Robust estimation of the hierarchical model for responses and response times

Van der Linden's (2007, Psychometrika, 72, 287) hierarchical model for responses and response times in tests has numerous applications in psychological assessment. The success of these applications requires the parameters of the model to have been estimated without bias. The data used for model fitting, however, are often contaminated, for example, by rapid guesses or lapses of attention. This distorts the parameter estimates. In the present paper, a novel estimation approach is proposed that is robust against contamination. The approach consists of two steps. In the first step, the response time model is fitted on the basis of a robust estimate of the covariance matrix. In the second step, the item response model is extended to a mixture model, which allows for a proportion of irregular responses in the data. The parameters of the mixture model are then estimated with a modified marginal maximum likelihood estimator. The modified marginal maximum likelihood estimator downweights responses of test-takers with unusual response time patterns. As a result, the estimator is resistant to several forms of data contamination. The robustness of the approach is investigated in a simulation study. An application of the estimator is demonstrated with real data.     

Establishing the Duration of Crimes: An Individual Differences and Eye‐Tracking Investigation into Time Estimation

The time available for viewing a perpetrator at a crime scene predicts successful person recognition in subsequent identity line‐ups. This time is usually unknown and must be derived from eyewitnesses' duration estimates. This study therefore compared the estimates that different individuals provide for crimes. We then attempted to determine the accuracy of these durations by measuring observers' general time estimation ability with a set of estimator videos. Observers differed greatly in their ability to estimate time, but individual duration estimates correlated strongly for crime and estimator materials. This indicates that it might be possible to infer unknown durations of events, such as criminal incidents, from a person's ability to estimate known durations. We also measured observers' eye movements to a perpetrator during crimes. Only fixations on a perpetrator's face related to eyewitness accuracy, but these fixations did not correlate with exposure estimates for this person. The implications of these findings are discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic comparison of missing data procedures for estimating factor loadings

Large sample properties of four methods of handling multivariate missing data are compared. The criterion for comparison is how well the loadings from a single factor model can be estimated. It is shown that efficiencies of the methods depend on the pattern or arrangement of missing data, and an evaluation study is used to generate predictive efficiency equations to guide one's choice of an estimating procedure. A simple regression-type estimator is introduced which shows high efficiency relative to the maximum likelihood method over a large range of patterns and covariance matrices.     

Modelling inter-individual differences in latent within-person variation: The confirmatory factor level variability model

Psychological theories often produce hypotheses that pertain to individual differences in within-person variability. To empirically test the predictions entailed by such hypotheses with longitudinal data, researchers often use multilevel approaches that allow them to model between-person differences in the mean level of a certain variable and the residual within-person variance. Currently, these approaches can be applied only when the data stem from a single variable. However, it is common practice in psychology to assess not just a single measure but rather several measures of a construct. In this paper we describe a model in which we combine the single-indicator model with confirmatory factor analysis. The new model allows individual differences in latent mean-level factors and latent within-person variability factors to be estimated. Furthermore, we show how the model's parameters can be estimated with a maximum likelihood estimator, and we illustrate the approach using an example that involves intensive longitudinal data.     

Organizational Justice Dimensions: Validation of an Arabic Measure

There is a continued debate regarding the dimensions of organizational justice. The present project investigated the dimensionality of organizational justice and the validity of an Arabic measure of organizational justice for a Kuwaiti samples. The first study sample consisted of 1,184 Kuwaitis (619 males and 565 females) from two groups: 728 employees and 456 teachers working in the public sector. The second study sample consisted of 373 participants (190 employees and 183 teachers). The instrument items were based on a careful review of the organizational justice literature to ensure relevance to the sample culture. Confirmatory factor analyses (CFA) using WLSMV estimator is used. WLSMV method is more appropriate for our data because variables are measured on an ordinal scale. WLSMV is considered a less bias estimator compared with the standard maximum likelihood in case of ordinal data. CFA analyses identified the four distinctive factors of distributive, procedural, interpersonal, and informational organizational justice. The four‐factor model fit the data significantly better than one‐, two‐ or three‐factor models. Moreover, the study revealed that these four dimensions of organizational justice were significantly correlated with the four relevant outcomes of instrumentality, organizational commitment, organizational citizenship behavior, and collective esteem. Using the Arabic version of Colquitt's ( 2001 ) instrument (Fischer et al., 2011 ), the second study presented an evidence of concurrent validity of the new Arabic scale. The present study confirmed the four‐factor dimensionality of organizational justice. Results of the current study may raise the issue of development of scales versus translation of well‐ developed ones. Theoretical and practical implications of the results are discussed.     

