Examining Normality in Incomplete Data Sets: A Note on a Multiple Testing Approach

A multiple testing procedure for examining the assumption of normality that is often made in analyses of incomplete data sets is outlined. The method is concerned with testing normality within each missingness pattern and arriving at an overall statement about normality using the available data. The approach is readily applied in empirical research with missing data using the popular software Mplus, Stata, and R. The procedure can be used to ascertain a main assumption underlying frequent applications of maximum likelihood in incomplete data modeling with continuous outcomes. The discussed approach is illustrated with numerical examples.     

Examining the Missing Completely at Random Mechanism in Incomplete Data Sets: A Multiple Testing Approach

A multiple testing procedure for examining implications of the missing completely at random (MCAR) mechanism in incomplete data sets is discussed. The approach uses the false discovery rate concept and is concerned with testing group differences on a set of variables. The method can be used for ascertaining violations of MCAR and disproving this mechanism in empirical behavioral and social research. The procedure can also be employed when locating violations of MCAR in observed measures is of interest. The outlined approach is illustrated with data from a cognitive intervention study.     

Analysis of Longitudinal Studies With Missing Data Using Covariance Structure Modeling With Full-Information Maximum Likelihood

A didactic discussion of covariance structure modeling in longitudinal studies with missing data is presented. Use of the full-information maximum likelihood method is considered for model fitting, parameter estimation, and hypothesis testing purposes, particularly when interested in patterns of temporal change as well as its covariates and predictors. The approach is illustrated with an application of the popular level-and-shape model to data from a cognitive intervention study of elderly adults.     

Scale Reliability Evaluation Under Multiple Assumption Violations

A latent variable modeling approach to evaluate scale reliability under realistic conditions in empirical behavioral and social research is discussed. The method provides point and interval estimation of reliability of multicomponent measuring instruments when several assumptions are violated. These assumptions include missing data, correlated errors, nonnormality, lack of unidimensionality, and data not missing at random. The procedure can be readily used to aid scale construction and development efforts in applied settings, and is illustrated using data from an educational study.     

Reducing the length of questionnaires through structurally incomplete designs: An illustration

This paper serves as an illustration of the usefulness of structurally incomplete designs as an approach to reduce the length of educational questionnaires. In structurally incomplete test designs, respondents only fill out a subset of the total item set, while all items are still provided to the whole sample. The scores on the unadministered items are subsequently dealt with by using methods for the estimation of missing data. Two structurally incomplete test designs — one recording two thirds, and the other recording a half of the potentially complete data — were applied to the complete item scores on 8 educational psychology scales. The incomplete item scores were estimated with missing data method Data Augmentation. Complete and estimated test data were compared at the estimates of total scores, reliability, and predictive validity of an external criterion. The reconstructed data yielded estimates that were very close to the values in the complete data. As expected the statistical uncertainty was higher in the design that recorded fewer item scores. It was concluded that the procedure of applying incomplete test designs and subsequently dealing with the missing values is very fruitful for reducing questionnaire length.     

Handling Missing Data in the Modeling of Intensive Longitudinal Data

Myriad approaches for handling missing data exist in the literature. However, few studies have investigated the tenability and utility of these approaches when used with intensive longitudinal data. In this study, we compare and illustrate two multiple imputation (MI) approaches for coping with missingness in fitting multivariate time-series models under different missing data mechanisms. They include a full MI approach, in which all dependent variables and covariates are imputed simultaneously, and a partial MI approach, in which missing covariates are imputed with MI, whereas missingness in the dependent variables is handled via full information maximum likelihood estimation. We found that under correctly specified models, partial MI produces the best overall estimation results. We discuss the strengths and limitations of the two MI approaches, and demonstrate their use with an empirical data set in which children’s influences on parental conflicts are modeled as covariates over the course of 15 days (Schermerhorn, Chow, & Cummings, 2010).     

Detecting Individual Differences in Change: Methods and Comparisons

This study examined and compared various statistical methods for detecting individual differences in change. Considering 3 issues including test forms (specific vs. generalized), estimation procedures (constrained vs. unconstrained), and nonnormality, we evaluated 4 variance tests including the specific Wald variance test, the generalized Wald variance test, the specific likelihood ratio (LR) variance test, and the generalized LR variance test under both constrained and unconstrained estimation for both normal and nonnormal data. For the constrained estimation procedure, both the mixture distribution approach and the alpha correction approach were evaluated for their performance in dealing with the boundary problem. To deal with the nonnormality issue, we used the sandwich standard error (SE) estimator for the Wald tests and the Satorra–Bentler scaling correction for the LR tests. Simulation results revealed that testing a variance parameter and the associated covariances (generalized) had higher power than testing the variance solely (specific), unless the true covariances were zero. In addition, the variance tests under constrained estimation outperformed those under unconstrained estimation in terms of higher empirical power and better control of Type I error rates. Among all the studied tests, for both normal and nonnormal data, the robust generalized LR and Wald variance tests with the constrained estimation procedure were generally more powerful and had better Type I error rates for testing variance components than the other tests. Results from the comparisons between specific and generalized variance tests and between constrained and unconstrained estimation were discussed.     

