认知诊断模型的标准误与置信区间估计：并行自助法

认知诊断模型的标准误(Standard Error, SE; 或方差—协方差矩阵)与置信区间(Confidence Interval, CI)在模型参数估计不确定性的度量、项目功能差异检验、项目水平上的模型比较、Q矩阵检验以及探索属性层级关系等领域有重要的理论与实践价值。本研究提出了两种新的SE和CI计算方法：并行参数化自助法和并行非参数化自助法。模拟研究发现：模型完全正确设定时, 在高质量及中等质量项目条件下, 这两种方法在计算模型参数的SE和CI时均有好的表现; 模型参数存在冗余时, 在高质量及中等质量项目条件下, 对于大部分允许存在的模型参数而言, 其SE和CI有好的表现。通过实证数据展示了新方法的价值及计算效率提升效果。     

Equivalent models in covariance structure analysis

Defining equivalent models as those that reproduce the same set of covariance matrices, necessary and sufficient conditions are stated for the local equivalence of two expanded identified modelsM ₁ andM ₂ when fitting the more restricted modelM ₀. Assuming several regularity conditions, the rank deficiency of the Jacobian matrix, composed of derivatives of the covariance elements with respect to the union of the free parameters ofM ₁ andM ₂ (which characterizes modelM ₁₂), is a necessary and sufficient condition for the local equivalence ofM ₁ andM ₂. This condition is satisfied, in practice, when the analysis dealing with the fitting ofM ₀, predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting ofM ₁ orM ₂, orM ₁₂ are all equal for any set of sample data, except on differences due to rounding errors.This research was supported by the Foundation of Social-Cultural Sciences which is subsidized by the Dutch Scientific Organization (N.W.O.) under project number 500-278-003. The author wishes to thank Anne Boomsma, Ivo Molenaar, Albert Satorra, and Tom Snijders for their stimulating and crucial comments during the research, and the Editor, Paul Bekker, Henk Broer, and anonymous reviewers for their helpful suggestions.     

Maximal reliability and power in covariance structure models

In covariance structure modelling, the non‐centrality parameter of the asymptotic chi‐squared distribution is typically used as an indicator of asymptotic power for hypothesis tests. When a latent linear regression is of interest, the contribution to power by the maximal reliability coefficient, which is associated with used latent variable indicators, is examined and this relationship is further explicated in the case of congeneric measures. It is also shown that item parcelling may reduce power of tests of latent regression parameters. Recommendations on weights for parcelling to avoid power loss are provided, which are found to be those of optimal linear composites with maximal reliability.     

Testing differences between nested covariance structure models: Power analysis and null hypotheses

For comparing nested covariance structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit of the models. A modification of the standard null hypothesis of zero difference in fit is proposed allowing for testing an interval hypothesis that the difference in fit between models is small, rather than zero. These developments are combined yielding a procedure for estimating power of a test of a null hypothesis of small difference in fit versus an alternative hypothesis of larger difference.     

Estimation of covariance structure models with parameters subject to functional restraints

This paper demonstrates the feasibility of using the penalty function method to estimate parameters that are subject to a set of functional constraints in covariance structure analysis. Both types of inequality and equality constraints are studied. The approaches of maximum likelihood and generalized least squares estimation are considered. A modified Scoring algorithm and a modified Gauss-Newton algorithm are implemented to produce the appropriate constrained estimates. The methodology is illustrated by its applications to Heywood cases in confirmatory factor analysis, quasi-Weiner simplex model, and multitrait-multimethod matrix analysis.The author is indebted to several anonymous reviewers for creative suggestions for improvement of this paper. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

Standard errors and confidence intervals for correlations corrected for indirect range restriction: A simulation study comparing analytic and bootstrap methods

