On accuracy in reliability estimation

This study in parametric test theory deals with the statistics of reliability estimation when scores on two parts of a test follow a binormal distribution with equal (Case 1) or unequal (Case 2) expectations. In each case biased maximum-likelihood estimators of reliability are obtained and converted into unbiased estimators. Sampling distributions are derived. Second moments are obtained and utilized in calculating mean square errors of estimation as a measure of accuracy. A rank order of four estimators is established. There is a uniformly best estimator. Tables of absolute and relative accuracies are provided for various reliability parameters and sample sizes.     

A note on testing for homogeneity among effect sizes sharing a common control group

L. V. Hedges and I. Olkin (1985) presented a statistic to test for homogeneity among correlated effect sizes and L. J. Gleser and I. Olkin (1994) presented a large-sample approximation to the covariance matrix of the correlated effect sizes. This article presents a more exact expression for this covariance matrix, assuming normally distributed data but not large samples, for the situation where effect sizes are correlated because a single control group was compared with more than one treatment group. After the correlation between effect sizes has been estimated, the standard Q statistic for correlated effect sizes can be used to test for homogeneity. This method is illustrated using results from schizophrenia research.     

Estimating Success in Predicting a Variable with Nominal Measurement Using Other Variables with Nominal Measurements

In this article we are concerned with the situation where one is estimating the outcome of a variable Y, with nominal measurement, on the basis of the outcomes of several predictor variables, X 1, X 2, ..., X r, each with nominal measurement. We assume that we have a random sample from the population. Here we are interested in estimating p, the probability of successfully predicting a new Y from the population, given the X measurements for this new observation. We begin by proposing an estimator, p_a, which is the success rate in predicting Y from the current sample. We show that this estimator is always biased upwards. We then propose a second estimator, p_b, which divides the original sample into two groups, a holdout group and a training group, in order to estimate p. We show that procedures such as these are always biased downwards, no matter how we divide the original sample into the two groups. Because one of these estimators tends to overestimate p while the other tends to underestimate p, we propose as a heuristic solution to use the mean of these two estimators, p_c, as an estimator for p. We then perform several simulation studies to compare the three estimators with respect to both bias and MSE. These simulations seem to confirm that $ p c is a better estimator than either of the other two.     

Estimating the mean effect size in meta-analysis: Bias,precision, and mean squared error of different weighting methods

Although use of the standardized mean difference in meta-analysis is appealing for several reasons, there are some drawbacks. In this article, we focus on the following problem: that a precision-weighted mean of the observed effect sizes results in a biased estimate of the mean standardized mean difference. This bias is due to the fact that the weight given to an observed effect size depends on this observed effect size. In order to eliminate the bias, Hedges and Olkin (1985) proposed using the mean effect size estimate to calculate the weights. In the article, we propose a third alternative for calculating the weights: using empirical Bayes estimates of the effect sizes. In a simulation study, these three approaches are compared. The mean squared error (MSE) is used as the criterion by which to evaluate the resulting estimates of the mean effect size. For a meta-analytic dataset with a small number of studies, theMSE is usually smallest when the ordinary procedure is used, whereas for a moderate or large number of studies, the procedures yielding the best results are the empirical Bayes procedure and the procedure of Hedges and Olkin, respectively.     

Model-Implied Instrumental Variable—Generalized Method of Moments (MIIV-GMM) Estimators for Latent Variable Models

The common maximum likelihood (ML) estimator for structural equation models (SEMs) has optimal asymptotic properties under ideal conditions (e.g., correct structure, no excess kurtosis, etc.) that are rarely met in practice. This paper proposes model-implied instrumental variable – generalized method of moments (MIIV-GMM) estimators for latent variable SEMs that are more robust than ML to violations of both the model structure and distributional assumptions. Under less demanding assumptions, the MIIV-GMM estimators are consistent, asymptotically unbiased, asymptotically normal, and have an asymptotic covariance matrix. They are “distribution-free,” robust to heteroscedasticity, and have overidentification goodness-of-fit J-tests with asymptotic chi-square distributions. In addition, MIIV-GMM estimators are “scalable” in that they can estimate and test the full model or any subset of equations, and hence allow better pinpointing of those parts of the model that fit and do not fit the data. An empirical example illustrates MIIV-GMM estimators. Two simulation studies explore their finite sample properties and find that they perform well across a range of sample sizes.     

