Likelihood-Based Confidence Intervals in Exploratory Factor Analysis

In exploratory or unrestricted factor analysis, all factor loadings are free to be estimated. In oblique solutions, the correlations between common factors are free to be estimated as well. The purpose of this article is to show how likelihood-based confidence intervals can be obtained for rotated factor loadings and factor correlations, by applying maximum likelihood factor analysis subject to scaling and rotation constraints. As an illustrative example, an oblique 5-factor model will be fitted to the variance-covariance matrix of the 30 personality facets measured by the Revised NEO Personality Inventory, and confidence intervals will be estimated for all factor loadings and factor correlations, as well as for the associated reliability and validity coefficients.     

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.     

Monte Carlo Confidence Intervals for Complex Functions of Indirect Effects

One challenge in mediation analysis is to generate a confidence interval (CI) with high coverage and power that maintains a nominal significance level for any well-defined function of indirect and direct effects in the general context of structural equation modeling (SEM). This study discusses a proposed Monte Carlo extension that finds the CIs for any well-defined function of the coefficients of SEM such as the product of k coefficients and the ratio of the contrasts of indirect effects, using the Monte Carlo method. Finally, we conduct a small-scale simulation study to compare CIs produced by the Monte Carlo, nonparametric bootstrap, and asymptotic-delta methods. Based on our simulation study, we recommend researchers use the Monte Carlo method to test a complex function of indirect effects.     

Confidence Intervals for True Scores Under an Answer-Until-Correct Scoring Procedure

Under an answer-until-correct scoring procedure, many measurement problems can be solved when certain cognitive models of examinee behavior can be assumed (Wilcox, 1983). Point estimates of true score under these models are available, but the problem of obtaining a confidence interval has never been addressed. Two simple methods for obtaining a confidence interval are suggested that give good results when the sample size is reasonably large, say, greater than or equal to 20, and when true score is not too close to zero or one. A third procedure is suggested that can also be used to get slightly better results where again the sample size is assumed to be reasonably large and true score is not too close to zero or one. For small sample sizes or situations where true score is close to zero or one, a fourth procedure is described that always gives conservative results.     

Constructing Approximate Confidence Intervals for Parameters With Structural Equation Models

Confidence intervals (CIs) for parameters are usually constructed based on the estimated standard errors. These are known as Wald CIs. This article argues that likelihood-based CIs (CIs based on likelihood ratio statistics) are often preferred to Wald CIs. It shows how the likelihood-based CIs and the Wald CIs for many statistics and psychometric indexes can be constructed with the use of phantom variables (Rindskopf, 1984 Rindskopf, D. 1984. Using phantom and imaginary latent variables to parameterize constraints in linear structural models. Psychometrika, 49: 37–47. [Crossref], [Web of Science ®] , [Google Scholar]) in some of the current structural equation modeling (SEM) packages. The procedures to form CIs for the differences in correlation coefficients, squared multiple correlations, indirect effects, coefficient alphas, and reliability estimates are illustrated. A simulation study on the Pearson correlation is used to demonstrate the advantages of the likelihood-based CI over the Wald CI. Issues arising from this SEM approach and extensions of this approach are discussed.     

Confidence Intervals for Weighted Composite Scores Under the Compound Binomial Error Model

Reporting confidence intervals with test scores helps test users make important decisions about examinees by providing information about the precision of test scores. Although a variety of estimation procedures based on the binomial error model are available for computing intervals for test scores, these procedures assume that items are randomly drawn from a undifferentiated universe of items, and therefore might not be suitable for tests developed according to a table of specifications. To address this issue, four interval estimation procedures that use category subscores for the computation of confidence intervals are presented in this article. All four estimation procedures assume that subscores instead of test scores follow a binomial distribution (i.e., compound binomial error model). The relative performance of the four compound binomial–based interval estimation procedures is compared to each other and to the better known normal approximation and Wilson score procedures based on the binomial error model.     

Bootstrap方法在置信区间和标准差估计中的应用

Bootstrap方法是上世纪80年代出现和发展起来的一种新型再抽样统计方法，在统计各领域已有广泛应用．把Bootstrap方法应用于置信区间和标准差估计，在某些情况下，使得置信区间和标准差估计得到改进和改善。     

Inference by Eye: Pictures of Confidence Intervals and Thinking About Levels of Confidence

A picture of a 95% confidence interval (CI) implicitly contains pictures of CIs of all other levels of confidence, and information about the p‐value for testing a null hypothesis. This article discusses pictures, taken from interactive software, that suggest several ways to think about the level of confidence of a CI, p‐values, and what conclusions can be drawn from inspecting a CI.     

Likelihood-Based Confidence Intervals for a Parameter With an Upper or Lower Bound

The precision of estimates in many statistical models can be expressed by a confidence interval (CI). CIs based on standard errors (SEs) are common in practice, but likelihood-based CIs are worth consideration. In comparison to SEs, likelihood-based CIs are typically more difficult to estimate, but are more robust to model (re)parameterization. In latent variable models, some parameters might take on values outside of their interpretable range. Therefore, it is desirable to place a bound to keep the parameter interpretable. For likelihood-based CI, a correction is needed when a parameter is bounded. The correction is known (Wu & Neale, 2012), but is difficult to implement in practice. A novel automatic implementation that is simple for an applied researcher to use is introduced. A simulation study demonstrates the accuracy of the correction using a latent growth curve model and the method is illustrated with a multilevel confirmatory factor analysis.     

