A comparison of inclusive and restrictive strategies in modern missing data procedures.

Two classes of modem missing data procedures, maximum likelihood (ML) and multiple imputation (MI), tend to yield similar results when implemented in comparable ways. In either approach, it is possible to include auxiliary variables solely for the purpose of improving the missing data procedure. A simulation was presented to assess the potential costs and benefits of a restrictive strategy, which makes minimal use of auxiliary variables, versus an inclusive strategy, which makes liberal use of such variables. The simulation showed that the inclusive strategy is to be greatly preferred. With an inclusive strategy not only is there a reduced chance of inadvertently omitting an important cause of missingness, there is also the possibility of noticeable gains in terms of increased efficiency and reduced bias, with only minor costs. As implemented in currently available software, the MI approach tends to encourage the use of a restrictive strategy, whereas the MI approach makes it relatively simple to use an inclusive strategy. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Reconsidering formative measurement.

The relationship between observable responses and the latent constructs they are purported to measure has received considerable attention recently, with particular focus on what has become known as formative measurement. This alternative to reflective measurement in the area of theory-testing research is examined in the context of the potential for interpretational confounding and a construct's ability to function as a point variable within a larger model. Although these issues have been addressed in the traditional reflective measurement context, the authors suggest that they are particularly relevant in evaluating formative measurement models. On the basis of this analysis, the authors conclude that formative measurement is not an equally attractive alternative to reflective measurement and that whenever possible, in developing new measures or choosing among alternative existing measures, researchers should opt for reflective measurement. In addition, the authors provide guidelines for researchers dealing with existing formative measures. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The impact of nonnormality on full information maximum-likelihood estimation for structural equation models with missing data.

A Monte Carlo simulation examined full information maximum-likelihood estimation (FIML) in structural equation models with nonnormal indicator variables. The impacts of 4 independent variables were examined (missing data algorithm, missing data rate, sample size, and distribution shape) on 4 outcome measures (parameter estimate bias, parameter estimate efficiency, standard error coverage, and model rejection rates). Across missing completely at random and missing at random patterns, FIML parameter estimates involved less bias and were generally more efficient than those of ad hoc missing data techniques. However, similar to complete-data maximum-likelihood estimation in structural equation modeling, standard errors were negatively biased and model rejection rates were inflated. Simulation results suggest that recently developed correctives for missing data (e.g., rescaled statistics and the bootstrap) can mitigate problems that stem from nonnormal data. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The role of hyperarousal in the manifestation of posttraumatic psychological distress following injury.

The authors examined the temporal relation among posttraumatic stress disorder symptom clusters, using data derived from a longitudinal study of survivors of orofacial injury (N = 264). They conducted cross-lagged panel analyses, with self-reported symptom data collected at 1, 6, and 12 months postinjury. Results demonstrate that hyperarousal was a potent predictor of subsequent symptoms of reexperiencing and avoidance as well as hyperarousal. By contrast, neither reexperiencing nor avoidance was significantly related to other symptom clusters other than themselves over time. These findings underscore the distinctive nature of hyperarousal in the manifestation of posttraumatic psychological distress over time. Implications for theory, clinical intervention, and future research are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

New approaches to missing data in psychological research: Introduction to the special section.

Traditional approaches to missing data (e.g., listwise deletion) can lead to less than optimal results in terms of bias, statistical power, or both. This article introduces the 3 articles in the special section of Psychological Methods, which consider multiple imputation and maximum-likelihood methods, new approaches to missing data that can often yield improved results. Computer software is now available to implement these new methods. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Expected versus observed information in SEM with incomplete normal and nonnormal data.

Maximum likelihood is the most common estimation method in structural equation modeling. Standard errors for maximum likelihood estimates are obtained from the associated information matrix, which can be estimated from the sample using either expected or observed information. It is known that, with complete data, estimates based on observed or expected information are consistent. The situation changes with incomplete data. When the data are missing at random (MAR), standard errors based on expected information are not consistent, and observed information should be used. A less known fact is that in the presence of nonnormality, the estimated information matrix also enters the robust computations (both standard errors and the test statistic). Thus, with MAR nonnormal data, the use of the expected information matrix can potentially lead to incorrect robust computations. This article summarizes the results of 2 simulation studies that investigated the effect of using observed versus expected information estimates of standard errors and test statistics with normal and nonnormal incomplete data. Observed information is preferred across all conditions. Recommendations to researchers and software developers are outlined. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Perceived coping as a mediator between attachment and psychological distress: A structural equation modeling approach.

This study examined perceived coping (perceived problem-solving ability and progress in coping with problems) as a mediator between adult attachment (anxiety and avoidance) and psychological distress (depression, hopelessness, anxiety, anger, and interpersonal problems). Survey data from 515 undergraduate students were analyzed using structural equation modeling. Results indicated that perceived coping fully mediated the relationship between attachment anxiety and psychological distress and partially mediated the relationship between attachment avoidance and psychological distress. These findings suggest not only that it is important to consider attachment anxiety or avoidance in understanding distress but also that perceived coping plays an important role in these relationships. Implications for these more complex relations are discussed for both counseling interventions and further research. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Using analysis of covariance (ANCOVA) with fallible covariates.

