A Version of Quadratic Regression with Interpretable Parameters

UAV-based remote sensing of turbidity in coastal environment for regulatory monitoring and assessment

The adoption of Unmanned Aerial Vehicle (UAV) remote sensing for the regulatory monitoring of turbidity plumes induced by land reclamation operations remains a difficult task. Compared to UAV remote sensing on ambient turbidity in estuaries and rivers, such monitoring of construction-induced turbidity plumes requires significantly higher spatial resolutions and accuracy as well as wider turbidity ranges with nonlinear reflectance. In this study, a pilot-scale deployment of UAV-based hyperspectral sensing is carried out for this objective, with specific new elements developed to overcome the challenges and minimise the uncertainties involved. In particular, Machine learning (ML) models for the turbidity determination were trained by the large dataset collected to better capture the non-linearity of the relationship between the water leaving reflectance and turbidity level. The models achieve a good accuracy with a R2 score of 0.75 that is deemed acceptable in view of the uncertainties associated with construction and land reclamation work.

Reducing Incidence of Nonpositive Definite Covariance Matrices in Mixed Effect Models

Deciding which random effects to retain is a central decision in mixed effect models. Recent recommendations advise a maximal structure whereby all theoretically relevant random effects are retained. Nonetheless, including many random effects often leads to nonpositive definiteness. A typical remedy is to simplify the random effect structure by removing random effects or associated covariances. However, this practice is known to bias estimates of remaining covariance parameters and compromise fixed effect inferences. Cholesky decompositions frequently are suggested as an alternative and are automatically implemented in some software. Instead of Cholesky decompositions, we describe factor analytic structures as an approach to avoid nonpositive definiteness. This approach is occasionally employed in biosciences like plant breeding, but, ironically, has not been established in behavioral sciences despite the close historical connection with factor analysis in these fields. We discuss how a factor analytic structure facilitates estimation and conduct simulations to compare convergence and performance to simplifying the random effects structure or Cholesky decomposition approaches. Results show a lower rate of nonpositive definiteness with the factor analytic structure than Cholesky decomposition and suggest that factor analytic covariance structure may be useful to combating nonpositive definiteness, especially in models with many random effects.

Parenting costs time: Changes in pair bond maintenance across pregnancy and infant rearing in a monogamous primate (Plecturocebus cupreus)

Relationships support social animals' health, but maintaining relationships is challenging. When transitioning to parenthood, new parents balance pair-bond maintenance with infant care. We studied pair-bond maintenance via affiliation in 22 adult titi monkey pairs (Plecturocebus cupreus) for 16 months centered around their first offspring's birth. Pair affiliation peaked during pregnancy, decreased across the postpartum period, and rose after reaching minimum affiliation 32.6 weeks postpartum. Pairs in which fathers carry infants more than average had lower affiliation at the infant's birth and return to an increase in affiliation sooner. Parents of infants who were slow to independence had higher rates of affiliation. Titi monkey infants actively prefer their fathers; mothers may avoid their infant-carrying mate, suggesting infants play an active role in parental affiliative decline. Our data supports previous findings that affiliation between partners declines following an infant's birth, but demonstrates new knowledge about the extent and duration of affiliative decline.

When I'm 64: Age-related variability in over 40,000 online cognitive test takers

Objectives

Age-related differences in cognition are typically assessed by comparing groups of older to younger participants, but little is known about the continuous trajectory of cognitive changes across age, or when a shift to older adulthood occurs. We examined the pattern of mean age differences and variability on episodic memory and executive function measures over the adult life span, in a more fine-grained way than past group or life-span comparisons.

Method

We used a sample of over 40,000 people aged 18–90 who completed psychometrically validated online tests measuring episodic memory and executive functions (the Cogniciti Brain Health Assessment).

