An Empirical Evaluation of Alternative Methods of Estimation for Confirmatory Factor Analysis With Ordinal Data.

Confirmatory factor analysis (CFA) is widely used for examining hypothesized relations among ordinal variables (e.g., Likert-type items). A theoretically appropriate method fits the CFA model to polychoric correlations using either weighted least squares (WLS) or robust WLS. Importantly, this approach assumes that a continuous, normal latent process determines each observed variable. The extent to which violations of this assumption undermine CFA estimation is not well-known. In this article, the authors empirically study this issue using a computer simulation study. The results suggest that estimation of polychoric correlations is robust to modest violations of underlying normality. Further, WLS performed adequately only at the largest sample size but led to substantial estimation difficulties with smaller samples. Finally, robust WLS performed well across all conditions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The Integration of Continuous and Discrete Latent Variable Models: Potential Problems and Promising Opportunities.

Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes in SEMM: misspecification of the structural model, nonnormal continuous measures, and nonlinear relationships among observed and/or latent variables. When the objective of a SEMM analysis is the identification of latent classes, these conditions should be considered as alternative hypotheses and results should be interpreted cautiously. However, armed with greater knowledge about the estimation of SEMMs in practice, researchers can exploit the flexibility of the model to gain a fuller understanding of the phenomenon under study. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Statistical and Substantive Checking in Growth Mixture Modeling: Comment on Bauer and Curran (2003).

This commentary discusses the D. J. Bauer and P. J. Curran (see record 2003-09632-007) investigation of growth mixture modeling. Single-class modeling of nonnormal outcomes is compared with modeling with multiple latent trajectory classes. New statistical tests of multiple-class models are discussed. Principles for substantive investigation of growth mixture model results are presented and illustrated by an example of high school dropout predicted by low mathematics achievement development in Grades 7-10. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Information-theoretic latent distribution modeling: Distinguishing discrete and continuous latent variable models.

Distinguishing between discrete and continuous latent variable distributions has become increasingly important in numerous domains of behavioral science. Here, the authors explore an information-theoretic approach to latent distribution modeling, in which the ability of latent distribution models to represent statistical information in observed data is emphasized. The authors conclude that loss of statistical information with a decrease in the number of latent values provides an attractive basis for comparing discrete and continuous latent variable models. Theoretical considerations as well as the results of 2 Monte Carlo simulations indicate that information theory provides a sound basis for modeling latent distributions and distinguishing between discrete and continuous latent variable models in particular. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Modeling stage-sequential change in ordered categorical responses.

Although few would dispute the usefulness of looking at behavioral change from a stage-sequential perspective, until recently the lack of appropriate modeling techniques has hampered rigorous empirical tests of stage theories. In particular, for behavioral measurements that are ordinal, there is a need for methods that represent the underlying change processes in the form of qualitative and discontinuous shifts. This article introduces a stage-sequential ordinal model by postulating that at any point in time there are a finite number of latent stages. Panel members may shift among these stages over time. The authors show that the stage-sequential model provides a general approach for both the analysis of ordinal time-dependent data and tests of various competing theories and hypotheses about psychological change processes. An analysis of a 5-year study concerning attitudes toward alcohol consumption by teenagers is presented to illustrate the modeling approach. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Measurement of psychological disorders using cognitive diagnosis models.

Cognitive diagnosis models are constrained (multiple classification) latent class models that characterize the relationship of questionnaire responses to a set of dichotomous latent variables. Having emanated from educational measurement, several aspects of such models seem well suited to use in psychological assessment and diagnosis. This article presents the development of a new cognitive diagnosis model for use in psychological assessment--the DINO (deterministic input; noisy "or" gate) model--which, as an illustrative example, is applied to evaluate and diagnose pathological gamblers. As part of this example, a demonstration of the estimates obtained by cognitive diagnosis models is provided. Such estimates include the probability an individual meets each of a set of dichotomous Diagnostic and Statistical Manual of Mental Disorders (text revision [DSM-IV-TR]; American Psychiatric Association, 2000) criteria, resulting in an estimate of the probability an individual meets the DSM-IV-TR definition for being a pathological gambler. Furthermore, a demonstration of how the hypothesized underlying factors contributing to pathological gambling can be measured with the DINO model is presented, through use of a covariance structure model for the tetrachoric correlation matrix of the dichotomous latent variables representing DSM-IV-TR criteria. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Reporting results of latent growth modeling and multilevel modeling analyses: Some recommendations for rehabilitation psychology.

Objective: There has been a general increase in interest and use of modeling techniques that treat data as nested, whether it is people nested within larger units, such as families or treatment centers, or observations nested under people. The popularity can be witnessed by noting the number of new textbooks and articles related to latent growth curve modeling and multilevel modeling. This paper discusses both of these techniques in the context of longitudinal research designs, with the main purposes of highlighting some benefits and issues related to the use of these models and outlining guidelines for reporting results from studies using multilevel modeling or latent growth modeling. Implications: These longitudinal analytic techniques can be greatly beneficial to researchers conducting rehabilitation studies, but there are several issues related to their use and reporting that need to be taken into consideration. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Growth modeling with nonignorable dropout: Alternative analyses of the STAR*D antidepressant trial.

