Bootstrap‐corrected ADF test statistics in covariance structure analysis

The asymptotically distribution free (ADF) method is often used to estimate parameters or test models without a normal distribution assumption on variables, both in covariance structure analysis and in correlation structure analysis. However, little has been done to study the differences in behaviors of the ADF method in covariance versus correlation structure analysis. The behaviors of 3 test statistics frequently used to evaluate structural equation models with nonnormally distributed variables, χ2 test TAGLS and its small-sample variants TYB and TF(AGLS) were compared. Results showed that the ADF method in correlation structure analysis with test statistic TAGLS performs much better at small sample sizes than the corresponding test for covariance structures. In contrast, test statistics TYB and TF(AGLS) under the same conditions generally perform better with covariance structures than with correlation structures. It is proposed that excessively large and variable condition numbers of weight matrices are a cause of poor behavior of ADF test statistics in small samples, and results showed that these condition numbers are systematically increased with substantial increase in variance as sample size decreases. Implications for research and practice are discussed.

In this chapter, we bring together the notions of measurement error discussed in Chaps. 3 and 4 with the structural modeling of simultaneous relationships presented in Chap. 6. We demonstrate that a bias is introduced when estimating the relationship between two variables measured with error if that measurement error is ignored. We then present a methodology for estimating the parameters of structural relationships between variables that are not observed directly: analysis of covariance structures. We focus on the role of the measurement model as discussed in Chap. 4 with the confirmatory factor analytic model.

Even though data sets in psychology are seldom normal, the statistics used to evaluate covariance structure models are typically based on the assumption of multivariate normality. Consequently, many conclusions based on normal theory methods are suspect. In this paper, we develop test statistics that can be correctly applied to the normal theory maximum likelihood estimator. We propose three new asymptotically distribution-free (ADF) test statistics that technically must yield improved behaviour in samples of realistic size, and use Monte Carlo methods to study their actual finite sample behaviour. Results indicate that there exists an ADF test statistic that also performs quite well in finite sample situations. Our analysis shows that various forms of ADF test statistics are sensitive to model degrees of freedom rather than to model complexity. A new index is proposed for evaluating whether a rescaled statistic will be robust. Recommendations are given regarding the application of each test statistic.

Research studying robustness of maximum likelihood (ML) statistics in covariance structure analysis has concluded that test statistics and standard errors are biased under severe non-normality. An estimation procedure known as asymptotic distribution free (ADF), making no distributional assumption, has been suggested to avoid these biases. Corrections to the normal theory statistics to yield more adequate performance have also been proposed. This study compares the performance of a scaled test statistic and robust standard errors for two models under several non-normal conditions and also compares these with the results from ML and ADF methods. Both ML and ADF test statistics performed rather well in one model and considerably worse in the other. In general, the scaled test statistic seemed to behave better than the ML test statistic and the ADF statistic performed the worst. The robust and ADF standard errors yielded more appropriate estimates of sampling variability than the ML standard errors, which were usually downward biased, in both models under most of the non-normal conditions. ML test statistics and standard errors were found to be quite robust to the violation of the normality assumption when data had either symmetric and platykurtic distributions, or non-symmetric and zero kurtotic distributions.

The two-sample problem for Cronbach's coefficient [Formula: see text], as an estimate of test or composite score reliability, has attracted little attention compared to the extensive treatment of the one-sample case. It is necessary to compare the reliability of a test for different subgroups, for different tests or the short and long forms of a test. In this paper, we study statistical procedures of comparing two coefficients [Formula: see text] and [Formula: see text]. The null hypothesis of interest is [Formula: see text], which we test against one-or two-sided alternatives. For this purpose, resampling-based permutation and bootstrap tests are proposed for two-group multivariate non-normal models under the general asymptotically distribution-free (ADF) setting. These statistical tests ensure a better control of the type-I error, in finite or very small sample sizes, when the state-of-affairs ADF large-sample test may fail to properly attain the nominal significance level. By proper choice of a studentized test statistic, the resampling tests are modified in order to be valid asymptotically even in non-exchangeable data frameworks. Moreover, extensions of this approach to other designs and reliability measures are discussed as well. Finally, the usefulness of the proposed resampling-based testing strategies is demonstrated in an extensive simulation study and illustrated by real data applications.