Analysis of covariance (ANCOVA) is used widely in psychological research implementing nonexperimental designs. However, when covariates are fallible (i.e., measured with error), which is the norm, researchers must choose from among 3 inadequate courses of action: (a) know that the assumption that covariates are perfectly reliable is violated but use ANCOVA anyway (and, most likely, report misleading results); (b) attempt to employ 1 of several measurement error models with the understanding that no research has examined their relative performance and with the added practical difficulty that several of these models are not available in commonly used statistical software; or (c) not use ANCOVA at all. First, we discuss analytic evidence to explain why using ANCOVA with fallible covariates produces bias and a systematic inflation of Type I error rates that may lead to the incorrect conclusion that treatment effects exist. Second, to provide a solution for this problem, we conduct 2 Monte Carlo studies to compare 4 existing approaches for adjusting treatment effects in the presence of covariate measurement error: errors-in-variables (EIV; Warren, White, & Fuller, 1974), Lord's (1960) method, Raaijmakers and Pieters's (1987) method (R&P), and structural equation modeling methods proposed by S?rbom (1978) and Hayduk (1996). Results show that EIV models are superior in terms of parameter accuracy, statistical power, and keeping Type I error close to the nominal value. Finally, we offer a program written in R that performs all needed computations for implementing EIV models so that ANCOVA can be used to obtain accurate results even when covariates are measured with error. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Is formative measurement really measurement? Reply to Bollen (2007) and Bagozzi (2007).

R. D. Howell, E. Breivik, and J. B. Wilcox (2007; see record 2007-07830-006) examined the use of formative measurement models in theory testing in the social sciences. K. A. Bollen (2007; see record 2007-07830-007) and R. P. Bagozzi (2007; see record 2007-07830-008) have provided comments on this work. In this article, the authors reply to the commentators and suggest that the conclusions reached in the original article and the basis for those conclusions remain sound. They address the issue of misspecification raised by Bollen (2007) and the alternative to their realist philosophy of measurement offered by Bagozzi (2007). They conclude that misspecification as construed by Bollen (2007) will typically be undetectable in practice and cannot be distinguished from interpretational confounding. This can result in substantively different constructs retaining the same name from study to study, hindering the accumulation of knowledge. They further conclude that traditional reflective measurement is a better option for researchers in theory testing. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Introduction to the Special Section on Structural Equation Modeling.

Structural equation modeling (SEM) is a frequently used data-analytic technique in psychopathology research. This popularity is due to the unique capabilities and broad applicability of SEM and to recent advances in model and software development. Unfortunately, the popularity and accessibility of SEM is matched by its complexities and ambiguities. Thus, users are often faced with difficult decisions regarding a variety of issues. This special section is designed to increase the effective use of SEM by reviewing recently developed modeling capabilities, identifying common problems in application, and recommending appropriate strategies for analysis and evaluation. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Missing Data Techniques for Structural Equation Modeling.

As with other statistical methods, missing data often create major problems for the estimation of structural equation models (SEMs). Conventional methods such as listwise or pairwise deletion generally do a poor job of using all the available information. However, structural equation modelers are fortunate that many programs for estimating SEMs now have maximum likelihood methods for handling missing data in an optimal fashion. In addition to maximum likelihood, this article also discusses multiple imputation. This method has statistical properties that are almost as good as those for maximum likelihood and can be applied to a much wider array of models and estimation methods. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The prospective relationship of interpersonal forgiveness and psychological distress symptoms among college women.

Forgiveness has frequently been theorized to be related to decreased psychological distress, and longitudinal survey research is important for the examination of this relationship. The prospective relation of forgiveness to psychological distress symptoms (i.e., depression, anxiety, and stress) at a later time point (an average of 36 weeks later) was examined in a sample of 182 female undergraduate students. Through use of structural equation modeling, it was observed that offense-specific (as compared with dispositional) forgiveness toward an offender of a self-identified interpersonal transgression was significantly negatively related to psychological distress symptoms at Time 2, above and beyond the impact of symptom levels at Time 1. Perceived severity and time since the offense at Time 1 were examined as possible moderators of this relationship; time since offense was found to moderate the relationship between forgiveness and change in psychological distress symptoms between Time 1 and Time 2. Implications for acceptance-based interventions and prevention of psychopathology are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Testing parameters in structural equation modeling: Every "one" matters.

