependent variables: A case study of female labour supply

We propose a general strategy to estimate semi-parametrically simultaneous equations with limited dependent variables. First, each reduced form (RF) is estimated with various semi-parametric methods. Second, the specification of each RF is tested to select an appropriate method. Third, the structural form (SF) equations are estimated using minimum distance methods and the restrictions among the SF and RF parameters. A case study of female labour supply is presented.     

Robustness issues in structural equation modeling: a review of recent developments

In structural equation modeling the statistician needs assumptions inorder (1) to guarantee that the estimates are consistent for the parameters of interest, and (2) to evaluate precision of the estimates and significance level of test statistics. With respect to purpose (1), the typical type of analyses (ML and WLS) are robust against violation of distributional assumptions; i.e., estimates remain consistent or any type of WLS analysis and distribution of z. (It should be noted, however, that (1) is sensitive to structural misspecification.) A typical assumption used for purpose (2), is the assumption that the vector z of observable follows a multivariate normal distribution.In relation to purpose (2), distributional misspecification may have consequences for efficiency, as well as power of test statistics (see Satorra, 1989a); that is, some estimation methods may bemore precise than others for a given specific distribution of z. For instance, ADF-WLS is asymptotically optimal under a variety of distributions of z, while the asymptotic optimality of NT-WLS may be lost when the data is non-normal     

Hierarchical analysis: Classification with ordinal object dissimilarities

Summary Lerman [1970] has demonstrated, that the dissimilarity indices normally used in data analysis are identical up to strictly monotone transformationsf:R ₊R ₊ if the data are nominal and each set of attribute scores is finite.In that case he proposes to use a preorder between pairs of objects to express similarity or dissimilarity, in order to avoid inconsistent classification results that might occur, if clustering schemes which are not monotone invariant are applied to a quantitative index. Here it is shown, how a hierarchy on the objects can be calculated if such a preorder relation is given.     

On finite sample properties of alternative estimators of coefficients in a structural equation with many instruments

We compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases. We have found that the LIML estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large, that is, the micro-econometric models with “many instruments” in the terminology of recent econometric literature.     

Influence of freight transport purchasing processes on logistical variables related to CO2 emissions: a case study in Sweden

Emissions from freight transport stem from logistical variables such as vehicle utilisation, fuel efficiency, and distance. The purpose is to determine how shippers’ freight transport purchasing processes influence logistical variables. A multiple case study of freight transport purchasing processes was conducted, based on interviews with transport purchasers and providers. Three causes of influence of shippers’ purchasing processes on logistical variables were found: specific requirements, network structure of transport providers, and scope of contract. Specifications by purchasers, especially time requirements, influence several logistical variables (‘mode used’, ‘length of haul’, ‘load factor’, ‘empty running’, and ‘fuel efficiency’). This paper clarifies the implications of transport purchasing on CO₂ emissions in terms of logistical variables, which are understood in transportation research and practice. It describes the effects of shippers’ requirements on transport providers’ execution of transport. The results provide a foundation for shippers to discuss their influence on logistical variables with transport providers.     

Just-in-time purchasing: a case study

This article presents a case study on how Hutchinson Technology Incorporated (HTI) organized and implemented just-in-time (JIT) purchasing.     

Nonlinearity measures: a case study

Summary An important problem in applied statistics is fitting a given model function f (β) with unknown parameters β to a data vector y. Minimizing the residual sum of squares provides the least squares estimates of β. If f (β) is linear in β the precision of these estimates is well-known. In a nonlinear case approximate (though asymptotically exact) confidence statements can be made. B eale [1] introduced measures of nonlinearity which can be used to indicate when approximate confidence statements are appropriate. G uttman and M eeter [2] showed that in some, severely nonlinear, cases Beale's measures do not give the right indication. In this paper two new nonlinearity measures are introduced and their use is illustrated on a practical problem described by W itt [3]. A more detailed discussion of the theoretical background can be found in references [1] and [2].     

The informative sample size for dynamic multiple equation systems with moving average errors

Abstract  The problem considered here, is that of finding suitable conditions for dynamic economic systems that exclude the existence of observationally equivalent structures. Here observational equivalence refers to equality of distributions or first and second moments of a small finite sample from the observable process. It is shown, that under these conditions we may act as if the lagged endogenous variables are nonrandom exogenous variables, when global identifiability is investigated.