Fitting Nonlinear Latent Growth Curve Models With Individually Varying Time Points

Individual growth trajectories of psychological phenomena are often theorized to be nonlinear. Additionally, individuals’ measurement schedules might be unique. In a structural equation framework, latent growth curve model (LGM) applications typically have either (a) modeled nonlinearity assuming some degree of balance in measurement schedules, or (b) accommodated truly individually varying time points, assuming linear growth. This article describes how to fit 4 popular nonlinear LGMs (polynomial, shape-factor, piecewise, and structured latent curve) with truly individually varying time points, via a definition variable approach. The extension is straightforward for certain nonlinear LGMs (e.g., polynomial and structured latent curve) but in the case of shape-factor LGMs requires a reexpression of the model, and in the case of piecewise LGMs requires introduction of a general framework for imparting piecewise structure, along with tools for its automation. All 4 nonlinear LGMs with individually varying time scores are demonstrated using an empirical example on infant weight, and software syntax is provided. The discussion highlights some advantages of modeling nonlinear growth within structural equation versus multilevel frameworks, when time scores individually vary.     

Analyzing change in school psychology research

We look at three common research scenarios, one in the behavioral domain (i.e., disruptive behaviors) one in the cognitive domain (i.e., academic achievement), and one in the affective domain (i.e., anxiety and stress) for which school psychologists are asked to address important questions related to change. We list measurement and statistical considerations across these scenarios, including whether variables are manifest or latent, the scales of measurement, the dimensionality of measures, the units of analysis, the sample size, and the frequency or duration of time, given the primary nature of the variables under study. We suggest that researchers carefully consider whether assumptions can be met employing classical general linear models, or whether contemporary alternatives, such as hierarchical linear modeling (HLM), should be recommended for the more appropriate handling of data. © 2007 Wiley Periodicals, Inc. Psychol Schs 44: 535–542, 2007.     

Inferring Longitudinal Relationships Between Variables: Model Selection Between the Latent Change Score and Autoregressive Cross-Lagged Factor Models

This research focuses on the problem of model selection between the latent change score (LCS) model and the autoregressive cross-lagged (ARCL) model when the goal is to infer the longitudinal relationship between variables. We conducted a large-scale simulation study to (a) investigate the conditions under which these models return statistically (and substantively) different results concerning the presence of bivariate longitudinal relationships, and (b) ascertain the relative performance of an array of model selection procedures when such different results arise. The simulation results show that the primary sources of differences in parameter estimates across models are model parameters related to the slope factor scores in the LCS model (specifically, the correlation between the intercept factor and the slope factor scores) as well as the size of the data (specifically, the number of time points and sample size). Among several model selection procedures, correct selection rates were higher when using model fit indexes (i.e., comparative fit index, root mean square error of approximation) than when using a likelihood ratio test or any of several information criteria (i.e., Akaike’s information criterion, Bayesian information criterion, consistent AIC, and sample-size-adjusted BIC).     

Predictions of Individual Change Recovered With Latent Class or Random Coefficient Growth Models

Popular longitudinal models allow for prediction of growth trajectories in alternative ways. In latent class growth models (LCGMs), person-level covariates predict membership in discrete latent classes that each holistically define an entire trajectory of change (e.g., a high-stable class vs. late-onset class vs. moderate-desisting class). In random coefficient growth models (RCGMs, also known as latent curve models), however, person-level covariates separately predict continuously distributed latent growth factors (e.g., an intercept vs. slope factor). This article first explains how complex and nonlinear interactions between predictors and time are recovered in different ways via LCGM versus RCGM specifications. Then a simulation comparison illustrates that, aside from some modest efficiency differences, such predictor relationships can be recovered approximately equally well by either model—regardless of which model generated the data. Our results also provide an empirical rationale for integrating findings about prediction of individual change across LCGMs and RCGMs in practice.     

Automated Latent Growth Curve Model Fitting: A Segmentation and Knot Selection Approach

Latent growth curve models are widely used in the social and behavioral sciences to study complex developmental patterns of change over time. The trajectories of these developmental patterns frequently exhibit distinct segments in the studied variables. Latent growth models with piecewise functions for repeated measurements of variables have become increasingly popular for modeling such developmental trajectories. A major problem with using piecewise models is determining the precise location of the point where the change in the process has occurred and uncovering the related number of segments. The purpose of this paper is to introduce an optimization procedure that can be used to determine both the segments and location of the knots in piecewise linear latent growth models. The procedure is illustrated using empirical data in order to detect the number of segments and change points. The results demonstrate the capabilities of the procedure for fitting latent growth curve models.     

