Nonlinear Growth Mixture Models With Fractional Polynomials: An Illustration With Early Childhood Mathematics Ability

Applications of growth mixture modeling have become widespread in the fields of medicine, public health, and the social sciences for modeling linear and nonlinear patterns of change in longitudinal data with presumed heterogeneity with respect to latent group membership. However, in contrast to linear approaches, there has been relatively less focus on methods for modeling nonlinear change. We introduce a nonlinear mixture modeling approach for estimating change trajectories that rely on the use of fractional polynomials within a growth mixture modeling framework. Fractional polynomials allow for more parsimonious and flexible models in comparison to conventional polynomial models. The procedures are illustrated through the use of math ability scores obtained from 499 children over a period of 3 years, with 4 measurement occasions. Techniques for identifying the best empirically derived growth mixture model solution are also described and illustrated by way of substantive example and a simulation.     

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.     

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.     

On the Estimation of Nonlinear Mixed-Effects Models and Latent Curve Models for Longitudinal Data

Nonlinear models are effective tools for the analysis of longitudinal data. These models provide a flexible means for describing data that follow complex forms of change. Exponential and logistic functions that include a parameter to represent an asymptote, for instance, are useful for describing responses that tend to level off with time. There are forms of nonlinear latent curve models and nonlinear mixed-effects model that are equivalent, and so given the same set of data, growth function, distributional assumptions, and method of estimation, the 2 models yield equivalent results. There are also forms that are strikingly different and can yield different interpretations for a given set of data. This article discusses cases in which nonlinear mixed-effects models and nonlinear latent curve models are equivalent and those in which they are different and clarifies the estimation needs of the different models. Examples based on empirical data help to illustrate these points.     

Specifying Piecewise Latent Trajectory Models for Longitudinal Data

Piecewise latent trajectory models for longitudinal data are useful in a wide variety of situations, such as when a simple model is needed to describe nonlinear change, or when the purpose of the analysis is to evaluate hypotheses about change occurring during a particular period of time within a model for a longer overall time frame, such as change that occurs following onset of a treatment or some other event. However, the specification of various forms of piecewise models has not been fully explicated for the structural equation modeling (SEM) framework. This article describes piecewise models as a straightforward extension of the basic SEM model for linear growth, which makes them relatively easy both to specify and to interpret. After presenting models for 2 linear slopes (or pieces) in detail, the article discusses extensions that include additional linear slopes (i.e., a 3-piece model) or a quadratic factor (i.e., a hybrid linear-quadratic model).     

Determining the Number of Latent Classes in Single- and Multiphase Growth Mixture Models

Stage-sequential (or multiphase) growth mixture models are useful for delineating potentially different growth processes across multiple phases over time and for determining whether latent subgroups exist within a population. These models are increasingly important as social behavioral scientists are interested in better understanding change processes across distinctively different phases, such as before and after an intervention. One of the less understood issues related to the use of growth mixture models is how to decide on the optimal number of latent classes. The performance of several traditionally used information criteria for determining the number of classes is examined through a Monte Carlo simulation study in single- and multiphase growth mixture models. For thorough examination, the simulation was carried out in 2 perspectives: the models and the factors. The simulation in terms of the models was carried out to see the overall performance of the information criteria within and across the models, whereas the simulation in terms of the factors was carried out to see the effect of each simulation factor on the performance of the information criteria holding the other factors constant. The findings not only support that sample size adjusted Bayesian Information Criterion would be a good choice under more realistic conditions, such as low class separation, smaller sample size, or missing data, but also increase understanding of the performance of information criteria in single- and multiphase growth mixture models.     

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.     

Examining the Institutional Transformation Process: The Importance of Sensemaking,Interrelated Strategies,and Balance

This study develops elements of a transformational change framework that is theoretically and empirically grounded and is context based through case studies of 6 institutions over a 4-year period. The 3 key findings include: (a) 5 core strategies for transformational change; (b) the characteristic that makes them the essential, sensemaking; and (c) the interrelationship among core and secondary strategies, the nonlinear process of change, and the need for balance among strategies. Two major conclusions are developed from the study findings: (a) the efficacy for researchers of combining multiple conceptual models for understanding change processes; and (b) the importance of social cognition models for future studies of transformational change based on the significance of sensemaking.     