A frequent topic of psychological research is the estimation of the correlation between two variables from a sample that underwent a selection process based on a third variable. Due to indirect range restriction, the sample correlation is a biased estimator of the population correlation, and a correction formula is used. In the past, bootstrap standard error and confidence intervals for the corrected correlations were examined with normal data. The present study proposes a large-sample estimate (an analytic method) for the standard error, and a corresponding confidence interval for the corrected correlation. Monte Carlo simulation studies involving both normal and non-normal data were conducted to examine the empirical performance of the bootstrap and analytic methods. Results indicated that with both normal and non-normal data, the bootstrap standard error and confidence interval were generally accurate across simulation conditions (restricted sample size, selection ratio, and population correlations) and outperformed estimates of the analytic method. However, with certain combinations of distribution type and model conditions, the analytic method has an advantage, offering reasonable estimates of the standard error and confidence interval without resorting to the bootstrap procedure's computer-intensive approach. We provide SAS code for the simulation studies.     

The robustness of estimates of total indirect effects in covariance structure models estimated by maximum

The large sample distribution of total indirect effects in covariance structure models in well known. Using Monte Carlo methods, this study examines the applicability of the large sample theory to maximum likelihood estimates oftotal indirect effects in sample sizes of 50, 100, 200, 400, and 800. Two models are studied. Model 1 is a recursive model with observable variables and Model 2 is a nonrecursive model with latent variables. For the large sample theory to apply, the results suggest that sample szes of 200 or more and 400 or more are required for models such as Model 1 and Model 2, respectively.For helpful comments on a previous draft of this paper, we are grateful to Gerhard Arminger, Clifford C. Clogg, and several anonymous reviewers.     

Pseudo maximum likelihood estimation and a test for misspecification in mean and covariance structure models

Using the theory of pseudo maximum likelihood estimation the asymptotic covariance matrix of maximum likelihood estimates for mean and covariance structure models is given for the case where the variables are not multivariate normal. This asymptotic covariance matrix is consistently estimated without the computation of the empirical fourth order moment matrix. Using quasi-maximum likelihood theory a Hausman misspecification test is developed. This test is sensitive to misspecification caused by errors that are correlated with the independent variables. This misspecification cannot be detected by the test statistics currently used in covariance structure analysis.For helpful comments on a previous draft of the paper we are indebted to Kenneth A. Bollen, Ulrich L. Küsters, Michael E. Sobel and the anonymous reviewers of Psychometrika. For partial research support, the first author wishes to thank the Department of Sociology at the University of Arizona, where he was a visiting professor during the fall semester 1987.     

WAIS-IV subtest covariance structure: conceptual and statistical considerations

D. Wechsler (2008b) reported confirmatory factor analyses (CFAs) with standardization data (ages 16-69 years) for 10 core and 5 supplemental subtests from the Wechsler Adult Intelligence Scale-Fourth Edition (WAIS-IV). Analyses of the 15 subtests supported 4 hypothesized oblique factors (Verbal Comprehension, Working Memory, Perceptual Reasoning, and Processing Speed) but also revealed unexplained covariance between Block Design and Visual Puzzles (Perceptual Reasoning subtests). That covariance was not included in the final models. Instead, a path was added from Working Memory to Figure Weights (Perceptual Reasoning subtest) to improve fit and achieve a desired factor pattern. The present research with the same data (N = 1,800) showed that the path from Working Memory to Figure Weights increases the association between Working Memory and Matrix Reasoning. Specifying both paths improves model fit and largely eliminates unexplained covariance between Block Design and Visual Puzzles but with the undesirable consequence that Figure Weights and Matrix Reasoning are equally determined by Perceptual Reasoning and Working Memory. An alternative 4-factor model was proposed that explained theory-implied covariance between Block Design and Visual Puzzles and between Arithmetic and Figure Weights while maintaining compatibility with WAIS-IV Index structure. The proposed model compared favorably with a 5-factor model based on Cattell-Horn-Carroll theory. The present findings emphasize that covariance model comparisons should involve considerations of conceptual coherence and theoretical adherence in addition to statistical fit.     

