Are Robust Standard Errors the Best Approach for Interval Estimation With Nonnormal Data in Structural Equation Modeling?

When the multivariate normality assumption is violated in structural equation modeling, a leading remedy involves estimation via normal theory maximum likelihood with robust corrections to standard errors. We propose that this approach might not be best for forming confidence intervals for quantities with sampling distributions that are slow to approach normality, or for functions of model parameters. We implement and study a robust analog to likelihood-based confidence intervals based on inverting the robust chi-square difference test of Satorra (2000). We compare robust standard errors and the robust likelihood-based approach versus resampling methods in confirmatory factor analysis (Studies 1 & 2) and mediation analysis models (Study 3) for both single parameters and functions of model parameters, and under a variety of nonnormal data generation conditions. The percentile bootstrap emerged as the method with the best calibrated coverage rates and should be preferred if resampling is possible, followed by the robust likelihood-based approach.     

Performance of Bootstrapping Approaches to Model Test Statistics and Parameter Standard Error Estimation in Structural Equation Modeling

Though the common default maximum likelihood estimator used in structural equation modeling is predicated on the assumption of multivariate normality, applied researchers often find themselves with data clearly violating this assumption and without sufficient sample size to utilize distribution-free estimation methods. Fortunately, promising alternatives are being integrated into popular software packages. Bootstrap resampling, which is offered in AMOS (Arbuckle, 1997), is one potential solution for estimating model test statistic p values and parameter standard errors under nonnormal data conditions. This study is an evaluation of the bootstrap method under varied conditions of nonnormality, sample size, model specification, and number of bootstrap samples drawn from the resampling space. Accuracy of the test statistic p values is evaluated in terms of model rejection rates, whereas accuracy of bootstrap standard error estimates takes the form of bias and variability of the standard error estimates themselves.     

Examining Normality in Incomplete Data Sets: A Note on a Multiple Testing Approach

A multiple testing procedure for examining the assumption of normality that is often made in analyses of incomplete data sets is outlined. The method is concerned with testing normality within each missingness pattern and arriving at an overall statement about normality using the available data. The approach is readily applied in empirical research with missing data using the popular software Mplus, Stata, and R. The procedure can be used to ascertain a main assumption underlying frequent applications of maximum likelihood in incomplete data modeling with continuous outcomes. The discussed approach is illustrated with numerical examples.     

Mapping of quantitative trait loci using the skew-normal distribution

In standard interval mapping (IM) of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. When this assumption of normality is violated, the most commonly adopted strategy is to use the previous model after data transformation. However, an appropriate transformation may not exist or may be difficult to find. Also this approach can raise interpretation issues. An interesting alternative is to consider a skew-normal mixture model in standard IM, and the resulting method is here denoted as skew-normal IM. This flexible model that includes the usual symmetric normal distribution as a special case is important, allowing continuous variation from normality to non-normality. In this paper we briefly introduce the main peculiarities of the skew-normal distribution. The maximum likelihood estimates of parameters of the skew-normal distribution are obtained by the expectation-maximization (EM) algorithm. The proposed model is illustrated with real data from an intercross experiment that shows a significant departure from the normality assumption. The performance of the skew-normal IM is assessed via stochastic simulation. The results indicate that the skew-normal IM has higher power for QTL detection and better precision of QTL location as compared to standard IM and nonparametric IM.     

Four New Corrected Statistics for SEM With Small Samples and Nonnormally Distributed Data

When the assumption of multivariate normality is violated and the sample sizes are relatively small, existing test statistics such as the likelihood ratio statistic and Satorra–Bentler’s rescaled and adjusted statistics often fail to provide reliable assessment of overall model fit. This article proposes four new corrected statistics, aiming for better model evaluation with nonnormally distributed data at small sample sizes. A Monte Carlo study is conducted to compare the performances of the four corrected statistics against those of existing statistics regarding Type I error rate. Results show that the performances of the four new statistics are relatively stable compared with those of existing statistics. In particular, Type I error rates of a new statistic are close to the nominal level across all sample sizes under a condition of asymptotic robustness. Other new statistics also exhibit improved Type I error control, especially with nonnormally distributed data at small sample sizes.     