A note on the unbiased estimation of the intraclass correlation

The intraclass correlation,, is a parameter featured in much psychological research. Two commonly used estimators of, the maximum likelihood and least squares estimators, are known to be negatively biased. Olkin and Pratt (1958) derived the minimum variance unbiased estimator of the intraclass correlation, but use of this estimator has apparently been impeded by the lack of a closed form solution. This note briefly reviews the unbiased estimator and gives a FORTRAN 77 subroutine to calculate it.The first author was supported by an All-University Fellowship from the University of Southern California.     

Bias and Efficiency in Structural Equation Modeling: Maximum Likelihood Versus Robust Methods

In the structural equation modeling literature, the normal-distribution-based maximum likelihood (ML) method is most widely used, partly because the resulting estimator is claimed to be asymptotically unbiased and most efficient. However, this may not hold when data deviate from normal distribution. Outlying cases or nonnormally distributed data, in practice, can make the ML estimator (MLE) biased and inefficient. In addition to ML, robust methods have also been developed, which are designed to minimize the effects of outlying cases. But the properties of robust estimates and their standard errors (SEs) have never been systematically studied. This article studies two robust methods and compares them against the ML method with respect to bias and efficiency using a confirmatory factor model. Simulation results show that robust methods lead to results comparable with ML when data are normally distributed. When data have heavy tails or outlying cases, robust methods lead to less biased and more efficient estimators than MLEs. A formula to obtain consistent SEs for one of the robust methods is also developed. The formula-based SEs for both robust estimators match the empirical SEs very well with medium-size samples. A sample of the Cross Racial Identity Scale with a 6-factor model is used for illustration. Results also confirm conclusions of the simulation study.     

Interval estimation of gamma for an<Emphasis Type="Italic">R</Emphasis>×<Emphasis Type="Italic">S</Emphasis> table

When the underlying responses are on an ordinal scale, gamma is one of the most frequently used indices to measure the strength of association between two ordered variables. However, except for a brief mention on the use of the traditional interval estimator based on Wald's statistic, discussion of interval estimation of the gamma is limited. Because it is well known that an interval estimator using Wald's statistic is generally not likely to perform well especially when the sample size is small, the goal of this paper is to find ways to improve the finite-sample performance of this estimator. This paper develops five asymptotic interval estimators of the gamma by employing various methods that are commonly used to improve the normal approximation of the maximum likelihood estimator (MLE). Using Monte Carlo simulation, this paper notes that the coverage probability of the interval estimator using Wald's statistic can be much less than the desired confidence level, especially when the underlying gamma is large. Further, except for the extreme case, in which the underlying gamma is large and the sample size is small, the interval estimator using a logarithmic transformation together with a monotonic function proposed here not only performs well with respect to the coverage probability, but is also more efficient than all the other estimators considered here. Finally, this paper notes that applying an ad hoc adjustment procedure—whenever any observed frequency equals 0, we add 0.5 to all cells in calculation of the cell proportions—can substantially improve the traditional interval estimator. This paper includes two examples to illustrate the practical use of interval estimators considered here.The authors wish to thank the Associate Editor and the two referees for many valuable comments and suggestions to improve the contents and clarity of this paper. The authors also want to thank Dr. C. D. Lin for his graphic assistance.     

Sampling behaviour in estimating predictive validity in the context of selection and latent variable modelling: A Monte Carlo study

While the effect of selection in predictive validity studies has long been recognized and discussed in psychometric studies, little consideration has been given to this problem in the context of latent variable models. In a recent paper, Muthén & Hsu (1993) proposed and compared estimators of predictive validity of a multifactorial test. Both selectivity and measurement error were considered in the estimation of predictive validity. The purpose of the present paper is to expand on Muthén & Hsu (1993) by examining and comparing the sampling behaviour of three estimators for predictive validity, LQL (listwise, quasi-likelihood estimator), FQL (full, quasi-likelihood estimator) and FS (factor score estimator), using a Monte Carlo approach. Effects of selection procedures, selection ratios and sample sizes on the sampling behaviours of the estimators are also investigated. The results show that FQL and FS are the two preferred estimators and each has different strengths and weaknesses. A real data application is presented to illustrate the practical implementation of the estimators.     