Structural Equation Modeling with Small Samples: Test Statistics

Structural equation modeling is a well-known technique for studying relationships among multivariate data. In practice, high dimensional nonnormal data with small to medium sample sizes are very common, and large sample theory, on which almost all modeling statistics are based, cannot be invoked for model evaluation with test statistics. The most natural method for nonnormal data, the asymptotically distribution free procedure, is not defined when the sample size is less than the number of nonduplicated elements in the sample covariance. Since normal theory maximum likelihood estimation remains defined for intermediate to small sample size, it may be invoked but with the probable consequence of distorted performance in model evaluation. This article studies the small sample behavior of several test statistics that are based on maximum likelihood estimator, but are designed to perform better with nonnormal data. We aim to identify statistics that work reasonably well for a range of small sample sizes and distribution conditions. Monte Carlo results indicate that Yuan and Bentler's recently proposed F-statistic performs satisfactorily.     

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.     

Joint Maximum Likelihood Estimation for High-Dimensional Exploratory Item Factor Analysis

Joint maximum likelihood (JML) estimation is one of the earliest approaches to fitting item response theory (IRT) models. This procedure treats both the item and person parameters as unknown but fixed model parameters and estimates them simultaneously by solving an optimization problem. However, the JML estimator is known to be asymptotically inconsistent for many IRT models, when the sample size goes to infinity and the number of items keeps fixed. Consequently, in the psychometrics literature, this estimator is less preferred to the marginal maximum likelihood (MML) estimator. In this paper, we re-investigate the JML estimator for high-dimensional exploratory item factor analysis, from both statistical and computational perspectives. In particular, we establish a notion of statistical consistency for a constrained JML estimator, under an asymptotic setting that both the numbers of items and people grow to infinity and that many responses may be missing. A parallel computing algorithm is proposed for this estimator that can scale to very large datasets. Via simulation studies, we show that when the dimensionality is high, the proposed estimator yields similar or even better results than those from the MML estimator, but can be obtained computationally much more efficiently. An illustrative real data example is provided based on the revised version of Eysenck’s Personality Questionnaire (EPQ-R).

     

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.     

Simplified Estimation and Testing in Unbalanced Repeated Measures Designs

In this paper we propose a simple estimator for unbalanced repeated measures design models where each unit is observed at least once in each cell of the experimental design. The estimator does not require a model of the error covariance structure. Thus, circularity of the error covariance matrix and estimation of correlation parameters and variances are not necessary. Together with a weak assumption about the reason for the varying number of observations, the proposed estimator and its variance estimator are unbiased. As an alternative to confidence intervals based on the normality assumption, a bias-corrected and accelerated bootstrap technique is considered. We also propose the naive percentile bootstrap for Wald-type tests where the standard Wald test may break down when the number of observations is small relative to the number of parameters to be estimated. In a simulation study we illustrate the properties of the estimator and the bootstrap techniques to calculate confidence intervals and conduct hypothesis tests in small and large samples under normality and non-normality of the errors. The results imply that the simple estimator is only slightly less efficient than an estimator that correctly assumes a block structure of the error correlation matrix, a special case of which is an equi-correlation matrix. Application of the estimator and the bootstrap technique is illustrated using data from a task switch experiment based on an experimental within design with 32 cells and 33 participants.