Measuring ambivalence to science

Ambivalence is a psychological state in which a person holds mixed feelings (positive and negative) towards some psychological object. Standard methods of attitude measurement, such as Likert and semantic differential scales, ignore the possibility of ambivalence; ambivalent responses cannot be distinguished from neutral ones. This neglect arises out of an assumption that positive and negative affects towards a particular psychological object are bipolar, i.e., unidimensional in opposite directions. This assumption is frequently untenable. Conventional item statistics and measures of test internal consistency are ineffective as checks on this assumption; it is possible for a scale to be multidimensional and still display apparent internal consistency. Factor analysis is a more effective procedure. Methods of measuring ambivalence are suggested, and implications for research are discussed.     

Standard Error of Linear Observed‐Score Equating for the NEAT Design With Nonnormally Distributed Data

In the nonequivalent groups with anchor test (NEAT) design, the standard error of linear observed‐score equating is commonly estimated by an estimator derived assuming multivariate normality. However, real data are seldom normally distributed, causing this normal estimator to be inconsistent. A general estimator, which does not rely on the normality assumption, would be preferred, because it is asymptotically accurate regardless of the distribution of the data. In this article, an analytical formula for the standard error of linear observed‐score equating, which characterizes the effect of nonnormality, is obtained under elliptical distributions. Using three large‐scale real data sets as the populations, resampling studies are conducted to empirically evaluate the normal and general estimators of the standard error of linear observed‐score equating. The effect of sample size (50, 100, 250, or 500) and equating method (chained linear, Tucker, or Levine observed‐score equating) are examined. Results suggest that the general estimator has smaller bias than the normal estimator in all 36 conditions; it has larger standard error when the sample size is at least 100; and it has smaller root mean squared error in all but one condition. An R program is also provided to facilitate the use of the general estimator.     

Growth Mixture Modeling With Nonnormal Distributions: Implications for Data Transformation

This study investigated the extent to which class-specific parameter estimates are biased by the within-class normality assumption in nonnormal growth mixture modeling (GMM). Monte Carlo simulations for nonnormal GMM were conducted to analyze and compare two strategies for obtaining unbiased parameter estimates: relaxing the within-class normality assumption and using data transformation on repeated measures. Based on unconditional GMM with two latent trajectories, data were generated under different sample sizes (300, 800, and 1500), skewness (0.7, 1.2, and 1.6) and kurtosis (2 and 4) of outcomes, numbers of time points (4 and 8), and class proportions (0.5:0.5 and 0.25:0.75). Of the four distributions, it was found that skew-t GMM had the highest accuracy in terms of parameter estimation. In GMM based on data transformations, the adjusted logarithmic method was more effective in obtaining unbiased parameter estimates than the use of van der Waerden quantile normal scores. Even though adjusted logarithmic transformation in nonnormal GMM reduced computation time, skew-t GMM produced much more accurate estimation and was more robust over a range of simulation conditions. This study is significant in that it considers different levels of kurtosis and class proportions, which has not been investigated in depth in previous studies. The present study is also meaningful in that investigated the applicability of data transformation to nonnormal GMM.     

A Study Of The Relationship Between Grade And Age And Variability

Power and stability of Type I error rates are investigated for the Box-Scheffé test of homogeneity of variance with varying subsample sizes under conditions of normality and nonnormality. The test is shown to be robust to violation of the normality assumption when sampling is from a leptokurtic population. Subsample sizes which produce maximum power are given for small, intermediate, and large sample situations. Suggestions for selecting subsample sizes which will produce maximum power for a given n are provided. A formula for estimating power in the equal n case is shown to give results agreeing with empirical results.     

A systematic procedure for constructing a valid microcomputer attitude scale

The purpose of this study was twofold: (1) to design a system for constructing Likert attitude scales as supported by the sociopsychological and measurement literature, and (2) using the design to assemble a microcomputer attitude scale for inservice and preservice teachers (n = 281). The results of the study: (1) a 15-step flow chart for designing reliable and valid attitude scales, and (2) a 23-item microcomputer Likert attitude scale with the following characteristics: (a) coefficient alpha 0.89, (b) range of adjusted item-total correlations from 0.29 to 0.62, (c) range of interitem correlations from 0.04 to 0.60, (d) correlation of 0.20 with a mathematics attitude scale and 0.02 with a reading attitude scale, and (e) favorable factor analysis and emotional intensity data.     