A New Approach to Comparing Several Equating Methods in the Context of the NEAT Design

The nonequivalent groups with anchor test (NEAT) design involves missing data that are missing by design. Three equating methods that can be used with a NEAT design are the frequency estimation equipercentile equating method, the chain equipercentile equating method, and the item-response-theory observed-score-equating method. We suggest an approach to perform a fair comparison of the three methods. The approach is then applied to compare the three equating methods using three data sets from operational tests. For each data set, we examine how the three equating methods perform when the missing data satisfy the assumptions made by only one of these equating methods. The chain equipercentile equating method is somewhat more satisfactory overall than the other methods.     

Testing Criterion Correlations With Scale Component Measurement Errors Using Latent Variable Modeling

A latent variable modeling method for testing criterion correlations with measurement error terms in multicomponent measuring instruments is outlined. The approach is based on an application of the Benjamini–Hochberg multiple testing procedure and can be used when assumptions of validity estimation related procedures need to be examined. The method also allows studying the extent to which criterion validity coefficients might be due to the relationship between a presumed underlying latent construct evaluated by a psychometric scale and a criterion variable, or could be a consequence of the relation between measurement error in the overall scale score and the criterion. The discussed procedure is widely applicable with popular latent variable modeling software, and is illustrated using a numerical example.     

A Bayesian Approach for Analyzing Hierarchical Data With Missing Outcomes Through Structural Equation Models

Structural equation models are widely appreciated in behavioral, social, and psychological research to model relations between latent constructs and manifest variables, and to control for measurement errors. Most applications of structural equation models are based on fully observed data that are independently distributed. However, hierarchical data with a correlated structure are common in behavioral research, and very often, missing data are encountered. In this article, we propose a 2-level structural equation model for analyzing hierarchical data with missing entries, and describe a Bayesian approach for estimation and model comparison. We show how to use WinBUGS software to get the solution conveniently. The proposed methodologies are illustrated through a simulation study, and a real application in relation to organizational and management research concerning the study of the interrelationships of the latent constructs about job satisfaction, job responsibility, and life satisfaction for citizens in 43 countries.     

Estimating Classification Accuracy and Consistency Indices for Multiple Measures with the Simple Structure MIRT Model

Multiple measures, such as multiple content domains or multiple types of performance, are used in various testing programs to classify examinees for screening or selection. Despite the popular usages of multiple measures, there is little research on classification consistency and accuracy of multiple measures. Accordingly, this study introduces an approach to estimate classification consistency and accuracy indices for multiple measures under four possible decision rules: (1) complementary, (2) conjunctive, (3) compensatory, and (4) pairwise combinations of the three. The current study uses the IRT-recursive-based approach with the simple-structure multidimensional IRT model (SS-MIRT) to estimate the classification consistency and accuracy for multiple measures. Theoretical formulations of the four decision rules with a binary decision (Pass/Fail) are presented. The estimation procedures are illustrated using an empirical data example based on SS-MIRT. In addition, this study applies the estimation procedures to the unidimensional IRT (UIRT) context, considering that UIRT is practically used more. This application shows that the proposed procedure of classification consistency and accuracy could be used with a UIRT model for individual measures as an alternative method of SS-MIRT.     

Parameter Estimation in Rasch Models for Examinee‐Selected Items

The examinee‐selected‐item (ESI) design, in which examinees are required to respond to a fixed number of items in a given set of items (e.g., choose one item to respond from a pair of items), always yields incomplete data (i.e., only the selected items are answered and the others have missing data) that are likely nonignorable. Therefore, using standard item response theory models, which assume ignorable missing data, can yield biased parameter estimates so that examinees taking different sets of items to answer cannot be compared. To solve this fundamental problem, in this study the researchers utilized the specific objectivity of Rasch models by adopting the conditional maximum likelihood estimation (CMLE) and pairwise estimation (PE) methods to analyze ESI data, and conducted a series of simulations to demonstrate the advantages of the CMLE and PE methods over traditional estimation methods in recovering item parameters in ESI data. An empirical data set obtained from an experiment on the ESI design was analyzed to illustrate the implications and applications of the proposed approach to ESI data.     