Results

Cognitive performance declined gradually over adulthood, and rapidly later in life on spatial working memory, processing speed, facilitation (but not interference), associative recognition, and set shifting. Both polynomial and segmented regression fit the data well, indicating a nonlinear pattern. Segmented regression revealed a shift from gradual to rapid decline that occurred in the early 60s. Variability between people (interindividual variability or diversity) and variability within a person across tasks (intraindividual variability or dispersion) also increased gradually until the 60s, and rapidly after. Confirmatory factor analysis revealed a single general factor (of variance shared between tasks) offered a good fit for performance across tasks.

Discussion

Life-span cognitive performance shows a nonlinear pattern, with gradual decline over early and mid-adulthood, followed by a transition in the 60s to notably accelerated, but more variable, decline. Some people show less decline than others, and some cognitive abilities show less within-person decline than others.

Estimating New Quantities from Longitudinal Test Scores to Improve Forecasts of Future Performance

Psychometric models for longitudinal test scores typically estimate quantities associated with single-administration tests, like ability at each time-point. However, models for longitudinal tests have not considered opportunities to estimate new quantities that are unavailable from single-administration tests. Specifically, we discuss dynamic measurement models - which combine aspects of longitudinal IRT, nonlinear growth models, and dynamic assessment - to directly estimate capacity, defined as the expected future score once the construct has fully developed. After discussing the history and connecting these areas into a single framework, we apply the model to verbal test scores from the Intergenerational Studies, which follow 494 people from 3 to 72 years old. The goal is to predict adult verbal scores (Age ≥ 34) from adolescent scores (Age ≤ 20). We held-out the adult data for prediction and compared predictions from traditional longitudinal IRT ability scores and proposed dynamic measurement capacity scores from models fit to the adolescent data. Results showed that the R² from capacity scores were 2.5 times larger than the R² from longitudinal IRT ability scores (43% vs. 16%), providing some evidence that exploring new quantities available from longitudinal testing could be worthwhile when an interest in testing is forecasting future performance.

Conditional Direction Dependence Analysis: Evaluating the Causal Direction of Effects in Linear Models with Interaction Terms

Direction dependence analysis (DDA) makes use of higher than second moment information of variables (x and y) to detect potential confounding and to probe the causal direction of linear variable relations (i.e., whether x → y or y → x better approximates the underlying causal mechanism). The "true" predictor is assumed to be a continuous nonnormal exogenous variable. Existing methods compatible with DDA, however, are of limited use when the relation of a focal predictor and an outcome is affected by a moderator. This study presents a conditional direction dependence analysis (CDDA) framework which enables researchers to evaluate the causal direction of conditional regression effects. Monte-Carlo simulations were used to evaluate two different moderation scenarios: Study 1 evaluates the performance of CDDA tests when a moderator affects the strength of the causal effect x → y. Study 2 evaluates cases in which the causal direction itself (x → y vs y → x) depends on moderator values. Study 3 evaluates the robustness of DDA tests in the presence of functional model misspecifications. Results suggest that significance tests compatible with CDDA are suitable in both moderation scenarios, i.e., CDDA allows one to discern regions of a moderator in which the causal direction is uniquely identifiable. An empirical example is provided to illustrate the approach.

Heterogeneity in the dynamic arousal and modulation of fear in young foster children

Guided by emotional security theory, we explored how child and context-related factors were associated with heterogeneity in young foster children's organized patterns of fear response to distress. Results from group-based trajectory modeling used to analyze observational data from a fear-eliciting task showed that children from our sample (mean age = 62 months, SD = 9) were classified into 3 specific fear regulation patterns differentiated by the emotional response parameters of onset intensity, peak intensity, and rise time. A descriptive examination of child's emotion knowledge, aggressive behaviors, and attention problems, as well as length of time in current foster home, placement transitions, and caregiver responsiveness and modeling showed class-specific differences in means. Moreover, the likelihood of class membership was significantly predicted by children's emotion knowledge, aggressive behaviors, and foster mothers' responsiveness and modeling of appropriate boundaries. Results show promising support for the implementation of individualized, child-directed interventions targeting specific patterns of response parameters of emotion regulation for young foster children. Further, parenting intervention services need to promote the emotion socialization skills of foster parents that are tailored toward each specific trajectory pattern of emotion arousal and modulation.

