Modeling Nonlinear Change via Latent Change and Latent Acceleration Frameworks: Examining Velocity and Acceleration of Growth Trajectories

A general equadion is presented, covering all arbitrary values for the true population splits, for obtaining the true population phi, given observed cell frequencies for a selected sample, and true population splits. A nongeneral solution is also offered, based on the use of the G Index. Demonstrations with hypothetical data are given.     

Predicting Latent Class Scores for Subsequent Analysis

Latent class regression models relate covariates and latent constructs such as psychiatric disorders. Though full maximum likelihood estimation is available, estimation is often in three steps: (i) a latent class model is fitted without covariates; (ii) latent class scores are predicted; and (iii) the scores are regressed on covariates. We propose a new method for predicting class scores that, in contrast to posterior probability-based methods, yields consistent estimators of the parameters in the third step. Additionally, in simulation studies the new methodology exhibited only a minor loss of efficiency. Finally, the new and the posterior probability-based methods are compared in an analysis of mobility/exercise.     

Solution Strategies and Achievement in Dutch Complex Arithmetic: Latent Variable Modeling of Change

In the Netherlands, national assessments at the end of primary school (Grade 6) show a decline of achievement on problems of complex or written arithmetic over the last two decades. The present study aims at contributing to an explanation of the large achievement decrease on complex division, by investigating the strategies students used in solving the division problems in the two most recent assessments carried out in 1997 and in 2004. The students’ strategies were classified into four categories. A data set resulted with two types of repeated observations within students: the nominal strategies and the dichotomous achievement scores (correct/incorrect) on the items administered.It is argued that latent variable modeling methodology is appropriate to analyze these data. First, latent class analyses with year of assessment as a covariate were carried out on the multivariate nominal strategy variables. Results showed a shift from application of the traditional long division algorithm in 1997, to the less accurate strategy of stating an answer without writing down any notes or calculations in 2004, especially for boys. Second, explanatory IRT analyses showed that the three main strategies were significantly less accurate in 2004 than they were in 1997.     

Bootstrap-Calibrated Interval Estimates for Latent Variable Scores in Item Response Theory

In most item response theory applications, model parameters need to be first calibrated from sample data. Latent variable (LV) scores calculated using estimated parameters are thus subject to sampling error inherited from the calibration stage. In this article, we propose a resampling-based method, namely bootstrap calibration (BC), to reduce the impact of the carryover sampling error on the interval estimates of LV scores. BC modifies the quantile of the plug-in posterior, i.e., the posterior distribution of the LV evaluated at the estimated model parameters, to better match the corresponding quantile of the true posterior, i.e., the posterior distribution evaluated at the true model parameters, over repeated sampling of calibration data. Furthermore, to achieve better coverage of the fixed true LV score, we explore the use of BC in conjunction with Jeffreys’ prior. We investigate the finite-sample performance of BC via Monte Carlo simulations and apply it to two empirical data examples.     

Measuring Patterns of Change in Personality Assessments: An Annotated Application of Latent Growth Curve Modeling

Latent growth curve (LGC) modeling within the framework of structural equation modeling (SEM) is now highly regarded as one of the most powerful and informative approaches to the analysis of longitudinal data (see, e.g., Curran &; Hussong, 2003 Curran, P. J. and Hussong, A. M. 2003. The use of latent trajectory models in psychopathology research. Journal of Abnormal Psychology, 112: 526–544. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Whereas LGC modeling enables researchers to test for differences in developmental trajectories across time, conventional repeated measures analyses do not provide this opportunity. Nonetheless, a review of studies reported in most psychology journals reveals scant application of this methodological approach. One possible explanation for this limited use of LGC modeling is a lack of knowledge related to its application. The intent of this article, then, is to address this deficiency by presenting an annotated application of LGC modeling to health psychology data. Based on a sample of 405 Hong Kong Chinese women who recently underwent breast cancer surgery, we walk the readers through SEM modeling procedures that test for differences in both the initial status and rate of change in Psychological Morbidity and Social Adjustment at 1, 4, and 8 months postsurgery. We interpret findings from both a methodological and a substantive perspective.     

