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.     

A Bootstrap Experiment on the Statistical Properties of Students' Ratings of Teaching Effectiveness

This paper examines the effects of two background variables in students' ratings of teaching effectiveness (SETs): class size and students' motivation (as surrogated by students' likelihood to respond randomly). Resampling simulation methodology has been employed to test the sensitivity of the SET scale for three hypothetical instructors (excellent, average, and poor). In an ideal scenario without confounding factors, SET statistics unmistakably distinguish the instructors. However, at different class sizes and levels of random responses, SET class averages are significantly biased. Results suggest that evaluations based on SET statistics should look at more than class averages. Resampling methodology (bootstrap simulation) is useful for SET research for scale sensitivity study, research results validation, and actual SET score analyses. Examples will be given on how bootstrap simulation can be applied to real-life SET data comparison.     

Measurement and Structural Model Class Separation in Mixture CFA: ML/EM Versus MCMC

Parameter recovery was assessed within mixture confirmatory factor analysis across multiple estimator conditions under different simulated levels of mixture class separation. Mixture class separation was defined in the measurement model (through factor loadings) and the structural model (through factor variances). Maximum likelihood (ML) via the EM algorithm was compared to a Markov chain Monte Carlo (MCMC) estimator condition using weak priors and a condition using tight priors. Results indicated that the MCMC weak condition produced the highest bias, particularly with a weak Dirichlet prior for the mixture class proportions. Specifically, the weak Dirichlet prior affected parameter estimates under all mixture class separation conditions, even with moderate and large sample sizes. With little knowledge about parameters, ML/EM should be used over MCMC weak. However, MCMC tight produced the lowest bias under all mixture class separation conditions and should be used if tight and accurate priors can be placed on parameters.     

Evaluation of a New Mean Scaled and Moment Adjusted Test Statistic for SEM

Recently a new mean scaled and skewness adjusted test statistic was developed for evaluating structural equation models in small samples and with potentially nonnormal data, but this statistic has received only limited evaluation. The performance of this statistic is compared to normal theory maximum likelihood and 2 well-known robust test statistics. A modification to the Satorra–Bentler scaled statistic is developed for the condition that sample size is smaller than degrees of freedom. The behavior of the 4 test statistics is evaluated with a Monte Carlo confirmatory factor analysis study that varies 7 sample sizes and 3 distributional conditions obtained using Headrick's fifth-order transformation to nonnormality. The new statistic performs badly in most conditions except under the normal distribution. The goodness-of-fit χ² test based on maximum-likelihood estimation performed well under normal distributions as well as under a condition of asymptotic robustness. The Satorra–Bentler scaled test statistic performed best overall, whereas the mean scaled and variance adjusted test statistic outperformed the others at small and moderate sample sizes under certain distributional conditions.     

Formulation of a DIMTEST Effect Size Measure (DESM) and Evaluation of the DESM Estimator Bias

DIMTEST is a widely used and studied method for testing the hypothesis of test unidimensionality as represented by local item independence. However, DIMTEST does not report the amount of multidimensionality that exists in data when rejecting its null. To provide more information regarding the degree to which data depart from unidimensionality, a DIMTEST-based Effect Size Measure (DESM) was formulated. In addition to detailing the development of the DESM estimate, the current study describes the theoretical formulation of a DESM parameter. To evaluate the efficacy of the DESM estimator according to test length, sample size, and correlations between dimensions, Monte Carlo simulations were conducted. The results of the simulation study indicated that the DESM estimator converged to its parameter as test length increased, and, as desired, its expected value did not increase with sample size (unlike the DIMTEST statistic in the case of multidimensionality). Also as desired, the standard error of DESM decreased as sample size increased.     

Assessing Dimensionality of a Set of Item Responses-Comparison of Different Approaches

This study compares the performance of three methodologies for assessing unidi-mensionality: DIMTEST, Holland and Rosenbaum's approach, and nonlinear factor analysis. Each method is examined and compared with other methods on simulated and real data sets. Seven data sets, all with 2,000 examinees, were generated: three unidimensional and four two-dimensional data sets. Two levels of correlation between abilities were considered:ρ= 3 and ρ= . 7. Eight different real data sets were used: Four of them were expected to be unidimensional, and the other four were expected to be two-dimensional. Findings suggest that all three methods correctly confirmed unidimensionality but differed in their ability to detect lack of unidimensionality. DIMTEST showed excellent power in detecting lack of unidimensionality; Holland and Rosenbaum's and nonlinear factor analysis approaches showed good power, provided the correlation between abilities was low.     