The growth of mathematics and reading skills in segregated and diverse schools: An opportunity-propensity analysis of a national database

The goal of the present study was to continue to build and refine the Opportunity-Propensity (O-P) model of achievement by using it to explain well-known achievement disparities between schools that differ in terms of their racial and ethnic composition. The O-P model categorizes predictors into antecedent factors (e.g., family SES, parent educational aspirations), opportunity factors (e.g., content coverage and teaching style), and propensity factors (e.g., prior knowledge and motivation). To refine the model further, the authors did the following: (a) added predictors in each category that have not been examined to date; (b) used hierarchical linear modeling to explain growth in knowledge between two assessment points; (c) identified similarities and differences in the models for two content areas (i.e., math and reading); and (d) identified similarities and differences in the models for two age levels (i.e., 3rd grade and 8th grade). Results showed that the combination of new and established predictors accounted for approximately 50% of the variance in the rate at which knowledge grew in both math and reading at both age levels. In addition, the variance explained by the racial and ethnic composition of schools when this variable was entered as the sole predictor of achievement was substantially reduced after antecedent, opportunity, and propensity factors were entered in subsequent models. However, the coefficients for certain school compositions remained significant even after such controls, and the models differed somewhat between content areas (math and reading), and age levels (3rd and 8th grade). Findings are discussed in terms of the implications of these results for building and refining the O-P model further.     

Evidence of a continuum in foundational expressive communication skills

Progress monitoring measurement is increasingly needed in early childhood to inform practitioners when an intervention change is needed and as a tool for accomplishing individualization and improving results for individual children. The Early Communication Indicator (ECI) is such a measure for infants and toddlers 6–42 months of age. A greater understanding of the ECI key skills (i.e., gestures, vocalizations, single- and multiple-word utterances) could lead to further improvements in the sensitivity and utility of the decisions made compared to ECIs composite total communication score. Thus, we examined the pattern of growth within and between the ECI's four foundational skills in a large sample of children served in Early Head Start. Results confirmed a unique pattern of growth and change within each skill trajectory in terms of (a) age at skill onset and (b) peaks in each trajectory defining an inflection point or change from acceleration to deceleration. Using these inflection points as intercepts with before and after trajectory slopes, we tested the fit of an adjacent-skills temporally ordered growth model. Results indicated good fit. Implications of a continuum of foundational ECI skills to future validation and decision making utility of the measure are discussed.     

Implementing Restricted Maximum Likelihood Estimation in Structural Equation Models

Structural equation modeling (SEM) is now a generic modeling framework for many multivariate techniques applied in the social and behavioral sciences. Many statistical models can be considered either as special cases of SEM or as part of the latent variable modeling framework. One popular extension is the use of SEM to conduct linear mixed-effects modeling (LMM) such as cross-sectional multilevel modeling and latent growth modeling. It is well known that LMM can be formulated as structural equation models. However, one main difference between the implementations in SEM and LMM is that maximum likelihood (ML) estimation is usually used in SEM, whereas restricted (or residual) maximum likelihood (REML) estimation is the default method in most LMM packages. This article shows how REML estimation can be implemented in SEM. Two empirical examples on latent growth model and meta-analysis are used to illustrate the procedures implemented in OpenMx. Issues related to implementing REML in SEM are discussed.     

Measurement Models,Estimation, and the Study of Change

The study of change is based on the idea that the score or index at each measurement occasion has the same meaning and metric across time. In tests or scales with multiple items, such as those common in the social sciences, there are multiple ways to create such scores. Some options include using raw or sum scores (i.e., sum of item responses or linear transformation thereof), using Rasch-scaled scores provided by the test developers, fitting item response models to the observed item responses and estimating ability or aptitude, and jointly estimating the item response and growth models. We illustrate that this choice can have an impact on the substantive conclusions drawn from the change analysis using longitudinal data from the Applied Problems subtest of the Woodcock–Johnson Psycho-Educational Battery–Revised collected as part of the National Institute of Child Health and Human Development's Study of Early Child Care. Assumptions of the different measurement models, their benefits and limitations, and recommendations are discussed.     