     

Random‐effects meta‐analysis of correlations: Monte Carlo evaluation of mean estimators

Several authors have cautioned against using Fisher's z‐transformation in random‐effects meta‐analysis of correlations, which seems to perform poorly in some situations, especially with substantial inter‐study heterogeneity. Attributing this performance largely to the direct z‐to‐r transformation (DZRT) of Fisher z results (e.g. point estimate of mean correlation), in a previous paper Hafdahl (2009) proposed point and interval estimators of the mean Pearson r correlation that instead use an integral z‐to‐r transformation (IZRT). The present Monte Carlo study of these IZRT Fisher z estimators includes comparisons with their DZRT counterparts and with estimators based on Pearson r correlations. The IZRT point estimator was usually more accurate and efficient than its DZRT counterpart and comparable to the two Pearson r point estimators – better in some conditions but worse in others. Coverage probability for the IZRT confidence intervals (CIs) was often near nominal, much better than for the DZRT CIs, and comparable to coverage for the Pearson r CIs; every approach's CI fell markedly below nominal in some conditions. The IZRT estimators contradict warnings about Fisher z estimators' poor performance. Recommendations for practising research synthesists are offered, and an Appendix provides computing code to implement the IZRT as in the real‐data example.     

More on least squares estimation of the transition matrix in a stationary first-order markov process from sample proportions data

Miller suggested ordinary least squares estimation of a constant transition matrix; Madansky proposed a relatively more efficient weighted least squares estimator which corrects for heteroscedasticity. In this paper an efficient generalized least squares estimator is derived which utilizes the entire covariance matrix of the distrubances. This estimator satisfies the condition that each row of the transition matrix must sum to unity. Madansky noted that estimates of the variances could be negative; a method for obtaining consistent non-negative estimates of the variances is suggested in this paper. The technique is applied to the hypothetical sample data used by Miller and Madansky.I am indebted to a referee for his thoughtful suggestions on content and style.     

Un score individuel de reconnaissance d'une série de stimuli intermédiaires entre deux sources : la perception catégorielle des expressions faciales

Morphing is a computerized tool allowing the generation of pictures which are « hybrids » of two sources-pictures, in controlled proportions. Over the last decade, this tool did contribute greatly to the knowledge of mechanisms underlying facial expressions recognition. Especially, it appeared that categorical perception is involved. Indeed, while the physical variation from one source to the other is continuous and linear, the perceptual variation takes the form of a sigmoïdal or threshold function. The same applies to facial identity recognition and to recognition of face's gender. The present paper shows that data collected to evidence categorical perception can also be used to assess accuracy of expression recognition, whatever categorical or not, and to compare, say, the performance of an individual subject to that of a control group. In short, the proportion of subjects of the control group making the same choice as the individual subject is used as an estimator of the probability that this subject is issued from a normal population. This procedure furnishes criteria to decide if a given response is correct or not, which is not possible when free choices are recorded. This adaptation can be generalized to every tool designed to evaluate categorical perception.     

The Impact of a Salesperson's In-House Conflicts and Influence Attempts on Buyer Commitment

This paper examines the link between a company's internal conflicts and its marketplace success. In particular, it examines how conflicts between a selling firm's sales and support personnel can lead to unintended variations in its external behavior and how these variations can affect the buyer's commitment. It also examines how salespeople may use a variety of informal influence tactics to moderate the impact of these conflicts on the firm's behavior. These relationships are tested using regression analysis and data from separate samples of sales and purchasing professionals. The data support the position that internal goal conflicts lead to variations in the firm's external behavior, that this relationship is moderated by a salesperson's use of influence tactics, and that these variations have a direct effect on the buyer's commitment.     