A Simple Analytic Confidence Interval for CFI Given Nonnormal Data

The comparative fit index (CFI) is one of the most widely-used fit indices in structural equation modeling (SEM). When applying the CFI to model evaluation, although it is universally recognized that the focus should be the population fit, in practice one often considers only the CFI value within a sample and neglects the uncertainty in point estimation. Confidence interval (CI) methods for CFI appeared only recently, but these methods assume multivariate normality, which often fails to hold in practice. In addition, the current methods are applications of the bootstrap and are thus computationally intensive. To better handle nonnormal data and simplify CI construction, in this paper we propose an analytic CI method for CFI without assuming normality. We then carry out simulation studies to compare the new and current methods at various levels of model misfit and nonnormality. Simulation results verify the effectiveness and advantages of the new method.     

Portfolio Selection Model with Derivative Securities

Traditional portfolio theory assumes that the return rate of portfolio follows normality ,However,this assumption is not true when derivative assets are incorporated,In this paper a portfolio selection model is devel-oped based on utility function which can captue asymmetries in random variable distributions.Other realistic conditions are also considered ,such as liabilities and integer decision variables,Since the resulting model is a complex mixed-integer nonlinear programming problem ,simulated annealing algorithm is applied for its solution.A numerical example is given and sensitivity analysis is conducted for the model.     

A Scaled F Distribution as an Approximation to the Distribution of Test Statistics in Covariance Structure Analysis

When the assumption of multivariate normality is violated or when a discrepancy function other than (normal theory) maximum likelihood is used in structural equation models, the null distribution of the test statistic may not be χ² distributed. Most existing methods to approximate this distribution only match up to 2 moments. In this article, we propose 2 additional approximation methods: a scaled F distribution that matches 3 moments simultaneously and a direct Monte Carlo–based weighted sum of i.i.d. χ² variates. We also conduct comprehensive simulation studies to compare the new and existing methods for both maximum likelihood and nonmaximum likelihood discrepancy functions and to separately evaluate the effect of sampling uncertainty in the estimated weights of the weighted sum on the performance of the approximation methods.     

Development and validation of an instrument for assessing attitudes of high school students about recycling

Recycling and its applications are growing significantly due to the great potential for solving a range of environmental problems in society. Nevertheless, there are currently very few instruments that can provide valid and reliable data on students’ attitudes toward recycling. In this regard, this article focuses on the development and validation of Recycling Attitude Scale (RAS). The items in the RAS were developed initially from the responses to three open-ended items by 53 tenth and eleventh grade students and literature review on recycling attitude. This initial form was pilot tested with 356 tenth and eleventh grade students and then subjected to exploratory factor analysis. Subsequently, the revised version of the scale was administrated to 694 tenth grade students, and the results were subjected to confirmatory factor analysis and reliability analysis. The RAS consists of 21 items in three subscales, with responses recorded on a four-point Likert scale, options ranging from strongly agree to strongly disagree. Cronbach’s alpha reliability coefficient (α) of the scale was found to be .87. The results indicate that the RAS a potentially valuable tool for both instructors and researchers in Turkey for the assessment of the attitudes toward recycling held by students in secondary education.     

Designing a user‐friendly microcomputer‐based laboratory package through the factor analysis of teacher evaluations

In this study we analyse how the experiences of chemistry teachers on the use of a Microcomputer‐Based Laboratory (MBL), gathered by a Likert‐scale instrument, can be utilized to develop the new package Empirica 2000. We used exploratory factor analysis to identify the essential features in a large set of questionnaire data to see how our previous MBL package, Empirica for Windows 4.0, looks from the point of view of experienced chemistry teachers. Together, a six‐factor solution explained 50.1% of the common variance and indicated the teachers' perspective on the use of a MBL package in chemical education. The factors were: ‘Versatility of the tool’, ‘User interface’, ‘Data presentation’, ‘Data acquisition’, ‘Set up’, and ‘Usability’. Based on the data, some conclusions concerning the software development and desired new features in the prototype software are discussed in the framework of each factor.     