This article uses a general latent variable framework to study a series of models for nonignorable missingness due to dropout. Nonignorable missing data modeling acknowledges that missingness may depend not only on covariates and observed outcomes at previous time points as with the standard missing at random assumption, but also on latent variables such as values that would have been observed (missing outcomes), developmental trends (growth factors), and qualitatively different types of development (latent trajectory classes). These alternative predictors of missing data can be explored in a general latent variable framework with the Mplus program. A flexible new model uses an extended pattern-mixture approach where missingness is a function of latent dropout classes in combination with growth mixture modeling. A new selection model not only allows an influence of the outcomes on missingness but allows this influence to vary across classes. Model selection is discussed. The missing data models are applied to longitudinal data from the Sequenced Treatment Alternatives to Relieve Depression (STAR*D) study, the largest antidepressant clinical trial in the United States to date. Despite the importance of this trial, STAR*D growth model analyses using nonignorable missing data techniques have not been explored until now. The STAR*D data are shown to feature distinct trajectory classes, including a low class corresponding to substantial improvement in depression, a minority class with a U-shaped curve corresponding to transient improvement, and a high class corresponding to no improvement. The analyses provide a new way to assess drug efficiency in the presence of dropout. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Ramsay-curve item response theory (RC-IRT) to detect and correct for nonnormal latent variables.

Popular methods for fitting unidimensional item response theory (IRT) models to data assume that the latent variable is normally distributed in the population of respondents, but this can be unreasonable for some variables. Ramsay-curve IRT (RC-IRT) was developed to detect and correct for this nonnormality. The primary aims of this article are to introduce RC-IRT less technically than it has been described elsewhere; to evaluate RC-IRT for ordinal data via simulation, including new approaches for model selection; and to illustrate RC-IRT with empirical examples. The empirical examples demonstrate the utility of RC-IRT for real data, and the simulation study indicates that when the latent distribution is skewed, RC-IRT results can be more accurate than those based on the normal model. Along with a plot of candidate curves, the Hannan-Quinn criterion is recommended for model selection. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Analysis of multivariate frequency data by graphical models and generalizations of the multidimensional row-column association model.

Models used to analyze cross-classifications of counts from psychological experiments must represent associations between multiple discrete variables and take into account attributes of stimuli, experimental conditions, or characteristics of subjects. The models must also lend themselves to psychological interpretations about underlying structures mediating the relationship between stimuli and responses. To meet these needs, the author extends the graphical latent variable models for nominal and/or ordinal data proposed by C. J. Anderson and J. K. Vermunt (2000) to situations in which dependencies between observed variables are not fully accounted for by the latent variables. The graphical models provide a unified framework for studying multivariate associations that include log-linear models and log-multiplicative association models as special cases. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Ordinal Process Dissociation and the Measurement of Automatic and Controlled Processes.

The process-dissociation equations (L. Jacoby, 1991) have been applied to results from inclusion and exclusion tasks to derive quantitative estimates of the influence of controlled and automatic processes on memory. This research has provoked controversies (e.g., T. Curran & D. Hintzman, 1995) regarding the validity of specific assumptions underlying the process-dissociation equations. In this article, the author explores the conclusions one can draw about the ordinal relations between automatic and controlled processes across experimental conditions from results in the inclusion and exclusion tasks. Given relatively neutral assumptions, this article presents and proves 6 theorems that allow investigators to draw conclusions about the ordinal relations between automatic and/or controlled processes across experimental conditions. An illustrative example is presented, and the current approach is compared with the original process-dissociation framework. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Parametric statistics and levels of measurement: Factorial designs and multiple regression.

Let Y be a continuous, ordinal measure of a latent variable Θ. In general, for factorial designs, an analysis of variance of the observed variable Y cannot be used to draw inferences about main effects and interactions on the latent variable Θ even when the standard normality and equality of variance assumptions hold. If Y is a continuous, ordinal measure of a latent variable Θ; X?,…, Xn are continuous, ordinal measures of latent variables Φ?,…, Φn; and the observed measures have a multivariate normal distribution, then a multiple regression analysis of the observed criterion measure Y and predictors X?,…, Xn can be used to test hypotheses about multivariate associations among the latent variables. Furthermore, the predicted values Y′ are unbiased estimates of quantities that are monotonically related to predicted values on the latent criterion variable Θ. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Omnibus Hypothesis Testing in Dominance-Based Ordinal Multiple Regression.

Often quantitative data in the social sciences have only ordinal justification. Problems of interpretation can arise when least squares multiple regression (LSMR) is used with ordinal data. Two ordinal alternatives are discussed, dominance-based ordinal multiple regression (DOMR) and proportional odds multiple regression. The Q2 statistic is introduced for testing the omnibus null hypothesis in DOMR. A simulation study is discussed that examines the actual Type I error rate and power of Q2 in comparison to the LSMR omnibus F test under normality and non-normality. Results suggest that Q2 has favorable sampling properties as long as the sample size-to-predictors ratio is not too small, and Q2 can be a good alternative to the omnibus F test when the response variable is non-normal. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Ordinal Representation of Numeric Quantities by Brown Capuchin Monkeys (Cebus apella).