两m相依样本均值差异的经验似然比置信区间

该文讨论了两m相依样本均值差异的经验似然的大样本性质,证明了经验似然比统计量依分布收敛于χ2随机变量,由此给出均值差异的经验似然置信区间。     

两m相依样本均值差异的经验似然比置信区间

该文讨论了两m相依样本均值差异的经验似然的大样本性质,证明了经验似然比统计量依分布收敛于x2随机变量,由此给出均值差异的经验似然置信区间.     

Confidence Intervals of Fit Indexes by Inverting a Bootstrap Test

Fit indexes are an important tool in the evaluation of model fit in structural equation modeling (SEM). Currently, the newest confidence interval (CI) for fit indexes proposed by Zhang and Savalei (2016) is based on the quantiles of a bootstrap sampling distribution at a single level of misspecification. This method, despite a great improvement over naive and model-based bootstrap methods, still suffers from unsatisfactory coverage. In this work, we propose a new method of constructing bootstrap CIs for various fit indexes. This method directly inverts a bootstrap test and produces a CI that involves levels of misspecification that would not be rejected in a bootstrap test. Similar in rationale to a parametric CI of root mean square error of approximation (RMSEA) based on a noncentral χ2 distribution and a profile-likelihood CI of model parameters, this approach is shown to have better performance than the approach of Zhang and Savalei (2016), with more accurate coverage and more efficient widths.     

Confidence Intervals for a Semiparametric Approach to Modeling Nonlinear Relations among Latent Variables

Compared to parametric models, nonparametric and semiparametric approaches to modeling nonlinearity between latent variables have the advantage of recovering global relationships of unknown functional form. Bauer (2005) Bauer, D. J. 2005. A semiparametric approach to modeling nonlinear relations among latent variables. Structural Equation Modeling, 12: 513–535. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] proposed an indirect application of finite mixtures of structural equation models where latent components are estimated in the service of more flexibly recovering characteristics of the latent aggregate regression function. This article develops and evaluates delta method and parametric bootstrap approaches for obtaining approximate confidence intervals for Bauer's semiparametric approach to modeling latent nonlinear functions. Coverage rates of these approximate point-wise confidence intervals or nonsimultaneous confidence bands are evaluated by Monte Carlo and recommendations for their use are suggested.     

Comparison of Approaches to Constructing Confidence Intervals for Mediating Effects Using Structural Equation Models

Regression mixture models, which have only recently begun to be used in applied research, are a new approach for finding differential effects. This approach comes at the cost of the assumption that error terms are normally distributed within classes. This study uses Monte Carlo simulations to explore the effects of relatively minor violations of this assumption. The use of an ordered polytomous outcome is then examined as an alternative that makes somewhat weaker assumptions, and finally both approaches are demonstrated with an applied example looking at differences in the effects of family management on the highly skewed outcome of drug use. Results show that violating the assumption of normal errors results in systematic bias in both latent class enumeration and parameter estimates. Additional classes that reflect violations of distributional assumptions are found. Under some conditions it is possible to come to conclusions that are consistent with the effects in the population, but when errors are skewed in both classes the results typically no longer reflect even the pattern of effects in the population. The polytomous regression model performs better under all scenarios examined and comes to reasonable results with the highly skewed outcome in the applied example. We recommend that careful evaluation of model sensitivity to distributional assumptions be the norm when conducting regression mixture models.     

Bernoulli分布中参数p的近似置信区间及应用

给出了Bernoulli分布中未知参数的4种不同形式的近似置信区间,包括基于Hoeffding不等式和Bernstein不等式两种新的置信区间,并通过模拟比较了置信区间在大样本和小样本情形下的优劣.     

Characterizing the Mathematics Anxiety Literature Using Confidence Intervals as a Literature Review Mechanism

ABSTRACT

The authors report the contextualization of effect sizes within mathematics anxiety research, and more specifically within research using the Mathematics Anxiety Rating Scale (MARS) and the MARS for Adolescents (MARS-A). The effect sizes from 45 studies were characterized by graphing confidence intervals (CIs) across studies involving (a) adults not participating in studies focusing on remedial or entry-level mathematics students or teachers, (b) remedial mathematics and entry-level college mathematics students, (c) preservice and inservice teachers, and (d) 7–12th-grade students and rising college students. The results also illustrate how CIs can be useful in research syntheses, because CIs (a) encourage meta-analytic thinking, (b) provide information about the research precision of a literature, and (c) provide plausible estimates for parameter values even if initial research expectations are wildly wrong.     

Planning for Mastery in Maths

Joanna Turnbull describes how her Maths Links materials can help low attainers in ordinary and special schools.     

New Confidence Intervals and Bias Comparisons Show That Maximum Likelihood Can Beat Multiple Imputation in Small Samples

When analyzing incomplete data, is it better to use multiple imputation (MI) or full information maximum likelihood (ML)? In large samples ML is clearly better, but in small samples ML’s usefulness has been limited because ML commonly uses normal test statistics and confidence intervals that require large samples. We propose small-sample t-based ML confidence intervals that have good coverage and are shorter than t-based confidence intervals under MI. We also show that ML point estimates are less biased and more efficient than MI point estimates in small samples of bivariate normal data. With our new confidence intervals, ML should be preferred over MI, even in small samples, whenever both options are available.     

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.