On the asymptotic standard error of a class of robust estimators of ability in dichotomous item response models

In item response theory, the classical estimators of ability are highly sensitive to response disturbances and can return strongly biased estimates of the true underlying ability level. Robust methods were introduced to lessen the impact of such aberrant responses on the estimation process. The computation of asymptotic (i.e., large‐sample) standard errors (ASE) for these robust estimators, however, has not yet been fully considered. This paper focuses on a broad class of robust ability estimators, defined by an appropriate selection of the weight function and the residual measure, for which the ASE is derived from the theory of estimating equations. The maximum likelihood (ML) and the robust estimators, together with their estimated ASEs, are then compared in a simulation study by generating random guessing disturbances. It is concluded that both the estimators and their ASE perform similarly in the absence of random guessing, while the robust estimator and its estimated ASE are less biased and outperform their ML counterparts in the presence of random guessing with large impact on the item response process.     

Some results on extensions and modifications of the Theil — Sen regression estimator

Many robust regression estimators have been proposed that have a high, finite‐sample breakdown point, roughly meaning that a large porportion of points must be altered to drive the value of an estimator to infinity. But despite this, many of them can be inordinately influenced by two properly placed outliers. With one predictor, an estimator that appears to correct this problem to a fair degree, and simultaneously maintain good efficiency when standard assumptions are met, consists of checking for outliers using a projection‐type method, removing any that are found, and applying the Theil — Sen estimator to the data that remain. When dealing with multiple predictors, there are two generalizations of the Theil — Sen estimator that might be used, but nothing is known about how their small‐sample properties compare. Also, there are no results on testing the hypothesis of zero slopes, and there is no information about the effect on efficiency when outliers are removed. In terms of hypothesis testing, using the more obvious percentile bootstrap method in conjunction with a slight modification of Mahalanobis distance was found to avoid Type I error probabilities above the nominal level, but in some situations the actual Type I error probabilities can be substantially smaller than intended when the sample size is small. An alternative method is found to be more satisfactory.     

The moral reasoning of juvenile delinquents: A meta-analysis

To test the hypothesized immaturity of juvenile delinquents' moral reasoning, the results of 15 studies of the moral reasoning of juvenile delinquents were integrated quantitatively using meta-analysis. Hedges and Olkin (1985) methods were used to (a) compute effect sizes, (b) test the homogeneity of the obtained effect sizes, and (c) test the statistical significance of the pooled mean effect size. The results supported the hypothesis that the moral reasoning of juvenile delinquents is immature. It was concluded that several other issues are in need of investigation.     

On the large sample distributions of modified sample biserial correlation coefficients

The asymptotic distributions of Brogden's and Lord's modified sample biserial correlation coefficients are derived. The asymptotic variances of these estimators are evaluated for bivariate normal populations and compared to the asymptotic variance of the maximum likelihood estimator.The author would like to thank the referees for several suggestions which improved the presentation of the paper.     

The performance of robust test statistics with categorical data

This paper reports on a simulation study that evaluated the performance of five structural equation model test statistics appropriate for categorical data. Both Type I error rate and power were investigated. Different model sizes, sample sizes, numbers of categories, and threshold distributions were considered. Statistics associated with both the diagonally weighted least squares (cat‐DWLS) estimator and with the unweighted least squares (cat‐ULS) estimator were studied. Recent research suggests that cat‐ULS parameter estimates and robust standard errors slightly outperform cat‐DWLS estimates and robust standard errors ( Forero, Maydeu‐Olivares, & Gallardo‐Pujol, 2009 ). The findings of the present research suggest that the mean‐ and variance‐adjusted test statistic associated with the cat‐ULS estimator performs best overall. A new version of this statistic now exists that does not require a degrees‐of‐freedom adjustment ( Asparouhov & Muthén, 2010 ), and this statistic is recommended. Overall, the cat‐ULS estimator is recommended over cat‐DWLS, particularly in small to medium sample sizes.     

An alternative two stage least squares (2SLS) estimator for latent variable equations

The Maximum-likelihood estimator dominates the estimation of general structural equation models. Noniterative, equation-by-equation estimators for factor analysis have received some attention, but little has been done on such estimators for latent variable equations. I propose an alternative 2SLS estimator of the parameters in LISREL type models and contrast it with the existing ones. The new 2SLS estimator allows observed and latent variables to originate from nonnormal distributions, is consistent, has a known asymptotic covariance matrix, and is estimable with standard statistical software. Diagnostics for evaluating instrumental variables are described. An empirical example illustrates the estimator. I gratefully acknowledge support for this research from the Sociology Program of the National Science Foundation (SES-9121564) and the Center for Advanced Study in the Behavioral Sciences, Stanford, California. This paper was presented at the Interdisciplinary Consortium for Statistical Applications at Indiana University at Bloomington (March 2, 1994) and at the RMD Conference on Causal Modeling at Purdue University, West Lafayette, Indiana (March 3-5, 1994).     