Modeling the processes of development via latent variable growth curve methodology

Recent advances in statistical methodology, in particular, latent growth modeling, allow for the testing of complex models regarding developmental trends from both an inter‐ and intraindividual perspective. An example application of latent growth curve methodology, analyzing the effects of gender and parental monitoring on developmental change in adolescent alcohol consumption, is presented. Furthermore, the analyses are conducted within a cohort‐sequential design, incorporating an approach to the analysis of missing data due to attrition. Findings are discussed with particular reference to the utility of latent growth curve models for assessing developmental processes at both the inter‐and intraindividual level across a variety of behavioral domains.     

A Second-Order Conditionally Linear Mixed Effects Model With Observed and Latent Variable Covariates

A conditionally linear mixed effects model is an appropriate framework for investigating nonlinear change in a continuous latent variable that is repeatedly measured over time. The efficacy of the model is that it allows parameters that enter the specified nonlinear time-response function to be stochastic, whereas those parameters that enter in a nonlinear manner are common to all subjects. In this article we describe how a variant of the Michaelis-Menten (M-M) function can be fit within this modeling framework using Mplus 6.0. We demonstrate how observed and latent covariates can be incorporated to help explain individual differences in growth characteristics. Features of the model including an explication of key analytic decision points are illustrated using longitudinal reading data. To aid in making this class of models accessible, annotated Mplus code is provided.     

Asymptotics for Nonlinear Transformations of Fractionally Integrated Time Series

The asymptotic theory for nonlinear transformations of fractionally integrated time series is developed.By the use of fractional Occupation Times Formula,various nonlinear functions of fractionally integrated series such as ARFIMA time series are studied,and the asymptotic distributions of the sample moments of such functions are obtained and analyzed.The transformations considered in this paper includes a variety of functions such as regular functions,integrable functions and asymptotically homogeneous functions that are often used in practical nonlinear econometric analysis.It is shown that the asymptotic theory of nonlinear transformations of original and normalized fractionally integrated processes is different from that of fractionally integrated processes,but is similar to the asymptotic theory of nonlinear transformations of integrated processes.     

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.     

A Second-Order Conditionally Linear Mixed Effects Model With Observed and Latent Variable Covariates

A conditionally linear mixed effects model is an appropriate framework for investigating nonlinear change in a continuous latent variable that is repeatedly measured over time. The efficacy of the model is that it allows parameters that enter the specified nonlinear time-response function to be stochastic, whereas those parameters that enter in a nonlinear manner are common to all subjects. In this article we describe how a variant of the Michaelis–Menten (M–M) function can be fit within this modeling framework using Mplus 6.0. We demonstrate how observed and latent covariates can be incorporated to help explain individual differences in growth characteristics. Features of the model including an explication of key analytic decision points are illustrated using longitudinal reading data. To aid in making this class of models accessible, annotated Mplus code is provided.     

The application of various nonlinear models to describe academic growth trajectories: an empirical analysis using four-wave longitudinal achievement data from a large urban school district

This study introduced various nonlinear growth models, including the quadratic conventional polynomial model, the fractional polynomial model, the Sigmoid model, the growth model with negative exponential functions, the multidimensional scaling technique, and the unstructured growth curve model. It investigated which growth models effectively describe student growth in math and reading using four-wave longitudinal achievement data. The objective of the study is to provide valuable information to researchers especially when they consider applying one of the nonlinear models to longitudinal studies. The results showed that the quadratic conventional polynomial model fit the data best. However, this model seemed to overfit the data and made statistical inference problematic concerning parameter estimates. Alternative nonlinear models with fewer parameters adequately fit the data and yielded consistent significance testing results under extreme multicollinearity. It indicates that the alternative models denoting somewhat simpler models would be selected over the conventional polynomial model with more fixed parameters. Other practical issues pertaining to these growth models are also discussed.     