Piecewise latent growth models: beyond modeling linear-linear processes

Piecewise latent growth models (LGMs) for linear-linear processes have been well-documented and studied in recent years. However, in the latent growth modeling literature, advancements to other functional forms as well as to multiple changepoints or knots have been nearly non-existent. This manuscript deals with three extensions. The first is to a piecewise latent growth model incorporating higher-order polynomials. The second is to extend the basic framework to three phases. The last extension is to inherently nonlinear functions. In these extensions, the changepoint(s) is a parameter to be estimated and may be fixed or allowed to vary across subjects as an application warrants. The approaches are developed and two illustrative empirical examples from psychology are used to highlight the methodological nuances. Annotated statistical software is provided to make these elaborations accessible to practitioners and methodologists.

Latent Growth and Dynamic Structural Equation Models

Latent growth models make up a class of methods to study within-person change—how it progresses, how it differs across individuals, what are its determinants, and what are its consequences. Latent growth methods have been applied in many domains to examine average and differential responses to interventions and treatments. In this review, we introduce the growth modeling approach to studying change by presenting different models of change and interpretations of their model parameters. We then apply these methods to examining sex differences in the development of binge drinking behavior through adolescence and into adulthood. Advances in growth modeling methods are then discussed and include inherently nonlinear growth models, derivative specification of growth models, and latent change score models to study stochastic change processes. We conclude with relevant design issues of longitudinal studies and considerations for the analysis of longitudinal data.

IV. DEVELOPMENTS IN THE ANALYSIS OF LONGITUDINAL DATA

Longitudinal data analytic techniques include a complex array of statistical techniques from repeated-measures analysis of variance, mixed-effects models, and time-series analysis, to longitudinal latent variable models (e.g., growth models, dynamic factor models) and mixture models (longitudinal latent profile analysis, growth mixture models). In this article, we focus our attention on the rationales of longitudinal research laid out by Baltes and Nesselroade (1979) and discuss the advancements in the analysis of longitudinal data since their landmark paper. We highlight the developments in growth and change analysis and its derivatives because these models best capture the rationales for conducting longitudinal research. We conclude with additional rationales of longitudinal research brought about by the development of new analytic techniques.

Using Integrative Data Analysis to Examine Changes in Alcohol Use and Changes in Sexual Risk Behavior Across Four Samples of STI Clinic Patients

BACKGROUND

Patients in sexually transmitted infection (STI) clinics report high levels of alcohol use, which are associated with risky sexual behavior. However, no studies have examined how changes in alcohol use relate to changes in sexual risk behavior.

PURPOSE

We used parallel process latent growth modeling to explore how changes in alcohol use related to changes in sexual behavior across four samples of clinic patients.

METHODS

Patients participating in HIV prevention trials from urban clinics in the Northeastern and Midwestern USA (N = 3761, 59 % male, 72 % Black) completed measures at 3-month intervals over 9-12 months. Integrative data analysis was used to create composite measures of alcohol use across samples. Sexual risk measures were counts of partners and unprotected sex acts. Parallel process models tested whether alcohol use changes were correlated with changes in the number of partners and unprotected sex.

RESULTS

Growth models with good fit showed decreases that slowed over time in sexual risk behaviors and alcohol use. Parallel process models showed positive correlations between levels of (rs = 0.17-0.40, ps < 0.001) and changes in (rs = 0.21-0.80, ps < 0.05) alcohol use and number of sexual partners across studies. There were strong associations between levels of (rs = 0.25-0.43, ps < 0.001) and changes in (rs = 0.24-0.57, ps < 0.01) alcohol use and unprotected sex in one study recruiting hazardous drinkers.