A problem with standard errors estimated by many structural equation modeling programs is described. In such programs, a parameter's standard error is sensitive to how the model is identified (i.e., how scale is set). Alternative but equivalent ways to identify a model may yield different standard errors, and hence different Z tests for a parameter, even though the identifications produce the same overall model fit. This lack of invariance due to model identification creates the possibility that different analysts may reach different conclusions about a parameter's significance level even though they test equivalent models on the same data. The authors suggest that parameters be tested for statistical significance through the likelihood ratio test, which is invariant to the identification choice. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Structural equation modeling of multitrait-multimethod data: Different models for different types of methods.

The question as to which structural equation model should be selected when multitrait-multimethod (MTMM) data are analyzed is of interest to many researchers. In the past, attempts to find a well-fitting model have often been data-driven and highly arbitrary. In the present article, the authors argue that the measurement design (type of methods used) should guide the choice of the statistical model to analyze the data. In this respect, the authors distinguish between (a) interchangeable methods, (b) structurally different methods, and (c) the combination of both kinds of methods. The authors present an appropriate model for each type of method. All models allow separating measurement error from trait influences and trait-specific method effects. With respect to interchangeable methods, a multilevel confirmatory factor model is presented. For structurally different methods, the correlated trait-correlated (method-1) model is recommended. Finally, the authors demonstrate how to appropriately analyze data from MTMM designs that simultaneously use interchangeable and structurally different methods. All models are applied to empirical data to illustrate their proper use. Some implications and guidelines for modeling MTMM data are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Three Cs in measurement models: Causal indicators, composite indicators, and covariates.

In the last 2 decades attention to causal (and formative) indicators has grown. Accompanying this growth has been the belief that one can classify indicators into 2 categories: effect (reflective) indicators and causal (formative) indicators. We argue that the dichotomous view is too simple. Instead, there are effect indicators and 3 types of variables on which a latent variable depends: causal indicators, composite (formative) indicators, and covariates (the “Three Cs”). Causal indicators have conceptual unity, and their effects on latent variables are structural. Covariates are not concept measures, but are variables to control to avoid bias in estimating the relations between measures and latent variables. Composite (formative) indicators form exact linear combinations of variables that need not share a concept. Their coefficients are weights rather than structural effects, and composites are a matter of convenience. The failure to distinguish the Three Cs has led to confusion and questions, such as, Are causal and formative indicators different names for the same indicator type? Should an equation with causal or formative indicators have an error term? Are the coefficients of causal indicators less stable than effect indicators? Distinguishing between causal and composite indicators and covariates goes a long way toward eliminating this confusion. We emphasize the key role that subject matter expertise plays in making these distinctions. We provide new guidelines for working with these variable types, including identification of models, scaling latent variables, parameter estimation, and validity assessment. A running empirical example on self-perceived health illustrates our major points. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

A general multilevel SEM framework for assessing multilevel mediation.

Several methods for testing mediation hypotheses with 2-level nested data have been proposed by researchers using a multilevel modeling (MLM) paradigm. However, these MLM approaches do not accommodate mediation pathways with Level-2 outcomes and may produce conflated estimates of between- and within-level components of indirect effects. Moreover, these methods have each appeared in isolation, so a unified framework that integrates the existing methods, as well as new multilevel mediation models, is lacking. Here we show that a multilevel structural equation modeling (MSEM) paradigm can overcome these 2 limitations of mediation analysis with MLM. We present an integrative 2-level MSEM mathematical framework that subsumes new and existing multilevel mediation approaches as special cases. We use several applied examples and accompanying software code to illustrate the flexibility of this framework and to show that different substantive conclusions can be drawn using MSEM versus MLM. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The Use of Latent Trajectory Models in Psychopathology Research.

Despite the recent surge in the development of powerful modeling strategies to test questions about individual differences in stability and change over time, these methods are not currently widely used in psychopathology research. In an attempt to further the dissemination of these new methods, the authors present a pedagogical introduction to the structural equation modeling based latent trajectory model, or LTM. They review several different types of LTMs, discuss matching an optimal LTM to a given question of interest, and highlight several issues that might be particularly salient for research in psychopathology. The authors augment each section with a review of published applications of these methods in psychopathology-related research to demonstrate the implementation and interpretation of LTMs in practice. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Fitting data to model: Structural equation modeling diagnosis using two scatter plots.

This article introduces two simple scatter plots for model diagnosis in structural equation modeling. One plot contrasts a residual-based M-distance of the structural model with the M-distance for the factor score. It contains information on outliers, good leverage observations, bad leverage observations, and normal cases. The other plot contrasts the residual-based M-distance with the quantile of a chi distribution. It allows the researcher to visually identify clusters of potential outliers. The article further studies the effect of the potential outliers on the overall model evaluation when they are removed according to the order of the clusters exhibited in the plot. Suggestions are provided on determining the outlier status of outstanding cases in real data analysis. Recommendations are also made on the choice of robust methods and maximum likelihood following outlier removal. (PsycINFO Database Record (c) 2010 APA, all rights reserved)