A new quantile estimator with weights based on a subsampling approach

Quantiles are widely used in both theoretical and applied statistics, and it is important to be able to deploy appropriate quantile estimators. To improve performance in the lower and upper quantiles, especially with small sample sizes, a new quantile estimator is introduced which is a weighted average of all order statistics. The new estimator, denoted NO, has desirable asymptotic properties. Moreover, it offers practical advantages over four estimators in terms of efficiency in most experimental settings. The Harrell–Davis quantile estimator, the default quantile estimator of the R programming language, the Sfakianakis–Verginis SV2 quantile estimator and a kernel quantile estimator. The NO quantile estimator is also utilized in comparing two independent groups with a percentile bootstrap method and, as expected, it is more successful than other estimators in controlling Type I error rates.     

Point-biserial correlation: Interval estimation,hypothesis testing,meta-analysis,and sample size determination

The point-biserial correlation is a commonly used measure of effect size in two-group designs. New estimators of point-biserial correlation are derived from different forms of a standardized mean difference. Point-biserial correlations are defined for designs with either fixed or random group sample sizes and can accommodate unequal variances. Confidence intervals and standard errors for the point-biserial correlation estimators are derived from the sampling distributions for pooled-variance and separate-variance versions of a standardized mean difference. The proposed point-biserial confidence intervals can be used to conduct directional two-sided tests, equivalence tests, directional non-equivalence tests, and non-inferiority tests. A confidence interval for an average point-biserial correlation in meta-analysis applications performs substantially better than the currently used methods. Sample size formulas for estimating a point-biserial correlation with desired precision and testing a point-biserial correlation with desired power are proposed. R functions are provided that can be used to compute the proposed confidence intervals and sample size formulas.     

On the Computation of the RMSEA and CFI from the Mean-And-Variance Corrected Test Statistic with Nonnormal Data in SEM

A new type of nonnormality correction to the RMSEA has recently been developed, which has several advantages over existing corrections. In particular, the new correction adjusts the sample estimate of the RMSEA for the inflation due to nonnormality, while leaving its population value unchanged, so that established cutoff criteria can still be used to judge the degree of approximate fit. A confidence interval (CI) for the new robust RMSEA based on the mean-corrected (“Satorra-Bentler”) test statistic has also been proposed. Follow up work has provided the same type of nonnormality correction for the CFI (Brosseau-Liard & Savalei, 2014). These developments have recently been implemented in lavaan. This note has three goals: a) to show how to compute the new robust RMSEA and CFI from the mean-and-variance corrected test statistic; b) to offer a new CI for the robust RMSEA based on the mean-and-variance corrected test statistic; and c) to caution that the logic of the new nonnormality corrections to RMSEA and CFI is most appropriate for the maximum likelihood (ML) estimator, and cannot easily be generalized to the most commonly used categorical data estimators.     

Robust inference with binary data

In this paper robustness properties of the maximum likelihood estimator (MLE) and several robust estimators for the logistic regression model when the responses are binary are analysed. It is found that the MLE and the classical Rao's score test can be misleading in the presence of model misspecification which in the context of logistic regression means either misclassification's errors in the responses, or extreme data points in the design space. A general framework for robust estimation and testing is presented and a robust estimator as well as a robust testing procedure are presented. It is shown that they are less influenced by model misspecifications than their classical counterparts. They are finally applied to the analysis of binary data from a study on breastfeeding.The author is partially supported by the Swiss National Science Foundation. She would like to thank Rand Wilcox, Eva Cantoni and Elvezio Ronchetti for their helpful comments on earlier versions of the paper, as well as Stephane Heritier for providing the routine to compute the OBRE.     

Information matrix estimation procedures for cognitive diagnostic models

Two new methods to estimate the asymptotic covariance matrix for marginal maximum likelihood estimation of cognitive diagnosis models (CDMs), the inverse of the observed information matrix and the sandwich-type estimator, are introduced. Unlike several previous covariance matrix estimators, the new methods take into account both the item and structural parameters. The relationships between the observed information matrix, the empirical cross-product information matrix, the sandwich-type covariance matrix and the two approaches proposed by de la Torre (2009, J. Educ. Behav. Stat., 34, 115) are discussed. Simulation results show that, for a correctly specified CDM and Q-matrix or with a slightly misspecified probability model, the observed information matrix and the sandwich-type covariance matrix exhibit good performance with respect to providing consistent standard errors of item parameter estimates. However, with substantial model misspecification only the sandwich-type covariance matrix exhibits robust performance.