CONCLUSIONS

Across four samples of clinic patients, reductions in alcohol use were associated with reductions in the number of sexual partners. HIV prevention interventions may be strengthened by addressing alcohol use.

Bayesian hierarchical piecewise regression models: a tool to detect trajectory divergence between groups in long-term observational studies

Background

Bayesian hierarchical piecewise regression (BHPR) modeling has not been previously formulated to detect and characterise the mechanism of trajectory divergence between groups of participants that have longitudinal responses with distinct developmental phases. These models are useful when participants in a prospective cohort study are grouped according to a distal dichotomous health outcome. Indeed, a refined understanding of how deleterious risk factor profiles develop across the life-course may help inform early-life interventions. Previous techniques to determine between-group differences in risk factors at each age may result in biased estimate of the age at divergence.

Methods

We demonstrate the use of Bayesian hierarchical piecewise regression (BHPR) to generate a point estimate and credible interval for the age at which trajectories diverge between groups for continuous outcome measures that exhibit non-linear within-person response profiles over time. We illustrate our approach by modeling the divergence in childhood-to-adulthood body mass index (BMI) trajectories between two groups of adults with/without type 2 diabetes mellitus (T2DM) in the Cardiovascular Risk in Young Finns Study (YFS).

Results

Using the proposed BHPR approach, we estimated the BMI profiles of participants with T2DM diverged from healthy participants at age 16 years for males (95% credible interval (CI):13.5–18 years) and 21 years for females (95% CI: 19.5–23 years). These data suggest that a critical window for weight management intervention in preventing T2DM might exist before the age when BMI growth rate is naturally expected to decrease. Simulation showed that when using pairwise comparison of least-square means from categorical mixed models, smaller sample sizes tended to conclude a later age of divergence. In contrast, the point estimate of the divergence time is not biased by sample size when using the proposed BHPR method.

Conclusions

BHPR is a powerful analytic tool to model long-term non-linear longitudinal outcomes, enabling the identification of the age at which risk factor trajectories diverge between groups of participants. The method is suitable for the analysis of unbalanced longitudinal data, with only a limited number of repeated measures per participants and where the time-related outcome is typically marked by transitional changes or by distinct phases of change over time.

Electronic supplementary material

The online version of this article (doi:10.1186/s12874-017-0358-9) contains supplementary material, which is available to authorized users.

Nonlinear Growth Models as Measurement Models: A Second-Order Growth Curve Model for Measuring Potential

Recent methodological work has highlighted the promise of nonlinear growth models for addressing substantive questions in the behavioral sciences. In this article, we outline a second-order nonlinear growth model in order to measure a critical notion in development and education: potential. Here, potential is conceptualized as having three components-ability, capacity, and availability-where ability is the amount of skill a student is estimated to have at a given timepoint, capacity is the maximum amount of ability a student is predicted to be able to develop asymptotically, and availability is the difference between capacity and ability at any particular timepoint. We argue that single timepoint measures are typically insufficient for discerning information about potential, and we therefore describe a general framework that incorporates a growth model into the measurement model to capture these three components. Then, we provide an illustrative example using the public-use Early Childhood Longitudinal Study-Kindergarten data set using a Michaelis-Menten growth function (reparameterized from its common application in biochemistry) to demonstrate our proposed model as applied to measuring potential within an educational context. The advantage of this approach compared to currently utilized methods is discussed as are future directions and limitations.

A Finite Mixture of Nonlinear Random Coefficient Models for Continuous Repeated Measures Data

Nonlinear random coefficient models (NRCMs) for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development (or growth) over time in measured variables. As an extension of this model, this study considers the finite mixture of NRCMs that combine features of NRCMs with the idea of finite mixture (or latent class) models. The efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations, thus allowing the simultaneous investigation of intra-individual (within-person) variability, inter-individual (between-person) variability, and subpopulation heterogeneity. Effectiveness of this model to work under real data analytic conditions was examined by executing a Monte Carlo simulation study. The simulation study was carried out using an R routine specifically developed for the purpose of this study. The R routine used maximum likelihood with the expectation-maximization algorithm. The design of the study mimicked the output obtained from running a two-class mixture model on task completion data.