Alternative test criteria in covariance structure analysis: A unified approach

In the context of covariance structure analysis, a unified approach to the asymptotic theory of alternative test criteria for testing parametric restrictions is provided. The discussion develops within a general framework that distinguishes whether or not the fitting function is asymptotically optimal, and allows the null and alternative hypothesis to be only approximations of the true model. Also, the equivalent of the information matrix, and the asymptotic covariance matrix of the vector of summary statistics, are allowed to be singular. When the fitting function is not asymptotically optimal, test statistics which have asymptotically a chi-square distribution are developed as a natural generalization of more classical ones. Issues relevant for power analysis, and the asymptotic theory of a testing related statistic, are also investigated.This research has been supported by the U.S.-Spanish Joint Committee for Cultural and Educational Cooperation, grant number V-B.854020. The author wishes to express his gratitude to P. M. Bentler who provided very helpful suggestions and research facilities—with an stimulating working environment—at the University of California, Los Angeles, where this work was undertaken. Thanks are also due to W. E. Saris who provided very valuable comments to earlier versions of this paper. Finally, it has also to be acknowledged the editor's and reviewers suggestions which led to substantial improvements of this article.     

Incorporating prior theory in covariance structure analysis: A bayesian approach

A Bayesian approach to the testing of competing covariance structures is developed. The method provides approximate posterior probablities for each model under consideration without prior specification of individual parameter distributions. The method is based on ayesian updating using cross-validated pseudo-likelihoods. Given that the observed variables are the samefor all competing models, the approximate posterior probabilities may be obtained easily from the chi square values and other known constants, using only a hand calculator. The approach is illustrated using and example which illustrates how the prior probabilities can alter the results concerning which model specification is preferred.     

Bootstrap approach to inference and power analysis based on three test statistics for covariance structure models

We study several aspects of bootstrap inference for covariance structure models based on three test statistics, including Type I error, power and sample‐size determination. Specifically, we discuss conditions for a test statistic to achieve a more accurate level of Type I error, both in theory and in practice. Details on power analysis and sample‐size determination are given. For data sets with heavy tails, we propose applying a bootstrap methodology to a transformed sample by a downweighting procedure. One of the key conditions for safe bootstrap inference is generally satisfied by the transformed sample but may not be satisfied by the original sample with heavy tails. Several data sets illustrate that, by combining downweighting and bootstrapping, a researcher may find a nearly optimal procedure for evaluating various aspects of covariance structure models. A rule for handling non‐convergence problems in bootstrap replications is proposed.     

Estimating standard errors in feature network models

Feature network models are graphical structures that represent proximity data in a discrete space while using the same formalism that is the basis of least squares methods employed in multidimensional scaling. Existing methods to derive a network model from empirical data only give the best‐fitting network and yield no standard errors for the parameter estimates. The additivity properties of networks make it possible to consider the model as a univariate (multiple) linear regression problem with positivity restrictions on the parameters. In the present study, both theoretical and empirical standard errors are obtained for the constrained regression parameters of a network model with known features. The performance of both types of standard error is evaluated using Monte Carlo techniques.     

Model modifications in covariance structure analysis: the problem of capitalization on chance

In applications of covariance structure modeling in which an initial model does not fit sample data well, it has become common practice to modify that model to improve its fit. Because this process is data driven, it is inherently susceptible to capitalization on chance characteristics of the data, thus raising the question of whether model modifications generalize to other samples or to the population. This issue is discussed in detail and is explored empirically through sampling studies using 2 large sets of data. Results demonstrate that over repeated samples, model modifications may be very inconsistent and cross-validation results may behave erratically. These findings lead to skepticism about generalizability of models resulting from data-driven modifications of an initial model. The use of alternative a priori models is recommended as a preferred strategy.     

Reading and listening comprehension in Broca's aphasia: Lexical versus syntactical errors

Ten agrammatic Broca's aphasics were presented with a series of four picture plates together with a spoken or written sentence stimulus. All sentence stimuli were of the structure, the + N + is/are + V + ing + the + N. The four pictures on each stimulus plate represented (a) the correct response, (b) a reversal of the stimulus sentence subject and object, (c) a change in the number of the subject of the stimulus sentence, and (d) a change in one of the major lexical items of the stimulus sentence. Subjects selected the correct picture most often. When they erred, they usually selected a subject-object reversal. Number errors were less frequent, and the patients seldom selected a change in major lexical item. This pattern occurred with both written and spoken sentences. These results were interpreted as reflecting the dependence of agrammatic Broca's aphasics on the semantic interpretation of the lexicon for decoding sentences in the face of deficits in syntactical-grammatical interpretation, irrespective of comprehension modality.     