Logistic Regression Procedure Using Penalized Maximum Likelihood Estimation for Differential Item Functioning

In the logistic regression (LR) procedure for differential item functioning (DIF), the parameters of LR have often been estimated using maximum likelihood (ML) estimation. However, ML estimation suffers from the finite-sample bias. Furthermore, ML estimation for LR can be substantially biased in the presence of rare event data. The bias of ML estimation due to small samples and rare event data can degrade the performance of the LR procedure, especially when testing the DIF of difficult items in small samples. Penalized ML (PML) estimation was originally developed to reduce the finite-sample bias of conventional ML estimation and also was known to reduce the bias in the estimation of LR for the rare events data. The goal of this study is to compare the performances of the LR procedures based on the ML and PML estimation in terms of the statistical power and Type I error. In a simulation study, Swaminathan and Rogers's Wald test based on PML estimation (PSR) showed the highest statistical power in most of the simulation conditions, and LRT based on conventional PML estimation (PLRT) showed the most robust and stable Type I error. The discussion about the trade-off between bias and variance is presented in the discussion section.     

Lower Level Mediation Effect Analysis in Two-Level Studies: A Note on a Multilevel Structural Equation Modeling Approach

This article presents a didactic discussion of a multilevel covariance structure modeling approach to estimation of lowest level mediation effect indexes in two-level studies. The procedure is useful when addressing questions about relations among total and indirect effects between variables of interest while accounting for the hierarchical structure of analyzed data. The discussed method also permits interval estimation and hypothesis tests with respect to related quantities of relevance when evaluating mediated effects with clustered data, and is illustrated on a two-level data set.     

Brief Research Report: Growth Models With Small Samples and Missing Data

Small samples are common in growth models due to financial and logistical difficulties of following people longitudinally. For similar reasons, longitudinal studies often contain missing data. Though full information maximum likelihood (FIML) is popular to accommodate missing data, the limited number of studies in this area have found that FIML tends to perform poorly with small-sample growth models. This report demonstrates that the fault lies not with how FIML accommodates missingness but rather with maximum likelihood estimation itself. We discuss how the less popular restricted likelihood form of FIML, along with small-sample-appropriate methods, yields trustworthy estimates for growth models with small samples and missing data. That is, previously reported small sample issues with FIML are attributable to finite sample bias of maximum likelihood estimation not direct likelihood. Estimation issues pertinent to joint multiple imputation and predictive mean matching are also included and discussed.     

Estimation of Reliability for Multiple-Component Measuring Instruments in Hierarchical Designs

A method for estimation of reliability for multiple-component measuring instruments with clustered data is outlined. The approach is applicable with hierarchical designs where individuals are nested within higher order units and exhibit possibly related performance on components of a scale of interest. The procedure is developed within the framework of multilevel covariance structure modeling. The described method is useful for point and interval estimation of the degree of consistency of measurement with congeneric composites in hierarchical populations, and is illustrated with an empirical example.     

Interval Estimation of Revision Effect on Scale Reliability via Covariance Structure Modeling

A didactic discussion of a procedure for interval estimation of change in scale reliability due to revision is provided, which is developed within the framework of covariance structure modeling. The method yields ranges of plausible values for the population gain or loss in reliability of unidimensional composites, which results from deletion or addition of scale components. The obtained confidence intervals can also be used for testing conventional as well as minimum effect hypotheses about the associated increment or drop in reliability. The approach yields as a by-product point and interval estimates for reliability of any instrument version, and is illustrated with two examples.     

A Primer on Maximum Likelihood Algorithms Available for Use With Missing Data

Maximum likelihood algorithms for use with missing data are becoming commonplace in microcomputer packages. Specifically, 3 maximum likelihood algorithms are currently available in existing software packages: the multiple-group approach, full information maximum likelihood estimation, and the EM algorithm. Although they belong to the same family of estimator, confusion appears to exist over the differences among the 3 algorithms. This article provides a comprehensive, nontechnical overview of the 3 maximum likelihood algorithms. Multiple imputation, which is frequently used in conjunction with the EM algorithm, is also discussed.     

Score Reporting for Examinees with Incomplete Data on Large‐Scale Educational Assessments

Technical difficulties occasionally lead to missing item scores and hence to incomplete data on computerized tests. It is not straightforward to report scores to the examinees whose data are incomplete due to technical difficulties. Such reporting essentially involves imputation of missing scores. In this paper, a simulation study based on data from three educational tests is used to compare the performances of six approaches for imputation of missing scores. One of the approaches, based on data mining, is the first application of its kind to the problem of imputation of missing data. The approach based on data mining and a multiple imputation approach based on chained equations led to the most accurate imputation of missing scores, and hence to most accurate score reporting. A simple approach based on linear regression performed the next best overall. Several recommendations are made regarding the reporting of scores to examinees with incomplete data.     

Efficacy of the indirect approach for estimating structural equation models with missing data: A comparison of five methods

Incomplete or missing data are routinely encountered in structural equation problems. Although current literature supports the use of a direct approach for modeling the missing values in a structural equation model, many situations are not applicable for the effective use of this approach. This leaves the use of an indirect approach for dealing with missing information. There is a general lack of knowledge regarding the efficacy of the use of the indirect approach in structural equation modeling. This article assesses the efficacy of five indirect methods for estimating parameters in a structural equation model with various levels of missing data.