Using simple and complex growth models to articulate developmental change: Matching theory to method

Growth curve modeling has become a mainstay in the study of development. In this article we review some of the flexibility provided by this technique for describing and testing hypotheses about: (1) intraindividual change across multiple occasions of measurement, and (2) interindividual differences in intraindividual change. Through empirical example we demonstrate how linear, quadratic, latent basis, exponential, and multiphase versions of the model can be specified using commonly available SEM/multilevel modeling software and illustrate and discuss how results are obtained and interpreted. Particularly, we underscore the “developmental theory” articulated by each model.

Modeling neuroendocrine stress reactivity in salivary cortisol: adjusting for peak latency variability

In this report, we present growth curve modeling (GCM) with landmark registration as an alternative statistical approach for the analysis of time series cortisol data. This approach addresses an often-ignored but critical source of variability in salivary cortisol analyses: individual and group differences in the time latency of post-stress peak concentrations. It allows for the simultaneous examination of cortisol changes before and after the peak while controlling for timing differences, and thus provides additional information that can help elucidate group differences in the underlying biological processes (e.g., intensity of response, regulatory capacity). We tested whether GCM with landmark registration is more sensitive than traditional statistical approaches (e.g., repeated measures ANOVA--rANOVA) in identifying sex differences in salivary cortisol responses to a psychosocial stressor (Trier Social Stress Test--TSST) in healthy adults (mean age 23). We used plasma ACTH measures as our "standard" and show that the new approach confirms in salivary cortisol the ACTH finding that males had longer peak latencies, higher post-stress peaks but a more intense post-peak decline. This finding would have been missed if only saliva cortisol was available and only more traditional analytic methods were used. This new approach may provide neuroendocrine researchers with a highly sensitive complementary tool to examine the dynamics of the cortisol response in a way that reduces risk of false negative findings when blood samples are not feasible.

Nonlinear random-effects mixture models for repeated measures

A mixture model for repeated measures based on nonlinear functions with random effects is reviewed. The model can include individual schedules of measurement, data missing at random, nonlinear functions of the random effects, of covariates and of residuals. Individual group membership probabilities and individual random effects are obtained as empirical Bayes predictions. Although this is a complicated model that combines a mixture of populations, nonlinear regression, and hierarchical models, it is straightforward to estimate by maximum likelihood using SAS PROC NLMIXED. Many different models can be studied with this procedure. The model is more general than those that can be estimated with most special purpose computer programs currently available because the response function is essentially any form of nonlinear regression. Examples and sample code are included to illustrate the method.

Modeling Intraindividual Change in Nonlinear Growth Models with Latent Change Scores

Three central goals of longitudinal research are the modeling of intraindividual change, the examination of interindividual differences in intraindividual change, and the evaluation of determinants of intraindividual change ( Baltes & Nesselroade, 1979 ). The latent growth model is a commonly fit statistical model to examine these goals. However, the latent growth model has difficulty in this examination when change trajectories are nonlinear with respect to time and multiple latent variables impact intraindividual change. We consider a latent growth modeling approach based upon latent change scores ( McArdle, 2001 , 2009 ), which yields information related to these goals of longitudinal research when change trajectories are nonlinear. We illustrate this approach with longitudinal data from the Berkeley Guidance Study regarding lifespan changes in verbal ability.