The Latent Change Score Model: A More Flexible Approach to Modeling Time in Self-Regulated Learning

Purpose

This research advances understanding of empirical time modeling techniques in self-regulated learning research. We intuitively explain several such methods by situating their use in the extant literature. Further, we note key statistical and inferential assumptions of each method while making clear the inferential consequences of inattention to such assumptions.

Design/Methodology/Approach

Using a population model derived from a recent large-scale review of the training and work learning literature, we employ a Monte Carlo simulation fitting six variations of linear mixed models, seven variations of latent common factor models, and a single latent change score model to 1500 simulated datasets.

Findings

The latent change score model outperformed all six of the linear mixed models and all seven of the latent common factor models with respect to (1) estimation precision of the average learner improvement, (2) correctly rejecting a false null hypothesis about such average improvement, and (3) correctly failing to reject true null hypothesis about between-learner differences (i.e., random slopes) in average improvement.

Implications

The latent change score model is a more flexible method of modeling time in self-regulated learning research, particularly for learner processes consistent with twenty-first-century workplaces. Consequently, defaulting to linear mixed or latent common factor modeling methods may have adverse inferential consequences for better understanding self-regulated learning in twenty-first-century work.

Originality/Value

Ours is the first study to critically, rigorously, and empirically evaluate self-regulated learning modeling methods and to provide a more flexible alternative consistent with modern self-regulated learning knowledge.

     

Review of Issues About Classical Change Scores: A Multilevel Modeling Perspective on Some Enduring Beliefs

Change scores obtained in pretest–posttest designs are important for evaluating treatment effectiveness and for assessing change of individual test scores in psychological research. However, over the years the use of change scores has raised much controversy. In this article, from a multilevel perspective, we provide a structured treatise on several persistent negative beliefs about change scores and show that these beliefs originated from the confounding of the effects of within-person change on change-score reliability and between-person change differences. We argue that psychometric properties of change scores, such as reliability and measurement precision, should be treated at suitable levels within a multilevel framework. We show that, if examined at the suitable levels with such a framework, the negative beliefs about change scores can be renounced convincingly. Finally, we summarize the conclusions about change scores to dispel the myths and to promote the potential and practical usefulness of change scores.     

Multilevel Reliability Measures of Latent Scores Within an Item Response Theory Framework

Abstract

This paper evaluated multilevel reliability measures in two-level nested designs (e.g., students nested within teachers) within an item response theory framework. A simulation study was implemented to investigate the behavior of the multilevel reliability measures and the uncertainty associated with the measures in various multilevel designs regarding the number of clusters, cluster sizes, and intraclass correlations (ICCs), and in different test lengths, for two parameterizations of multilevel item response models with separate item discriminations or the same item discrimination over levels. Marginal maximum likelihood estimation (MMLE)-multiple imputation and Bayesian analysis were employed to evaluate the accuracy of the multilevel reliability measures and the empirical coverage rates of Monte Carlo (MC) confidence or credible intervals. Considering the accuracy of the multilevel reliability measures and the empirical coverage rate of the intervals, the results lead us to generally recommend MMLE-multiple imputation. In the model with separate item discriminations over levels, marginally acceptable accuracy of the multilevel reliability measures and empirical coverage rate of the MC confidence intervals were found in a limited condition, 200 clusters, 30 cluster size, .2 ICC, and 40 items, in MMLE-multiple imputation. In the model with the same item discrimination over levels, the accuracy of the multilevel reliability measures and the empirical coverage rate of the MC confidence intervals were acceptable in all multilevel designs we considered with 40 items under MMLE-multiple imputation. We discuss these findings and provide guidelines for reporting multilevel reliability measures.     