响应面法优化小球藻粗脂肪提取工艺研究

利用响应面分析法优化小球藻粗脂肪的超声提取工艺,以粗脂肪提取率为评价指标,在考察单因素试验基础上,利用Box-Behnken中心组合试验和响应面分析法确定最佳提取工艺条件.小球藻粗脂肪提取的最佳工艺条件为：液料比为43∶1 ml/g,超声时间为22rmin,超声温度为35℃,在此条件下粗脂肪的提取率为8.27％.     

H-Likelihood Approach to Factor Analysis for Ordinal Data

Marginal likelihood-based methods are commonly used in factor analysis for ordinal data. To obtain the maximum marginal likelihood estimator, the full information maximum likelihood (FIML) estimator uses the (adaptive) Gauss–Hermite quadrature or stochastic approximation. However, the computational burden increases rapidly as the number of factors increases, which renders FIML impractical for large factor models. Another limitation of the marginal likelihood-based approach is that it does not allow inference on the factors. In this study, we propose a hierarchical likelihood approach using the Laplace approximation that remains computationally efficient in large models. We also proposed confidence intervals for factors, which maintains the level of confidence as the sample size increases. The simulation study shows that the proposed approach generally works well.     

Deciding on the Number of Classes in Latent Class Analysis and Growth Mixture Modeling: A Monte Carlo Simulation Study

Mixture modeling is a widely applied data analysis technique used to identify unobserved heterogeneity in a population. Despite mixture models' usefulness in practice, one unresolved issue in the application of mixture models is that there is not one commonly accepted statistical indicator for deciding on the number of classes in a study population. This article presents the results of a simulation study that examines the performance of likelihood-based tests and the traditionally used Information Criterion (ICs) used for determining the number of classes in mixture modeling. We look at the performance of these tests and indexes for 3 types of mixture models: latent class analysis (LCA), a factor mixture model (FMA), and a growth mixture models (GMM). We evaluate the ability of the tests and indexes to correctly identify the number of classes at three different sample sizes (n = 200, 500, 1,000). Whereas the Bayesian Information Criterion performed the best of the ICs, the bootstrap likelihood ratio test proved to be a very consistent indicator of classes across all of the models considered.     

On the Likelihood Ratio Test for the Number of Factors in Exploratory Factor Analysis

In the exploratory factor analysis, when the number of factors exceeds the true number of factors, the likelihood ratio test statistic no longer follows the chi-square distribution due to a problem of rank deficiency and nonidentifiability of model parameters. As a result, decisions regarding the number of factors may be incorrect. Several researchers have pointed out this phenomenon, but it is not well known among applied researchers who use exploratory factor analysis. We demonstrate that overfactoring is one cause for the well-known fact that the likelihood ratio test tends to find too many factors.     

Standardized Parameters in Misspecified Structural Equation Models: Empirical Performance in Point Estimates,Standard Errors,and Confidence Intervals

In practice, models always have misfit, and it is not well known in what situations methods that provide point estimates, standard errors (SEs), or confidence intervals (CIs) of standardized structural equation modeling (SEM) parameters are trustworthy. In this article we carried out simulations to evaluate the empirical performance of currently available methods. We studied maximum likelihood point estimates, as well as SE estimators based on the delta method, nonparametric bootstrap (NP-B), and semiparametric bootstrap (SP-B). For CIs we studied Wald CI based on delta, and percentile and BC_a intervals based on NP-B and SP-B. We conducted simulation studies using both confirmatory factor analysis and SEM models. Depending on (a) whether point estimate, SE, or CI is of interest; (b) amount of model misfit; (c) sample size; and (d) model complexity, different methods can be the one that renders best performance. Based on the simulation results, we discuss how to choose proper methods in practice.     

Information Functions of Rank-2PL Models for Forced-Choice Questionnaires

This paper presents the item and test information functions of the Rank two-parameter logistic models (Rank-2PLM) for items with two (pair) and three (triplet) statements in forced-choice questionnaires. The Rank-2PLM model for pairs is the MUPP-2PLM (Multi-Unidimensional Pairwise Preference) and, for triplets, is the Triplet-2PLM. Fisher's information and directional information are described, and the test information for Maximum Likelihood (ML), Maximum A Posterior (MAP), and Expected A Posterior (EAP) trait score estimates is distinguished. Expected item/test information indexes at various levels are proposed and plotted to provide diagnostic information on items and tests. The expected test information indexes for EAP scores may be difficult to compute due to a typical test's vast number of item response patterns. The relationships of item/test information with discrimination parameters of statements, standard error, and reliability estimates of trait score estimates are discussed and demonstrated using real data. Practical suggestions for checking the various expected item/test information indexes and plots are provided.     