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

We propose the use of the latent change and latent acceleration frameworks for modeling nonlinear growth in structural equation models. Moving to these frameworks allows for the direct identification of rates of change and acceleration in latent growth curves-information available indirectly through traditional growth curve models when change patterns are nonlinear with respect to time. To illustrate this approach, exponential growth models in the three frameworks are fit to longitudinal response time data from the Math Skills Development Project ( Mazzocco & Meyers, 2002 , 2003 ). We highlight the additional information gained from fitting growth curves in these frameworks as well as limitations and extensions of these approaches.

A comparison of methods for estimating quadratic effects in nonlinear structural equation models

Two Monte Carlo simulations were performed to compare methods for estimating and testing hypotheses of quadratic effects in latent variable regression models. The methods considered in the current study were (a) a 2-stage moderated regression approach using latent variable scores, (b) an unconstrained product indicator approach, (c) a latent moderated structural equation method, (d) a fully Bayesian approach, and (e) marginal maximum likelihood estimation. Of the 5 estimation methods, it was found that overall the methods based on maximum likelihood estimation and the Bayesian approach performed best in terms of bias, root-mean-square error, standard error ratios, power, and Type I error control, although key differences were observed. Similarities as well as disparities among methods are highlight and general recommendations articulated. As a point of comparison, all 5 approaches were fit to a reparameterized version of the latent quadratic model to educational reading data.

Long-term antecedents and outcomes of perceived control

Perceived control plays an important role in shaping development throughout adulthood and old age. Using data from the adult lifespan sample of the national German Socio-Economic Panel (SOEP; N > 10,000, covering 25 years of measurement), we explored long-term antecedents, correlates, and outcomes of perceived control and examined if associations differ with age. Targeting correlates and antecedents of control, findings indicated that higher concurrent levels of social participation, life satisfaction, and self-rated health as well as more positive changes in social participation over the preceding 11 years were each predictive of between-person differences in perceived control. Targeting health outcomes of control, survival analyses revealed that perceived control predicted 14-year hazard ratio for disability (n = 996 became disabled) and mortality (n = 1,382 died). The effect for mortality, but not for disability, was independent of sociodemographic and psychosocial factors. Overall, we found very limited support for age-differential associations. Our results provide further impetus to thoroughly examine processes involved in antecedent-consequent relations among perceived control, facets of social life, well-being, and health.

Modeling Time-Dependent Association in Longitudinal Data: A Lag as Moderator Approach

We describe a straightforward, yet novel, approach to examine time-dependent association between variables. The approach relies on a measurement-lag research design in conjunction with statistical interaction models. We base arguments in favor of this approach on the potential for better understanding the associations between variables by describing how the association changes with time. We introduce a number of different functional forms for describing these lag-moderated associations, each with a different substantive meaning. Finally, we use empirical data to demonstrate methods for exploring functional forms and model fitting based on this approach.

Nonlinear growth curves in developmental research

Developmentalists are often interested in understanding change processes, and growth models are the most common analytic tool for examining such processes. Nonlinear growth curves are especially valuable to developmentalists because the defining characteristics of the growth process such as initial levels, rates of change during growth spurts, and asymptotic levels can be estimated. A variety of growth models are described beginning with the linear growth model and moving to nonlinear models of varying complexity. A detailed discussion of nonlinear models is provided, highlighting the added insights into complex developmental processes associated with their use. A collection of growth models are fit to repeated measures of height from participants of the Berkeley Growth and Guidance Studies from early childhood through adulthood.

Nonlinear growth curves in developmental research

Developmentalists are often interested in understanding change processes, and growth models are the most common analytic tool for examining such processes. Nonlinear growth curves are especially valuable to developmentalists because the defining characteristics of the growth process such as initial levels, rates of change during growth spurts, and asymptotic levels can be estimated. A variety of growth models are described beginning with the linear growth model and moving to nonlinear models of varying complexity. A detailed discussion of nonlinear models is provided, highlighting the added insights into complex developmental processes associated with their use. A collection of growth models are fit to repeated measures of height from participants of the Berkeley Growth and Guidance Studies from early childhood through adulthood.