Power Computations for Polynomial Change Models in Block-Randomized Designs

Education experiments frequently assign students to treatment or control conditions within schools. Longitudinal components added in these studies (e.g., students followed over time) allow researchers to assess treatment effects in average rates of change (e.g., linear or quadratic). We provide methods for a priori power analysis in three-level polynomial change models for block-randomized designs. We discuss unconditional models and models with covariates at the second and third level. We illustrate how power is influenced by the number of measurement occasions, the sample sizes at the second and third levels, and the covariates at the second and third levels.     

An Adaptive Bayesian Lasso Approach with Spike-and-Slab Priors to Identify Multiple Linear and Nonlinear Effects in Structural Equation Models

In applied research, such as with motivation theories, typically many variables are theoretically implied predictors of an outcome and several interactions are assumed (e.g., Watt, 2004). However, estimation problems that might arise when several interaction and/or quadratic effects are analyzed simultaneously have not been investigated because simulation studies on interaction effects in the structural equation modeling framework have mainly focused on small models that contain single interaction effects. In this article, we show that traditional approaches can provide estimates with low accuracy when complex models are estimated. We introduce an adaptive Bayesian lasso approach with spike-and-slab priors that overcomes this problem. Using a complex model in a simulation study, we show that the parameter estimates of the proposed approach are more accurate in situations with high multicollinearity or low reliability compared with a standard Bayesian lasso approach and typical frequentist approaches (i.e., unconstrained product indicator approach and latent moderated structures approach).     

Observed Score Linear Equating with Covariates

This paper examined observed score linear equating in two different data collection designs, the equivalent groups design and the nonequivalent groups design, when information from covariates (i.e., background variables correlated with the test scores) was included. The main purpose of the study was to examine the effect (i.e., bias, variance, and mean squared error) on the estimators of including this additional information. A model for observed score linear equating with covariates first was suggested. As a second step, the model was used in a simulation study to show that the use of covariates such as gender and education can increase the accuracy of an equating by reducing the mean squared error of the estimators. Finally, data from two administrations of the Swedish Scholastic Assessment Test were used to illustrate the use of the model.     

Using LISREL to Fit Nonlinear Latent Curve Models

Latent curve models offer a flexible approach to the study of longitudinal data when the form of change in a response is nonlinear. This article considers such models that are conditionally linear with regard to the random coefficients at the 2nd level. This framework allows fixed parameters to enter a model linearly or nonlinearly, and random coefficients at the 2nd level may only enter linearly. Beginning with LISREL 8.80 for Windows, such models can be fitted, giving users greater flexibility in model specification. An example with LISREL syntax is provided.     

Detecting Appropriate Trajectories of Growth in Latent Growth Models: The Performance of Information-Based Criteria

Latent growth modeling (LGM) is a popular and flexible technique that may be used when data are collected across several different measurement occasions. Modeling the appropriate growth trajectory has important implications with respect to the accurate interpretation of parameter estimates of interest in a latent growth model that may impact educational policy decisions. A Monte Carlo simulation study was conducted to examine the accuracy of six information-based criteria (i.e., AIC, CAIC, AICC, BIC, nBIC, and HQIC) when selecting among various growth trajectories modeled using LGM under different sample size, number of time points, and growth trajectory scenarios. The accuracy of the information criteria generally improved as sample size increased. The cubic and linear growth models were distinguished most accurately by the information criteria. All of the nonlinear models were more easily distinguished as the number of time points increased. The comparative performance of the six information criteria was dependent upon the manipulated conditions. Implications of the findings are discussed.     

Quantum mechanics and conceptual change in high school chemistry textbooks

The presentation of sophisticated atomic theory (quantum mechanics) in secondary chemistry texts is not accompanied by sufficient evidence or applications to promote its rational acceptance as determined by a model of conceptual change. Eight secondary chemistry texts were analyzed for four elements associated with a conceptual change model: dissatisfaction, intelligibility, plausibility, and fruitfulness. These elements were not present in sufficient quantities to promote conceptual change, i.e., to have quantum mechanics rationally accepted over simple atomic models such as the Bohr theory. Recommendations include modifying future textbooks to include conceptual change elements; having secondary chemistry teachers emphasize these elements beyond the presentation in their textbooks; and using the conceptual change model to analyze other instructional materials, especially those presenting scientific theories. © 1997 John Wiley & Sons, Inc. J Res Sci Teach 34: 535–545, 1997.     