Latent Growth Modeling for Logistic Response Functions

Throughout much of the social and behavioral sciences, latent growth modeling (latent curve analysis) has become an important tool for understanding individuals' longitudinal change. Although nonlinear variations of latent growth models appear in the methodological and applied literature, a notable exclusion is the treatment of growth following logistic (sigmoidal; S-shape) response functions. Such trajectories are assumed in a variety of psychological and educational settings where learning occurs over time, and yet applications using the logistic model in growth modeling methodology have been sparse. The logistic function, in particular, may not be utilized as often because software options remain limited. In this article we show how a specialized version of the logistic function can be modeled using conventional structural equation modeling software. The specialization is a reparameterization of the logistic function whose new parameters correspond to scientifically interesting characteristics of the growth process. In addition to providing an example using simulated data, we show how this nonlinear functional form can be fit using transformed subject-level data obtained through a learning task from an air traffic controller simulation experiment. LISREL syntax for the empirical example is provided.     

The Role of Conditional Likelihoods in Latent Variable Modeling

Psychometrika - In psychometrics, the canonical use of conditional likelihoods is for the Rasch model in measurement. Whilst not disputing the utility of conditional likelihoods in measurement, we...     

Modeling Growth in Latent Variables Using a Piecewise Function

Latent growth curve models with piecewise functions for continuous repeated measures data have become increasingly popular and versatile tools for investigating individual behavior that exhibits distinct phases of development in observed variables. As an extension of this framework, this research study considers a piecewise function for describing segmented change of a latent construct over time where the latent construct is itself measured by multiple indicators gathered at each measurement occasion. The time of transition from one phase to another is not known a priori and thus is a parameter to be estimated. Utility of the model is highlighted in 2 ways. First, a small Monte Carlo simulation is executed to show the ability of the model to recover true (known) growth parameters, including the location of the point of transition (or knot), under different manipulated conditions. Second, an empirical example using longitudinal reading data is fitted via maximum likelihood and results discussed. Mplus (Version 6.1) code is provided in Appendix C to aid in making this class of models accessible to practitioners.     

Simultaneous Decision on the Number of Latent Clusters and Classes for Multilevel Latent Class Models

The Multilevel Latent Class Model (MLCM) proposed by Vermunt (2003) has been shown to be an excellent framework for analyzing nested data with assumed discrete latent constructs. The nonparametric version of MLCM assumes 2 levels of discrete latent components to describe the dependency observed in data. Model selection is an important step in any statistical modeling. The task of model selection for MLCM amounts to the decision on the number of discrete latent components at both higher and lower levels and is more challenging than standard Latent Class Models. In this article, simulation studies were conducted to systematically examine the effects of sample sizes, clusters/classes distinctness, and the number of latent clusters and classes on the performance of various information criteria in recovering the true latent structure. Results of the simulation studies are summarized and presented. The final section presents the remarks and recommendations about the simultaneous decision regarding the number of latent classes and clusters when applying MLCMs to analyze empirical data.     

Doubly Robust-Type Estimation for Covariate Adjustment in Latent Variable Modeling

Due to the difficulty in achieving a random assignment, a quasi-experimental or observational study design is frequently used in the behavioral and social sciences. If a nonrandom assignment depends on the covariates, multiple group structural equation modeling, that includes the regression function of the dependent variables on the covariates that determine the assignment, can provide reasonable estimates under the condition of correct specification of the regression function. However, it is usually difficult to specify the correct regression function because the dimensions of the dependent variables and covariates are typically large. Therefore, the propensity score adjustment methods have been proposed, since they do not require the specification of the regression function and have been applied to several applied studies. However, these methods produce biased estimates if the assignment mechanism is incorrectly specified. In order to make a more robust inference, it would be more useful to develop an estimation method that integrates the regression approach with the propensity score methodology. In this study we propose a doubly robust-type estimation method for marginal multiple group structural equation modeling. This method provides a consistent estimator if either the regression function or the assignment mechanism is correctly specified. A simulation study indicates that the proposed estimation method is more robust than the existing methods. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B), 187-30406.     

