Testing main effects and interactions in latent curve analysis

A key strength of latent curve analysis (LCA) is the ability to model individual variability in rates of change as a function of 1 or more explanatory variables. The measurement of time plays a critical role because the explanatory variables multiplicatively interact with time in the prediction of the repeated measures. However, this interaction is not typically capitalized on in LCA because the measure of time is rather subtly incorporated via the factor loading matrix. The authors' goal is to demonstrate both analytically and empirically that classic techniques for probing interactions in multiple regression can be generalized to LCA. A worked example is presented, and the use of these techniques is recommended whenever estimating conditional LCAs in practice.     

Response time analysis in cognitive tasks with increasing difficulty

In the present study, speeded tasks with differing assumed difficulties of the trials are regarded as a special class of simple cognitive tasks. Exploratory latent growth modeling with data-driven shape of a growth curve and nonlinear structured latent curve modeling with predetermined monotonically increasing functions were used to analyze individual response-time change with an increase in task difficulty and to investigate latent response-time growth variables in relation to cognitive ability. A task that required participants to state the number of dots in a set was used as an example of a simple cognitive task with increasing difficulty; the response latencies obtained in this task and the magnitude of RT-IQ correlations changed depending on the complexity of the trial.Consistent with theoretical predictions, the individual speed of baseline performance was only weakly negatively correlated with each of the three measures of cognitive ability used in the study. The rate of change in response times with increasing task difficulty, estimated as a generalized growth parameter in the exploratory latent growth model, was significantly negatively related to cognitive ability. Response times of less intelligent individuals increased more rapidly in the task with increasing difficulty than the response times of highly intelligent individuals. Within the nonlinear structured latent curve modeling, the growth of response times with an increase in task difficulty was best described by a function that approached an oblique asymptote after an initial interval of convexity. A slope of this oblique asymptote was significantly negatively related to cognitive ability; for less intelligent individuals, the asymptote of response-time change was steeper than that for more intelligent individuals. In addition, a parameter that determined the shape of response-time growth trajectories on the interval of convexity was related to one of the three measures of cognitive ability.     

Aging and Religious Participation in Late Life

The aim of the study is to evaluate the relationship between two dimensions of religiosity and the process of aging. Secondary analysis of longitudinal data from the Florida Retirement Study was used to assess the trajectories of religious development over time. We analyzed data from six interview waves with 1,000 older adults aged 72 or over. A baseline model of growth processes only indicated significant variation and mean decline in religious attendance, but no significant variation nor mean change in religious beliefs over time. A final model including a set of 17 covariates was estimated, and the model fit statistics indicated very good fit for this latent growth curve model. The decline in mean religious attendance across time did not accompany a mean increase in religious beliefs as expected. There were numerous individual differences in the trajectory of decline for religious attendance, as well as in the initial levels of attendance and religious beliefs.     

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.     

Mixture distribution latent state-trait analysis: basic ideas and applications

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

A joint modeling framework of responses and response times to assess learning outcomes

Abstract

A general modeling framework of response accuracy and response times is proposed to track skill acquisition and provide additional diagnostic information on the change of latent speed in a learning environment. This framework consists of two types of models: a dynamic response model that captures the response accuracy and the change of discrete latent attribute profile upon factors such as practice, intervention effects, and other latent and observable covariates, and a dynamic response time model that describes the change of the continuous response latency due to change of latent attribute profile. These two types of models are connected through a parameter, describing the change rate of the latent speed through the learning process, and a covariate defined as a function of the latent attribute profile. A Bayesian estimation procedure is developed to calibrate the model parameters and measure the latent variables. The estimation algorithm is evaluated through several simulation studies under various conditions. The proposed models are applied to a real data set collected through a spatial rotation diagnostic assessment paired with learning tools.     

On Latent Growth Models for Composites and Their Constituents

Over the last decade and a half, latent growth modeling has become an extremely popular and versatile technique for evaluating longitudinal change and its determinants. Most common among the models applied are those for a single measured variable over time. This model has been extended in a variety of ways, most relevant for the current work being the multidomain and the second-order latent growth models. Whereas the former allows for growth function characteristics to be modeled for multiple outcomes simultaneously, with the degree of growth characteristics' relations assessed within the model (e.g., cross-domain slope factor correlations), the latter models growth in latent outcomes, each of which has effect indicators repeated over time. But what if one has an outcome that is believed to be formative relative to its indicator variables rather than latent? In this case, where the outcome is a composite of multiple constituents, modeling change over time is less straightforward. This article provides analytical and applied details for simultaneously modeling growth in composites and their constituent elements, including a real data example using a general computer self-efficacy questionnaire.     