A Sandwich-Type Standard Error Estimator of SEM Models with Multivariate Time Series

Structural equation models are increasingly used as a modeling tool for multivariate time series data in the social and behavioral sciences. Standard error estimators of SEM models, originally developed for independent data, require modifications to accommodate the fact that time series data are inherently dependent. In this article, we extend a sandwich-type standard error estimator of independent data to multivariate time series data. One required element of this estimator is the asymptotic covariance matrix of concurrent and lagged correlations among manifest variables, whose closed-form expression has not been presented in the literature. The performance of the adapted sandwich-type standard error estimator is evaluated using a simulation study and further illustrated using an empirical example.     

A Polychoric Instrumental Variable (PIV) Estimator for Structural Equation Models with Categorical Variables

This paper presents a new polychoric instrumental variable (PIV) estimator to use in structural equation models (SEMs) with categorical observed variables. The PIV estimator is a generalization of Bollen’s (Psychometrika 61:109–121, 1996) 2SLS/IV estimator for continuous variables to categorical endogenous variables. We derive the PIV estimator and its asymptotic standard errors for the regression coefficients in the latent variable and measurement models. We also provide an estimator of the variance and covariance parameters of the model, asymptotic standard errors for these, and test statistics of overall model fit. We examine this estimator via an empirical study and also via a small simulation study. Our results illustrate the greater robustness of the PIV estimator to structural misspecifications than the system-wide estimators that are commonly applied in SEMs. Kenneth Bollen gratefully acknowledges support from NSF SES 0617276, NIDA 1-RO1-DA13148-01, and DA013148-05A2. Albert Maydeu-Olivares was supported by the Department of Universities, Research and Information Society (DURSI) of the Catalan Government, and by grant BSO2003-08507 from the Spanish Ministry of Science and Technology. We thank Sharon Christ, John Hipp, and Shawn Bauldry for research assistance. The comments of the members of the Carolina Structural Equation Modeling (CSEM) group are greatly appreciated. An earlier version of this paper under a different title was presented by K. Bollen at the Psychometric Society Meetings, June, 2002, Chapel Hill, North Carolina.     

DECREASING BURNED CHILDREN'S PAIN BEHAVIOR: IMPACTING THE TRAUMA OF HYDROTHERAPY

We evaluated the effects of cartoon viewing with the use of a star feedback chart on two burned children's pain behavior during their physical therapy sessions. In addition, the degree to which the observational data corresponded with physical therapists' and mothers' ratings of the children's pain, fear, and cooperativeness was examined. Using a reversal single-subject design, the results showed that the children's pain behavior substantially decreased during experimental treatment sessions compared to their baseline levels. The rating scale data indicated that the physical therapist's and mother's rating of pain, anxiety, and cooperativeness were all correlated significantly with the observational data (p < .05). The contributions of respondent and operant conditioning to the occurrence and treatment of pain behavior in burned children are discussed.     

Is subordinate's loyalty a precondition of supervisor's benevolent leadership? The moderating effects of supervisor's altruistic personality and perceived organizational support

Based on Cheng's differential leadership theory, we investigated the relationship between a subordinate's loyalty to a supervisor (SLS) and the supervisor's benevolent leadership in Chinese organizations. We also explored two moderators of this relationship, the supervisor's altruistic personality and perceived organizational support (POS). Using survey research, we collected data from supervisor‐subordinate dyads in Taiwan and made 167 valid observations. The results showed that SLS positively relates to the supervisor's benevolent behaviours; however, this relationship is diminished by the supervisor's altruistic personality and POS. That is, when the supervisor has a high level of altruistic personality or POS, the association of SLS with the supervisor's benevolent leadership is weaker.     

Investigating the Impact of Uncertainty About Item Parameters on Ability Estimation

Asymptotic expansions of the maximum likelihood estimator (MLE) and weighted likelihood estimator (WLE) of an examinee’s ability are derived while item parameter estimators are treated as covariates measured with error. The asymptotic formulae present the amount of bias of the ability estimators due to the uncertainty of item parameter estimators. A numerical example is presented to illustrate how to apply the formulae to evaluate the impact of uncertainty about item parameters on ability estimation and the appropriateness of estimating ability using the regular MLE or WLE method.