基于似然比统计量的正态总体的检验

假设检验问题的关键是在一定的统计思想之下构造相关的统计量。极大似然思想是统计和实际应用中的一种重要思想。基于极大似然原理,构造似然比统计量,讨论正态总体的检验问题,其结果与传统的U检验、T检验、χ2检验一致。     

基于似然比统计量的正态总体的检验

假设检验问题的关键是在一定的统计思想之下构造相关的统计量。极大似然思想是统计和实际应用中的一种重要思想。基于极大似然原理,构造似然比统计量,讨论正态总体的检验问题,其结果与传统的U检验、T检验、χ2检验一致。     

Bootstrapping Confidence Intervals for Fit Indexes in Structural Equation Modeling

Bootstrapping approximate fit indexes in structural equation modeling (SEM) is of great importance because most fit indexes do not have tractable analytic distributions. Model-based bootstrap, which has been proposed to obtain the distribution of the model chi-square statistic under the null hypothesis (Bollen & Stine, 1992), is not theoretically appropriate for obtaining confidence intervals (CIs) for fit indexes because it assumes the null is exactly true. On the other hand, naive bootstrap is not expected to work well for those fit indexes that are based on the chi-square statistic, such as the root mean square error of approximation (RMSEA) and the comparative fit index (CFI), because sample noncentrality is a biased estimate of the population noncentrality. In this article we argue that a recently proposed bootstrap approach due to Yuan, Hayashi, and Yanagihara (YHY; 2007) is ideal for bootstrapping fit indexes that are based on the chi-square. This method transforms the data so that the “parent” population has the population noncentrality parameter equal to the estimated noncentrality in the original sample. We conducted a simulation study to evaluate the performance of the YHY bootstrap and the naive bootstrap for 4 indexes: RMSEA, CFI, goodness-of-fit index (GFI), and standardized root mean square residual (SRMR). We found that for RMSEA and CFI, the CIs under the YHY bootstrap had relatively good coverage rates for all conditions, whereas the CIs under the naive bootstrap had very low coverage rates when the fitted model had large degrees of freedom. However, for GFI and SRMR, the CIs under both bootstrap methods had poor coverage rates in most conditions.     

Empirical Validation of DIMTEST on Nonnormal Ability Distributions

DIMTEST is a nonparametric statistical test procedure for assessing unidimensionality of binary item response data. The development of Stout's statistic, T, used in the DIMTEST procedure, does not require the assumption of a particular parametric form for the ability distributions or the item response functions. The purpose of the present study was to empirically investigate the performance of the statistic T with respect to different shapes of ability distributions. Several nonnormal distributions, both symmetric and nonsymmetric, were considered for this purpose. Other factors varied in the study were test length, sample size, and the level of correlation between abilities. The results of Type I error and power studies showed that the test statistic T exhibited consistently similar performance for all different shapes of ability distributions investigated in the study, which confirmed the nonparametric nature of the statistic T.     

Power and Precision in Confirmatory Factor Analytic Tests of Measurement Invariance

This study investigates the effects of sample size, factor overdetermination, and communality on the precision of factor loading estimates and the power of the likelihood ratio test of factorial invariance in multigroup confirmatory factor analysis. Although sample sizes are typically thought to be the primary determinant of precision and power, the degree of factor overdetermination and the level of indicator communalities also play important roles. Based on these findings, no single rule of thumb regarding the ratio of sample size to number of indicators can ensure adequate power to detect a lack of measurement invariance.     

A SINGLE-TONE SERIAL HF MODEM SCHEME

A HF receiver for data transmission based on Kalman filter and channelestimator is proposed.The simulation results show that the performance of the proposedscheme is about 2 dB better than that of Decision-Feedback Equalizer based onSquare-Root Kalman Algorithm(SRKA/DFE)and its computational complexity islower than that of Maximum Likelihood Sequence Estimation(MLSE).     

Statistical Power to Detect the Correct Number of Classes in Latent Profile Analysis

Little research has examined factors influencing statistical power to detect the correct number of latent classes using latent profile analysis (LPA). This simulation study examined power related to interclass distance between latent classes given true number of classes, sample size, and number of indicators. Seven model selection methods were evaluated. None had adequate power to select the correct number of classes with a small (Cohen's d = .2) or medium (d = .5) degree of separation. With a very large degree of separation (d = 1.5), the Lo–Mendell–Rubin test (LMR), adjusted LMR, bootstrap likelihood ratio test, Bayesian Information Criterion (BIC), and sample-size-adjusted BIC were good at selecting the correct number of classes. However, with a large degree of separation (d = .8), power depended on number of indicators and sample size. Akaike's Information Criterion and entropy poorly selected the correct number of classes, regardless of degree of separation, number of indicators, or sample size.