Using techniques established by E. M. Brannon and H. S. Terrace (2000) with rhesus macaques (Macaca mulatta), the authors tested the ability of brown capuchins (Cebus apella) to order arrays of items ranging in quantity from 1 to 9. Three monkeys were trained on a touch screen to select the quantities 1-4 in ascending order. The monkeys exhibited successful transfer of this ability to novel representations of the quantities 1-4 and to pairs of the novel quantities 5-9. Patterns of responding with respect to numeric distance and magnitude were similar to those seen in human subjects, suggesting the use of similar psychological processes. The capuchins demonstrated an ordinal representation of quantity equivalent to that shown in Old World monkeys. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Mixture distribution latent state-trait analysis: Basic ideas and applications.

Extensions of latent state-trait models for continuous observed variables to mixture latent state-trait models with and without covariates of change are presented that can separate individuals differing in their occasion-specific variability. An empirical application to the repeated measurement of mood states (N = 501) revealed that a model with 2 latent classes fits the data well. The larger class (76%) consists of individuals whose mood is highly variable, whose general well-being is comparatively lower, and whose mood variability is influenced by daily hassles and uplifts. The smaller class (24%) represents individuals who are rather stable and happier and whose mood is influenced only by daily uplifts but not by daily hassles. A simulation study on the model without covariates with 5 sets of sample sizes and 5 sets of number of occasions revealed that the appropriateness of the parameter estimates of this model depends on number of observations (the higher the better) and number of occasions (the higher the better). Another simulation study estimated Type I and II errors of the Lo-Mendell-Rubin test. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The theoretical status of latent variables.

This article examines the theoretical status of latent variables as used in modern test theory models. First, it is argued that a consistent interpretation of such models requires a realist ontology for latent variables. Second, the relation between latent variables and their indicators is discussed. It is maintained that this relation can be interpreted as a causal one but that in measurement models for interindividual differences the relation does not apply to the level of the individual person. To substantiate intraindividual causal conclusions, one must explicitly represent individual level processes in the measurement model. Several research strategies that may be useful in this respect are discussed, and a typology of constructs is proposed on the basis of this analysis. The need to link individual processes to latent variable models for interindividual differences is emphasized. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Structural equation modeling with longitudinal data: Strategies for examining group differences and reciprocal relationships.

This article describes the use of structural equation modeling with latent variables to examine group differences and test competing models about cause–effect relationships in passive longitudinal designs. This approach is compared with several other statistical methods including analysis of cross-lagged panel correlations, regression analysis, and path analysis. The mechanics and advantages of structural equation modeling are illustrated using an example based on a 3-wave longitudinal study of adolescents' alcohol use. Within this example, the generalizability of the measurement model and structural model are assessed across gender and time, and competing models about the causes and consequences of adolescents' alcohol use are tested. The article concludes with a discussion of some of the strengths and limitations of using structural equation modeling with longitudinal data. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Analyzing longitudinal data with multilevel models: An example with individuals living with lower extremity intra-articular fractures.

Objective: The use and quality of longitudinal research designs has increased over the past 2 decades, and new approaches for analyzing longitudinal data, including multilevel modeling (MLM) and latent growth modeling (LGM), have been developed. The purpose of this article is to demonstrate the use of MLM and its advantages in analyzing longitudinal data. Research Method: Data from a sample of individuals with intra-articular fractures of the lower extremity from the University of Alabama at Birmingham's Injury Control Research Center are analyzed using both SAS PROC MIXED and SPSS MIXED. Results: The authors begin their presentation with a discussion of data preparation for MLM analyses. The authors then provide example analyses of different growth models, including a simple linear growth model and a model with a time-invariant covariate, with interpretation for all the parameters in the models. Implications: More complicated growth models with different between- and within-individual covariance structures and nonlinear models are discussed. Finally, information related to MLM analysis, such as online resources, is provided at the end of the article. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Measurement invariance versus selection invariance: Is fair selection possible?

This article shows that measurement invariance (defined in terms of an invariant measurement model in different groups) is generally inconsistent with selection invariance (defined in terms of equal sensitivity and specificity across groups). In particular, when a unidimensional measurement instrument is used and group differences are present in the location but not in the variance of the latent distribution, sensitivity and positive predictive value will be higher in the group at the higher end of the latent dimension, whereas specificity and negative predictive value will be higher in the group at the lower end of the latent dimension. When latent variances are unequal, the differences in these quantities depend on the size of group differences in variances relative to the size of group differences in means. The effect originates as a special case of Simpson's paradox, which arises because the observed score distribution is collapsed into an accept-reject dichotomy. Simulations show the effect can be substantial in realistic situations. It is suggested that the effect may be partly responsible for overprediction in minority groups as typically found in empirical studies on differential academic performance. A methodological solution to the problem is suggested, and social policy implications are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)