This article proposes a new general methodology for constructing nonparametric and semiparametric Asymptotically Distribution-Free (ADF) tests for semiparametric hypotheses in regression models for possibly dependent data coming from a strictly stationary process. Classical tests based on the difference between the estimated distributions of the restricted and unrestricted regression errors are not ADF. In this article, we introduce a novel transformation of this difference that leads to ADF tests with well-known critical values. The general methodology is illustrated with applications to testing for parametric models against nonparametric or semiparametric alternatives, and semiparametric constrained mean–variance models. Several Monte Carlo studies and an empirical application show that the finite sample performance of the proposed tests is satisfactory in moderate sample sizes.

The point estimate of sample coefficient alpha may provide a misleading impression of the reliability of the test score. Because sample coefficient alpha is consistently biased downward, it is more likely to yield a misleading impression of poor reliability. The magnitude of the bias is greatest precisely when the variability of sample alpha is greatest (small population reliability and small sample size). Taking into account the variability of sample alpha with an interval estimator may lead to retaining reliable tests that would be otherwise rejected. Here, the authors performed simulation studies to investigate the behavior of asymptotically distribution-free (ADF) versus normal-theory interval estimators of coefficient alpha under varied conditions. Normal-theory intervals were found to be less accurate when item skewness >1 or excess kurtosis >1. For sample sizes over 100 observations, ADF intervals are preferable, regardless of item skewness and kurtosis. A formula for computing ADF confidence intervals for coefficient alpha for tests of any size is provided, along with its implementation as an SAS macro.

Asymptotically distribution free (ADF) type test statistics play an important role in covariance structure analysis with nonnormal data. However, empirical studies have indicated that ADF-type test statistics rejected the correct models too frequently at all but very large sample sizes. This phenomenon may similarly occur for tests involving nested models. Several statistics related to ADF-type nested tests are given. The new test statistics have the same asymptotic properties as the original ADF-type test statistics. When used to evaluate model restrictions with small sample sizes, our statistics are more conservative in rejecting correct models. Numerical comparison of critical values are given for selected sample sizes and model parameter differences.

Covariance structure analysis is used to evaluate hypothesized influences among unmeasured latent and observed variables. As implemented, it is not robust to outliers and bad data. Several robust methods in model fitting and testing are proposed. These include direct estimation of M-estimators of structured parameters and a two-stage procedure based on robust M- and S-estimators of population covariances. The large sample properties of these estimators are obtained. The equivalence between a direct M-estimator and a two-stage estimator based on an M-estimator of population covariance is established when sampling from an elliptical distribution. Two test statistics are presented in judging the adequacy of a hypothesized model; both are asymptotically distribution free if using distribution free weight matrices. So these test statistics possess both finite sample and large sample robustness. The two-stage procedures can be easily adapted into standard software packages by modifying existing asymptotically distribution free procedures. To demonstrate the two-stage procedure, S-estimator and M-estimators under different weight functions are calculated for some real data sets.

International convention on Standards of Training, Certification and Watch keeping for seafarers (STCW) was amended in Manila, the Philippines in 2010. The amendments will come into force on 1 January 2017. To meet the competence requirements for marine engineers, the implementation of Engine-room Resource Management training (hereafter, ERM training) was adopted as mandatory. The role of a practical ERM training model has become more important for marine engineers working in a complex system. The purpose of this study is to examine which ERM training best suits engine-room simulation education. First, in this paper, the characteristics of total human errors were investigated using 298 marine accident records from the Japan Marine Accident Inquiry. Second, the common factors of these accident records, namely the latent factors, were determined using covariance structure analysis. Furthermore, System Dynamics (SD) was employed to find the relationships between the latent factors and the technical support to gather the clues for a practical ERM training method. The model includes a data set of actual cases from records of marine accident inquiries and a hypothetical data set. The results show that the support of the onshore management and behavior characteristics of engineers which influence each other in a delicate balance could be expressed in a chronological order.