An Analysis of the New York State Regents Examinations in Chemistry from 1927 to 1934

Gibbons and Chakraborti's (1991) interpretation of recent simulation results and their recommendations to researchers are misleading in some respects. The present note emphasizes that the Mann-Whitney test is not a suitable replacement of the Student t test when variances and sample sizes are unequal, irrespective of whether the assumption of normality is satisfied or violated. When both normality and homogeneity of variance are violated together, an effective procedure, not widely known to researchers in education and psychology, is the Fligner-Policello test or, alternatively, the Welch t' test in conjunction with transformation of the original scores to ranks.     

Improving the Root Mean Square Error of Approximation for Nonnormal Conditions in Structural Equation Modeling

In this study, the authors investigated incorporating adjusted model fit information into the root mean square error of approximation (RMSEA) fit index. Through Monte Carlo simulation, the usefulness of this adjusted index was evaluated for assessing model adequacy in structural equation modeling when the multivariate normality assumption underlying maximum likelihood estimation is violated. Adjustment to the RMSEA was considered in 2 forms: a rescaling adjustment via the Satorra-Bentler rescaled goodness-of-fit statistic and a bootstrap adjustment via the Bollen and Stine adjusted model p value. Both properly specified and misspecifed models were examined. The adjusted RMSEA was evaluated in terms of the average index value across study conditions and with respect to model rejection rates under tests of exact fit, close fit, and not-close fit.     

Approximations to the Distributions of Fit Indexes for Misspecified Structural Equation Models

Approximations to the distributions of goodness-of-fit indexes in structural equation modeling are derived with the assumption of multivariate normality and slight misspecification of models. The fit indexes considered in this article are Joreskog and Sorbom's goodness-of-fit index (GFI) and the adjusted GFI, McDonald's absolute GFI, Steiger and Lind's root mean squared error of approximation, Steiger's Γ₁ and Γ₂, Bentler and Bonett's normed fit index, Bollen's incremental fit index and ρ₁, Tucker and Lewis's index ρ₂, and Bentler's fit index (McDonald and Marsh's relative noncentrality index). An approximation to the asymptotic covariance matrix for the fit indexes is derived by using the delta method. Furthermore, approximations to the densities of the fit indexes are obtained from the transformations of the asymptotically noncentral chi-square distributed variable. A simulation is carried out to confirm the accuracy of the approximations.     

Transformation Structural Equation Models With Highly Nonnormal and Incomplete Data

As useful multivariate techniques, structural equation models have attracted significant attention from various fields. Most existing statistical methods and software for analyzing structural equation models have been developed based on the assumption that the response variables are normally distributed. Several recently developed methods can partially address violations of this assumption, but still encounter difficulties in analyzing highly nonnormal data. Moreover, the presence of missing data is a practical issue in substantive research. Simply ignoring missing data or improperly treating nonignorable missingness as ignorable could seriously distort statistical influence results. The main objective of this article is to develop a Bayesian approach for analyzing transformation structural equation models with highly nonnormal and missing data. Different types of missingness are discussed and selected via the deviance information criterion. The empirical performance of our method is examined via simulation studies. Application to a study concerning people’s job satisfaction, home life, and work attitude is presented.     

Ordinal Factor Analysis of Graded-Preference Questionnaire Data

We introduce a new comparative response format, suitable for assessing personality and similar constructs. In this “graded-block” format, items measuring different constructs are first organized in blocks of 2 or more; then, pairs are formed from items within blocks. The pairs are presented 1 at a time to enable respondents expressing the extent of preference for 1 item or the other using several graded categories. We model such data using confirmatory factor analysis (CFA) for ordinal outcomes. We derive Fisher information matrices for the graded pairs, and supply R code to enable computation of standard errors of trait scores. An empirical example illustrates the approach in low-stakes personality assessments and shows that similar results are obtained when using graded blocks of size 3 and a standard Likert format. However, graded-block designs might be superior when insufficient differentiation between items is expected (due to acquiescence, halo, or social desirability).