A general model for the analysis of multilevel data

A general model is developed for the analysis of multivariate multilevel data structures. Special cases of the model include repeated measures designs, multiple matrix samples, multilevel latent variable models, multiple time series, and variance and covariance component models.We would like to acknowledge the helpful comments of Ruth Silver. We also wish to thank the referees for helping to clarify the paper. This work was partly carried out with research funds provided by the Economic and Social Research Council (U.K.).     

Latent Growth Curve and Repeated Measures ANOVA Contrasts: What the Models are Telling You

The relationship between the latent growth curve and repeated measures ANOVA models is often misunderstood. Although a number of investigators have looked into the similarities and differences among these models, a cursory reading of the literature can give the impression that they are very different models. Here we show that each model represents a set of contrasts on the occasion means. We demonstrate that the fixed effects parameters of the estimated basis vector latent growth curve model are merely a transformation of the repeated measures ANOVA fixed effects parameters. We further show that differences in fit in models that estimate the same means structure can be due to the different error covariance structures implied by the model. We show these relationships both algebraically and through using data from a simulation.     

新世纪20年国内心理统计方法研究回顾

新世纪头20年, 国内心理学11本专业期刊一共发表了213篇统计方法研究论文。研究范围主要包括以下10类(按论文篇数排序)：结构方程模型、测验信度、中介效应、效应量与检验力、纵向研究、调节效应、探索性因子分析、潜在类别模型、共同方法偏差和多层线性模型。对各类做了简单的回顾与梳理。结果发现, 国内心理统计方法研究的广度和深度都不断增加, 研究热点在相互融合中共同发展; 但综述类论文比例较大, 原创性研究论文比例有待提高, 研究力量也有待加强。     

老年人认知功能的变化轨迹：基于潜变量增长模型的分析

摘 要 本研究采用中国老年健康影响因素跟踪调查（CLHLS）的4次数据（2002,2005,2008,2011）,对老年人认知功能的变化趋势以及影响因素进行了探讨。结果显示：（1）老年人认知功能在4次测查中呈非线性的下降趋势。（2）日常生活活动能力较低的个体其认知功能也较低;老年人读书年限越高,其认知功能水平越高;女性的认知功能水平低于男性;不饮酒的老年人认知功能低于饮酒的老年人。（3）读书年限与饮酒会正向预测模型的斜率。     

Investigating population heterogeneity with factor mixture models

Sources of population heterogeneity may or may not be observed. If the sources of heterogeneity are observed (e.g., gender), the sample can be split into groups and the data analyzed with methods for multiple groups. If the sources of population heterogeneity are unobserved, the data can be analyzed with latent class models. Factor mixture models are a combination of latent class and common factor models and can be used to explore unobserved population heterogeneity. Observed sources of heterogeneity can be included as covariates. The different ways to incorporate covariates correspond to different conceptual interpretations. These are discussed in detail. Characteristics of factor mixture modeling are described in comparison to other methods designed for data stemming from heterogeneous populations. A step-by-step analysis of a subset of data from the Longitudinal Survey of American Youth illustrates how factor mixture models can be applied in an exploratory fashion to data collected at a single time point.     

Relating latent class membership to external variables: An overview

In this article we provide an overview of existing approaches for relating latent class membership to external variables of interest. We extend on the work of Nylund-Gibson et al. (Structural Equation Modeling: A Multidisciplinary Journal, 2019, 26, 967), who summarize models with distal outcomes by providing an overview of most recommended modeling options for models with covariates and larger models with multiple latent variables as well. We exemplify the modeling approaches using data from the General Social Survey for a model with a distal outcome where underlying model assumptions are violated, and a model with multiple latent variables. We discuss software availability and provide example syntax for the real data examples in Latent GOLD.     

Analysis of multivariate mixed longitudinal data: A flexible latent process approach

Multivariate ordinal and quantitative longitudinal data measuring the same latent construct are frequently collected in psychology. We propose an approach to describe change over time of the latent process underlying multiple longitudinal outcomes of different types (binary, ordinal, quantitative). By relying on random‐effect models, this approach handles individually varying and outcome‐specific measurement times. A linear mixed model describes the latent process trajectory while equations of observation combine outcome‐specific threshold models for binary or ordinal outcomes and models based on flexible parameterized non‐linear families of transformations for Gaussian and non‐Gaussian quantitative outcomes. As models assuming continuous distributions may be also used with discrete outcomes, we propose likelihood and information criteria for discrete data to compare the goodness of fit of models assuming either a continuous or a discrete distribution for discrete data. Two analyses of the repeated measures of the Mini‐Mental State Examination, a 20‐item psychometric test, illustrate the method. First, we highlight the usefulness of parameterized non‐linear transformations by comparing different flexible families of transformation for modelling the test as a sum score. Then, change over time of the latent construct underlying directly the 20 items is described using two‐parameter longitudinal item response models that are specific cases of the approach.     