CONSISTENCY AND MAGNITUDE OF DIFFERENCES IN READING CURRICULUM‐BASED MEASUREMENT SLOPES IN BENCHMARK VERSUS STRATEGIC MONITORING

Differences in oral reading curriculum‐based measurement (R‐CBM) slopes based on two commonly used progress monitoring practices in field‐based data were compared in this study. Semester‐specific R‐CBM slopes were calculated for 150 Grade 1 and 2 students who completed benchmark (i.e., 3 R‐CBM probes collected 3 times per year) and strategic (i.e., one R‐CBM probe collected monthly) assessments. Slopes based on two adjacent benchmark assessments were positively correlated with slopes based on three monthly strategic assessments in the spring semester of Grade 1 but not in either Grade 2 semester, and significant differences were found between the slopes in all semesters. Consistent with another study showing that slopes are overestimated when single probes are administered per occasion, slopes were larger when based on strategic versus benchmark data in the current study, and the average discrepancies between slopes were greater‐than‐expected growth rates in all semesters. The current findings, based on field‐based data, illustrate the impact of variations in commonly used progress monitoring procedures on the precision of calculated slope estimates.     

The developmental influence of sex composition in preschool classrooms: Boys fare worse in preschool classrooms with more boys

A multilevel modeling approach was employed to investigate the relation between sex composition and developmental change in 70 urban preschool classrooms. The research represents a unique contribution as (1) few studies have examined the influence of sex composition during the preschool years, (2) it represents the first research to use a continuous (i.e., sex ratio) as opposed to binary (i.e., mixed- versus single-sex) indicator for classroom sex composition, and (3) the sample represents an important and often neglected group (i.e., low-income children from urban schools). A series of HLM models were run, addressing the nested nature of the data (children within classrooms), and relating classroom sex composition to developmental change using the cognitive, motor, and social subscales from the Child Observation Record (COR). Overall, there were no main effects at the classroom-level for sex composition. However, a cross-level interaction indicated that, while girls’ development was not influenced by classroom sex composition, boys in classrooms with proportionally more boys fared significantly worse in terms of development as assessed by combined score on the COR. More specifically, this interaction was significant when predicting the COR cognitive subscale, but nonsignificant when predicting the COR social and motor subscales. This was true when controlling for the number of students at the classroom-level, as well as child's age and baseline ability (i.e., Time 1 COR) at the child-level. Implications for early childhood education policy are discussed.     

Toward a typology of integration: Examining the documents model framework

Drawing on the Documents Model Framework, in two studies we investigate the types of multiple text models (e.g., mush model, separate representations model, documents model) reflected in students’ written responses, composed based on multiple texts. In among the first studies to holistically code written responses for the type of multiple text models they reflect, we find evidence for all of the multiple text representations specified in the Documents Model. We further examine the types of integration, or cross-textual connections, featured in students’ responses, including evidentiary (i.e., corroborating evidence), thematic (i.e., comparing main ideas across texts), and contextual (i.e., comparing meta-textual information, like author expertise) connections. Finally, we associate the types of multiple text models reflected in students’ written responses with various measures of integration, including the number of discourse connectives and citations included and self-reports of strategy use (Study 1), as well as with integrative utterances reported during a cued think-aloud (Study 2).     

Trajectories of Infants’ Biobehavioral Development: Timing and Rate of A‐Not‐B Performance Gains and EEG Maturation

This study examined how timing (i.e., relative maturity) and rate (i.e., how quickly infants attain proficiency) of A‐not‐B performance were related to changes in brain activity from age 6 to 12 months. A‐not‐B performance and resting EEG (electroencephalography) were measured monthly from age 6 to 12 months in 28 infants and were modeled using logistic and linear growth curve models. Infants with faster performance rates reached performance milestones earlier. Infants with faster rates of increase in A‐not‐B performance had lower occipital power at 6 months and greater linear increases in occipital power. The results underscore the importance of considering nonlinear change processes for studying infants’ cognitive development as well as how these changes are related to trajectories of EEG power.