Latent Class Modeling of Phases in the Development of Transitive Reasoning

Fuzzy trace theory posits that during development the use of verbatim information for solving transitive relationships shifts to the use of gist information. In cognitive developmental research that uses a cross-sectional design, the binomial mixture model is often used to identify such shifts. Because the binomial mixture model assumes equal task difficulty and uses the number of correctly solved tasks for data analysis, it may be too restrictive and the more flexible latent class model is adopted as an alternative. This model allows varying task difficulty and uses the pattern of task scores as input for data analysis. The binomial mixture model and the latent class model are compared theoretically, and applied to transitive reasoning test data obtained from a cross-sectional sample of 615 children. The latent class model is found to be more appropriate for identifying multiple phases. Three phases are distinguished which can be interpreted well by means of fuzzy trace theory. These phases do not encompass fixed age periods.     

Dynamic of Change in Pathological Personality Trait Dimensions: A Latent Change Analysis Among at-Risk Women

This study explores longitudinally a four-factor structure of pathological personality trait dimensions (PPTDs) to examine both its structural stability and intra-individual changes among PPTDs over time. Personality Disorder (PD) scales of the Millon Clinical Multiaxial Inventory-III were administered to 361 low-income women with various psychiatric conditions (drug dependence, depression), who were followed in a two-wave study over 5-years. Cross-sectional and longitudinal factor analyses outlined a robust factorial structure of PPTDs, extrinsically invariant over time, representing Negative Emotionality, Introversion, Antagonism and Impulsivity. Despite moderate rank-order stability in the PPTDs, results also indicated substantial intra-individual variability in the degree and direction of change, consistent with trajectories of change in participants’ clinical diagnoses. Results are discussed in light of current debates on the structure and dynamic of pathological personality.     

Latent Ability: Grades and Test Scores Systematically Underestimate the Intellectual Ability of Negatively Stereotyped Students

ABSTRACT— Past research has assumed that group differences in academic performance entirely reflect genuine differences in ability. In contrast, extending research on stereotype threat, we suggest that standard measures of academic performance are biased against non-Asian ethnic minorities and against women in quantitative fields. This bias results not from the content of performance measures, but from the context in which they are assessed—from psychological threats in common academic environments, which depress the performances of people targeted by negative intellectual stereotypes. Like the time of a track star running into a stiff headwind, such performances underestimate the true ability of stereotyped students. Two meta-analyses, combining data from 18,976 students in five countries, tested this latent-ability hypothesis. Both meta-analyses found that, under conditions that reduce psychological threat, stereotyped students performed better than nonstereotyped students at the same level of past performance. We discuss implications for the interpretation of and remedies for achievement gaps.     

The Stability of Social Desirability: A Latent Change Analysis

Social desirability has been shown to be stable in samples with higher school education. However, little is known about the stability of social desirability in more heterogeneous samples differing in school education. This study aimed to investigate the stability of social desirability and which factors predict interindividual differences in intraindividual change. As part of a randomized controlled trial, 1,243 job seekers with unhealthy alcohol use were systematically recruited at three job agencies. A total of 1,094 individuals (87.8%) participated in at least one of two follow‐ups (6 and 15 months after baseline) and constitute this study's sample. The Social Desirability Scale‐17 was applied. Two latent change models were conducted: Model 1 tested for interindividual differences in intraindividual change of social desirability between both follow‐ups; Model 2 included possible predictors (age, sex, education, current employment status) of interindividual differences in intraindividual change. Model 1 revealed a significant decrease of social desirability over time. Model 2 revealed school education to be the only significant predictor of change. These findings indicate that stability of social desirability may depend on school education. It may not be as stable in individuals with higher school education as in individuals with lower education.     

Studying Multivariate Change Using Multilevel Models and Latent Curve Models

In longitudinal research investigators often measure multiple variables at multiple points in time and are interested in investigating individual differences in patterns of change on those variables. In the vast majority of applications, researchers focus on studying change in one variable at a time. In this article we consider methods for studying relations1.1ips between patterns of change on different variables. We show how the multilevel modeling framework, which is often used to study univariate change, can be extended to the multivariate case to yield estimates of covariances of parameters representing aspects of change on different variables. We illustrate this approach using data from a study of physiological response to marital conflict in older married couples, showing a substantial correlation between rate of linear change on different stress-related hormones during conflict. We also consider how similar issues can be studied using extensions of latent curve models to the multivariate case, and we show how such models are related to multivariate multilevel models.