Terminal decline from within- and between-person perspectives, accounting for incident dementia

OBJECTIVE

The terminal cognitive decline hypothesis has been debated for almost 50 years. This hypothesis implies a change in rate of decline within an individual. Therefore, we examine the hypothesis from a within-person perspective using a time to death chronological structure.

METHOD

Scores on a Swedish version of the Wechsler Adult Intelligence Scale Information and Block Design scores from 461 OCTO-Twin Study participants with confirmed death dates were modeled using quadratic growth curve models including both age and distance from death at study entry, sex, education, and dementia diagnosis as covariates of initial performance and of linear and quadratic change over time.

RESULTS

Information scores showed statistically significant evidence of slight within-person acceleration of declines in the no dementia group. Individuals with incident dementia declined more quickly, and those who were closer to death at study baseline had a stronger acceleration. Block Design scores declined but did not show evidence of such acceleration either within or across individuals. Decline was faster in incident cases closer to death at study entry.

DISCUSSION

Within-person evidence of terminal decline is not as strong as previously published between-person results. Strategies for focusing models on longitudinal aspects of available data and the extent to which lack of within-person evidence for terminal decline may stem from common data limitations are discussed.

Non-linear Growth Models in Mplus and SAS

Non-linear growth curves or growth curves that follow a specified non-linear function in time enable researchers to model complex developmental patterns with parameters that are easily interpretable. In this paper we describe how a variety of sigmoid curves can be fit using the Mplus structural modeling program and the non-linear mixed-effects modeling procedure NLMIXED in SAS. Using longitudinal achievement data collected as part of a study examining the effects of preschool instruction on academic gain we illustrate the procedures for fitting growth models of logistic, Gompertz, and Richards functions. Brief notes regarding the practical benefits, limitations, and choices faced in the fitting and estimation of such models are included.

Structural equation modeling: strengths, limitations, and misconceptions

Because structural equation modeling (SEM) has become a very popular data-analytic technique, it is important for clinical scientists to have a balanced perception of its strengths and limitations. We review several strengths of SEM, with a particular focus on recent innovations (e.g., latent growth modeling, multilevel SEM models, and approaches for dealing with missing data and with violations of normality assumptions) that underscore how SEM has become a broad data-analytic framework with flexible and unique capabilities. We also consider several limitations of SEM and some misconceptions that it tends to elicit. Major themes emphasized are the problem of omitted variables, the importance of lower-order model components, potential limitations of models judged to be well fitting, the inaccuracy of some commonly used rules of thumb, and the importance of study design. Throughout, we offer recommendations for the conduct of SEM analyses and the reporting of results.

Analysis of nonlinear patterns of change with random coefficient models

Nonlinear patterns of change arise frequently in the analysis of repeated measures from longitudinal studies in psychology. The main feature of nonlinear development is that change is more rapid in some periods than in others. There generally also are strong individual differences, so although there is a general similarity of patterns for different persons over time, individuals exhibit substantial heterogeneity in their particular response. To describe data of this kind, researchers have extended the random coefficient model to accommodate nonlinear trajectories of change. It can often produce a statistically satisfying account of subject-specific development. In this review we describe and illustrate the main ideas of the nonlinear random coefficient model with concrete examples.

Structured latent curve models for the study of change in multivariate repeated measures

This article considers a structured latent curve model for multiple repeated measures. In a structured latent curve model, a smooth nonlinear function characterizes the mean response. A first-order Taylor polynomial taken with regard to the mean function defines elements of a restricted factor matrix that may include parameters that enter nonlinearly. Similar to factor scores, random coefficients are combined with the factor matrix to produce individual latent curves that need not follow the same form as the mean curve. Here the associations between change characteristics in multiple repeated measures are studied. A factor analysis model for covariates is included as a means of relating latent covariates to the factors characterizing change in different repeated measures. An example is provided.