Children's intuitive mathematics: the development of knowledge about nonlinear growth

This study is concerned with the development of children's intuitive understanding of nonlinear processes. The ability to estimate linear and exponential growth was examined in 7-, 9-, 11-, and 13-year-old children and adults (N=160). Whereas linear growth was judged correctly at all ages, estimations of exponential growth were in line with mathematically correct values only in 13-year-olds and adults. However, 9-year-olds already judged the result of exponential growth as being significantly higher than that of linear growth, and even a remarkable proportion of 7-year-olds showed such discrimination between the two types of functions. Results point to the existence of an early intuitive knowledge about the characteristics of nonlinear growth, long before those functions are taught in school.     

Theoretical constructs that explain and enhance learning: a longitudinal examination

One important line of inquiry in educational psychology involves the study of change of individuals' cognitive-motivational processes. The conjunctive use of longitudinal data with latent growth curve modeling procedures has, for example, allowed researchers to identify initial levels and to trace trajectories of theoretical variables such as self-efficacy over time. The study reported in this article proposed a conceptual model that depicted relations between a deep-learning approach, mastery goals and self-efficacy over time. A final sample of 195 second-year university students (100 females, 95 males) took part in this three-wave panel study. We used various inventories to test the initial states and rates of change of the three aforementioned constructs. As an a posteriori analysis, we included prior academic achievement as a possible predictor of change. The results ascertained from our analyses indicate an increase in growth of a deep-learning approach, mastery goals and self-efficacy across the two-year period. Importantly, a posteriori results accentuated the role of prior academic achievement as a predictor of the initial level of personal self-efficacy.     

Learning through transitions: The role of institutions

In this paper two models are proposed for analysing transitions in education. Firstly, transitions are the processes that follow ruptures perceived by people. They include learning, identity change, and meaning making processes. Secondly, processes of change are observed through a semiotic prism, articulating self-other-object-sense of the object for self, and located in a specific social frame. Transitions are thus analysed as reconfigurations of such semiotic prism. The paper proposes to highlight the role of institutions as social frames likely to facilitate, or constrain, such reconfigurations. The role of institutions in transitions is discussed through three case studies: the transition to vocational training, the transition out of a religious school, and the transition to work at war-time.     

From individual differences to dynamic pathways of development

A fruitful way to build upon French-language research on development of analogical and propositional processes in logical reasoning tasks is to use dynamic systems tools to describe and analyze relevant developmental pathways. Issues to address include (1) the characteristics of developmental transitions, such as hysteresis; (2) the nature of growth processes, such as hierarchical development or predator-prey interactions; and (3) the construction of effective scales for measuring change in logical reasoning.     

Trajectories of Ethnic–Racial Identity and Autonomy Among Mexican‐Origin Adolescent Mothers in the United States

This study examined trajectories of ethnic–racial identity (ERI) and autonomy development among Mexican‐origin adolescent females in the United States (N = 181; M_age at Wave 1 = 16.80 years, SD = 1.00) as they transitioned through the first 5 years of parenthood. Trajectories of ERI and autonomy also were examined in relation to psychosocial functioning. Unconditional latent growth models indicated significant growth in autonomy, ERI resolution, and ERI affirmation from middle to late adolescence. Conditional latent growth models indicated that autonomy and ERI exploration growth trajectories were positively associated with psychosocial adjustment. Although adolescent mothers are experiencing transitions that are not normative during adolescence, they also engage in normative developmental processes, and their engagement in such processes is linked with better adjustment.     

Estimating Individual Developmental Functions: Methods and Their Assumptions

This article introduces developmentalists to methods for estimating individual developmental functions from longitudinal data in a multilevel analysis. Quantitative growth curve models for estimating the developmental functions from various types of longitudinal data are discussed in the context of both an investigator's assumptions about individual development on the attribute and the design characteristics of the prospective study. General linear and inherently nonlinear models that estimate population, individual, and prototypic growth curves are illustrated and contrasted when they are fit to speech development data.