Abstract. The Maslach Burnout Inventory (MBI) is an instrument which has been used worldwide to assess occupational stress in human service professions. While the MBI has been used in school settings, an inventory specifically oriented to measure burnout in teachers was developed (MBI-Form Ed; Schwab, in Maslach & Jackson, 1986 ). A Catalan version of the MBI-Form Ed was administered to 631 elementary school teachers. Several models, including those posited by some studies using the English language MBI-Form Ed, were analyzed in order to test their cross-cultural validity. Confirmatory factor analyses (CFAs) were used to fit all models by maximum likelihood (ML) and asymptotically distribution-free (ADF) estimation methods to test the hypothesis that incorrect use of the ML estimation method could be the reason for model misfit in previous studies. Results show that there are no differences due to the estimation method and support the three-factor structure. Furthermore, a shortened three-factor model of 17 items shows a better fit and similar reliability, and its use seems plausible in the research and psychological evaluation of teachers' burnout in both English and Catalan language settings.

Traditionally, two approaches have been employed for structural equation modeling: covariance structure analysis and partial least squares. A third alternative, generalized structured component analysis, was introduced recently in the psychometric literature. The authors conduct a simulation study to evaluate the relative performance of these three approaches in terms of parameter recovery under different experimental conditions of sample size, data distribution, and model specification. In this study, model specification is the only meaningful condition in differentiating the performance of the three approaches in parameter recovery. Specifically, when the model is correctly specified, covariance structure analysis tends to recover parameters better than the other two approaches. Conversely, when the model is misspecified, generalized structured component analysis tends to recover parameters better. Finally, partial least squares exhibits inferior performance in parameter recovery compared with the other approaches. In particular, this tendency is salient when the model involves cross-loadings. Thus, generalized structured component analysis may be a good alternative to partial least squares for structural equation modeling and is recommended over covariance structure analysis unless correct model specification is ensured.

A major issue in the utilization of covariance structure analysis is model fit evaluation. Recent years have witnessed increasing interest in various test statistics and so-called fit indexes, most of which are actually based on or closely related to F 0, a measure of model fit in the population. This study aims to provide a systematic investigation about the performance of 4 available estimators of F 0. [Fcirc]01 is the conventional estimator and is based on noncentral chi-square approximation. [Fcirc]02 is newly proposed and does not assume noncentral chi-square approximation. [Fcirc]03 and [Fcirc]04 are variations of [Fcirc]02. A Monte Carlo simulation study is conducted to examine how these four estimators of F 0 perform across varying model misspecifications, data distributions, model sizes, and sample sizes. The results show that under normality all 4 quantities estimate F 0 equally well, and under nonnormality [Fcirc]02, [Fcirc]03, and [Fcirc]04 outperform [Fcirc]01. Issues related to these findings are discussed.

Working within the framework of a confirmatory factor analytic (CFA) model, this study adds another dimension to construct validation of both the Beck Depression Inventory-II (BDI-II; Beck, Steer, & Brown, 1996) and a Chinese version of the BDI-II (C-BDI-II; Chinese Behavioral Sciences Society, 2000). Specifically, we tested for measurement equivalence of the C-BDI-II with the original BDI-II across Hong Kong (N = 1771) and American (N = 501) adolescents, respectively. Provided with evidence of measurement equivalence, we then tested for differences in the latent factor means (i.e., subscale levels) of three first-order factors of negative attitude, performance difficulty, and somatic elements and one second-order factor of general depression. All procedures were based on analyses of mean and covariance structures (MACS) that took into account both the incompleteness and non-normality of the data. Findings revealed sound evidence of measurement equivalence of the C-BDI-II and BDI-II factorial structures, and latent factor means that pointed to higher levels of depressive symptoms for Hong Kong adolescents than for their American counterparts.

We propose an additive hazards model with latent variables to investigate the observed and latent risk factors of the failure time of interest. Each latent risk factor is characterized by correlated observed variables through a confirmatory factor analysis model. We develop a hybrid procedure that combines the expectation–maximization (EM) algorithm and the borrow-strength estimation approach to estimate the model parameters. We establish the consistency and asymptotic normality of the parameter estimators. Various nice features, including finite sample performance of the proposed methodology, are demonstrated by simulation studies. Our model is applied to a study concerning the risk factors of chronic kidney disease for Type 2 diabetic patients. Supplementary materials for this article are available online.