On the asymptotic distributions of two statistics for two-level covariance structure models within the class of elliptical distributions

Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes. The authors thank an associate editor and three referees for their constructive comments, which led to an improved version of the paper. This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of Notre Dame faculty research grant.     

Career maturity: A latent means and covariance structure analysis of Daidalos

A latent means and covariance structure analysis was conducted to assess the construct validity and construct comparability in the measuring of career maturity across boys and girls. The career maturity inventory has been coined Daidalos. The sample consisted of 2,443 high school students recruited from one county in Norway. Of these, 1,132 were males, and 1,311 were females. The mean age of the participants was 17 years of age. Goodness-of-fit statistics provided support for a five-factor first-order model in which the factor loadings, factor covariances and item intercepts were invariant across groups. Additionally, ΔS-B χ² was shown to be non-significant for the testing of invariance of the measurement model. Three significant differences in latent means were detected, with boys scoring higher on negative attitudes toward school or dropout intentions, and girls scoring higher on career uncertainty and need for world-of-work information.     

Power of the likelihood ratio test in covariance structure analysis

A procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived. The procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified. Using the noncentral Chi-square distribution, the power of the test is approximated by the asymptotic one for a sequence of local alternatives. The procedure is illustrated by an example. A Monte Carlo experiment also shows how good the approximation is for a specific case.This research was made possible by a grant from the Dutch Organization for Advancement of Pure Research (ZWO). The authors also like to acknowledge the helpful comments and suggestions from the editor and anonymous reviewers.     

Standard errors of fit indices using residuals in structural equation modeling

The asymptotic standard errors of the correlation residuals and Bentler's standardized residuals in covariance structures are derived based on the asymptotic covariance matrix of raw covariance residuals. Using these results, approximations of the asymptotic standard errors of the root mean square residuals for unstandardized or standardized residuals are derived by the delta method. Further, in mean structures, approximations of the asymptotic standard errors of residuals, standardized residuals and their summary statistics are derived in a similar manner. Simulations are carried out, which show that the asymptotic standard errors of the various types of residuals and the root mean square residuals in covariance, correlation and mean structures are close to actual ones.The author is indebted to the reviewers for their comments and suggestions which have led to an improvement of this work.     

Standard errors for the class of orthomax-rotated factor loadings: Some matrix results

The partial derivative matrices of the class of orthomax-rotated factor loadings with respect to the unrotated maximum likelihood factor loadings are derived. The reported results are useful for obtaining standard errors of the orthomax-rotated factor loadings, with or without row normalization (standardization) of the initial factor loading matrix for rotation. Using a numerical example, we verify our analytic formulas by comparing the obtained standard error estimates with that from some existing methods. Some advantages of the current approach are discussed.Authorship is determined by alphabetical order. The authors contributed equally to the research. Kentaro Hayashi is now at the Department of Mathematics, Bucknell University, Lewisburg, PA 17837 (email: khayashi@Bucknell.edu). Yiu-Fai Yung is now at the SAS Institute, Inc., SAS Campus Drive, Cary, NC 27513 (email: yiyung@wnt.sas.com).Part of the research was completed while Yiu-Fai Yung was a visiting scholar at the Department of Psychology, the Ohio State University. The visit was supported in part by grant N4856118101 from the NIMH and the Mason and Linda Stephenson Travel Award from the Department of Psychology, University of North Carolina at Chapel Hill. The authors are grateful to Michael Browne who suggested some relevant references and provided valuable comments on the research, and to Robert Cudeck who provided the FAS program for the numerical comparison. The expert comments by the reviewers are deeply appreciated.