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

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

Building an identifiable latent class model with covariate effects on underlying and measured variables

In recent years, latent class models have proven useful for analyzing relationships between measured multiple indicators and covariates of interest. Such models summarize shared features of the multiple indicators as an underlying categorical variable, and the indicators' substantive associations with predictors are built directly and indirectly in unique model parameters. In this paper, we provide a detailed study on the theory and application of building models that allow mediated relationships between primary predictors and latent class membership, but that also allow direct effects of secondary covariates on the indicators themselves. Theory for model identification is developed. We detail an Expectation-Maximization algorithm for parameter estimation, standard error calculation, and convergent properties. Comparison of the proposed model with models underlying existing latent class modeling software is provided. A detailed analysis of how visual impairments affect older persons' functioning requiring distance vision is used for illustration.This work was supported by National Institute on Aging (NIA) Program Project P01-AG-10184-03 and National Institutes of Mental Health grant R01-MH-56639-01A1. Dr. Bandeen-Roche is a Brookdale National Fellow. The authors wish to thank Drs. Gary Rubin and Sheila West for kindly making the Salisbury Eye Evaluation data available. We also thank the Editor, the Associate Editor, and three referees for their valuable comments.     

Treatment effects on count outcomes with non-normal covariates

The effects of a treatment or an intervention on a count outcome are often of interest in applied research. When controlling for additional covariates, a negative binomial regression model is usually applied to estimate conditional expectations of the count outcome. The difference in conditional expectations under treatment and under control is then defined as the (conditional) treatment effect. While traditionally aggregates of these conditional treatment effects (e.g., average treatment effects) are computed by averaging over the empirical distribution, a recently proposed moment-based approach allows for computing aggregate effects as a function of distribution parameters. The moment-based approach makes it possible to control for (latent) multivariate normally distributed covariates and provides more reliable inferences under certain conditions. In this paper we propose three different ways to account for non-normally distributed continuous covariates in this approach: an alternative, known non-normal distribution; a plausible factorization of the joint distribution; and an approximation using finite Gaussian mixtures. A saturated model is used for categorical covariates, making a distributional assumption obsolete. We further extend the moment-based approach to allow for multiple treatment conditions and the computation of conditional effects for categorical covariates. An illustrative example highlighting the key features of our extension is provided.     

An illustration of multilevel factor analysis

The factor analysis of repeated measures psychiatric data presents interesting challenges for researchers in terms of identifying the latent structure of an assessment instrument. Specifically, repeated measures contain both within and between individual sources of variance. Although a number of techniques exist for separating out these 2 sources of variance, all are problematic. Recently, researchers have proposed that exploratory multilevel factor analysis (MFA) be used to appropriately analyze the latent structure of repeated measures data. The chief objective of this report is to provide a didactic step-by-step guide on how MFA may be applied to psychiatric data. In the discussion, we describe difficulties associated with MFA and consider challenges in factor analyzing life event appraisals in psychiatric samples.     

时变效应模型及在密集追踪数据分析中的应用

密集追踪数据通常蕴含了心理过程的详细变化信息,反映了某些心理的复杂变化过程。时变效应模型用函数替代恒定的系数,可描述密集追踪数据中随时间推移心理的动态变化过程和时变效应,是分析复杂心理过程的有效方法。在介绍时变效应模型的原理后,通过模拟研究考察模型的表现,结果显示:(1)样本量增加可降低函数估计的误差;(2)惩罚样条法的节点数选择与函数的复杂度有关,函数越复杂,所需节点越多;(3)样本量与节点数对函数估计误差的交互效应不显著。进一步应探讨测量次数、数据分布形态、数据缺失等如何影响模型的表现。     

Loglinear Rasch models for the analysis of stability and change

Loglinear unidimensional and multidimensional Rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories. Reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change. Models of purely quantitative change in latent traits are proposed as well as models including structural change. A conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales Rasch models. In the context of longitudinal research, this renders possible the statistical test of homogeneity of change against subject-specific change in latent traits. Applications to two empirical data sets illustrate the use of the models.The author is greatly indebted to Ulf Böckenholt, Rolf Langeheine, and several anonymous reviewers for many helpful suggestions.