多阶段混合增长模型的影响因素：距离与形态

通过模拟研究, 考察潜类别距离和发展形态等因素对多阶段混合增长模型的模型选择和参数估计的影响：(1)潜类别距离越大, 模型选择和分类效果越好。(2)混合模型的选择, 应以一定样本量(至少200)为前提, 首先考虑BIC选出正确的分类模型, 再通过熵值、ARI等选择分类确定性较高的模型。(3)多阶段的发展形态对正确模型的选择和分类的确定性均有一定程度影响。(4)潜类别距离和样本量越大, 参数估计精度越高。(5)在判断分类准确性的指标中, ARI的选择更偏向于真实的模型。     

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.     

Identifying Students at Risk: An Examination of Computer-Adaptive Measures and Latent Class Growth Analysis

Multitiered systems of support depend on screening technology to identify students at risk. The purpose of this study was to examine the use of a computer-adaptive test and latent class growth analysis (LCGA) to identify students at risk in reading with focus on the use of this methodology to characterize student performance in screening. Participants included 3,699 students in Grades 3–5. Three time points of administration (fall, winter, and spring) of the computer-adaptive reading measure were selected. LCGA results indicated 6–7 classes, depending on grade, informed by level and growth in student performance that significantly predicted failure on the statewide test administered at the end of the year. The lowest-performing classes had failure rates above 90% across all grades. The results indicate that identifying homogeneous groups of learners through LCGA may be valuable as an approach to determining students who need additional instruction. Practical implications and future directions are discussed.     

An overview of Markov chain methods for the study of stage-sequential developmental processes

This article presents an overview of quantitative methodologies for the study of stage-sequential development based on extensions of Markov chain modeling. Four methods are presented that exemplify the flexibility of this approach: the manifest Markov model, the latent Markov model, latent transition analysis, and the mixture latent Markov model. A special case of the mixture latent Markov model, the so-called mover-stayer model, is used in this study. Unconditional and conditional models are estimated for the manifest Markov model and the latent Markov model, where the conditional models include a measure of poverty status. Issues of model specification, estimation, and testing using the Mplus software environment are briefly discussed, and the Mplus input syntax is provided. The author applies these 4 methods to a single example of stage-sequential development in reading competency in the early school years, using data from the Early Childhood Longitudinal Study--Kindergarten Cohort.     

On Inclusion of Covariates for Class Enumeration of Growth Mixture Models

In this article, we directly question the common practice in growth mixture model (GMM) applications that exclusively rely on the fitting model without covariates for GMM class enumeration. We provide theoretical and simulation evidence to demonstrate that exclusion of covariates from GMM class enumeration could be problematic in many cases. Based on our findings, we provide recommendations for examining the class enumeration by the fitting model without covariates and discuss the potential of covariate inclusion as a remedy for the weakness of GMM class enumeration without including covariates. A real example on the development of children's cumulative exposure to risk factors for adolescent substance use is provided to illustrate our methodological developments.     

Profiling retirees in the retirement transition and adjustment process: examining the longitudinal change patterns of retirees' psychological well-being

The author used role theory, continuity theory, and the life course perspective to form hypotheses regarding the different retirement transition and adjustment patterns and how different individual and contextual variables related to those patterns. The longitudinal data of 2 samples (n(1) = 994; n(2) = 1,066) from the Health and Retirement Survey were used. Three latent growth curve patterns of retirees' psychological well-being were identified as coexisting in the retiree samples through growth mixture modeling (GMM) analysis. On the basis of the latent class membership derived from GMM, retiree subgroups directly linked to different growth curve patterns were profiled with individual (e.g., bridge job status) and contextual variables (e.g., spouse working status). By recognizing the existence of multiple retiree subgroups corresponding to different psychological well-being change patterns, this study suggests that retirees do not follow a uniform adjustment pattern during the retirement process, which reconciles inconsistent previous findings. A resource perspective is further introduced to provide a more integrated theory for the current findings. The practical implications of this study are also discussed at both individual level and policy level.     

Examining Spirituality Over Time: Latent Growth Curve and Individual Growth Curve Analyses

SUMMARY

Research on spirituality and religiousness has gained growing attention in recent years; however, most studies have used cross-sectional designs. As research on this topic evolves, there has been increasing recognition of the need to examine these constructs and their effects through the use of longitudinal designs. Beyond repeated-measures ANOVA and OLS regression models, what tools are available to examine these constructs over time? The purpose of this paper is to provide an overview of two cutting-edge statistical techniques that will facilitate longitudinal investigations of spirituality and religiousness: latent growth curve analysis using structural equation modeling (SEM) and individual growth curve models. The SEM growth curve approach examines change at the group level, with change over time expressed as a single latent growth factor. In contrast, individual growth curve models consider longitudinal change at the level of the person. While similar results may be obtained using either method, researchers may opt for one over the other due to the strengths and weaknesses associated with these methods. Examples of applications of both approaches to longitudinal studies of spirituality and religiousness are presented and discussed, along with design and data considerations when employing these modeling techniques.     

Modeling Incomplete Longitudinal Substance Use Data Using Latent Variable Growth Curve Methodology

Longitudinal data sets typically suffer from attrition and other forms of missing data. When this common problem occurs, several researchers have demonstrated that correct maximum likelihood estimation with missing data can be obtained under mild assumptions concerning the missing data mechanism. With reasonable substantive theory, a mixture of cross-sectional and longitudinal methods developed within multiple-group structural equation modeling can provide a strong basis for inference about developmental change. Using an approach to the analysis of missing data, the present study investigated developmental trends in adolescent (N = 759) alcohol, marijuana, and cigarette use across a 5-year period using multiple-group latent growth modeling. An associative model revealed that common developmental trends existed for all three substances. Age and gender were included in the model as predictors of initial status and developmental change. Findings discuss the utility of latent variable structural equation modeling techniques and missing data approaches in the study of developmental change.     

Latent Growth Modeling for Logistic Response Functions

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

Growth mixture modelling in developmental psychology: overview and demonstration of heterogeneity in developmental trajectories of adolescent antisocial behaviour

Recent advances in statistical techniques for longitudinal data analysis have provided increased capabilities for elucidating individual differences in trajectories of change in child behaviours and abilities. However, most techniques still assume that there is a single underlying distribution with respect to changes over time, about which children are normally distributed. If there are multiple subgroups of youth following distinct developmental trajectories with unique predictors, however, the results of these statistical techniques may provide an incomplete analysis of the data. A newer class of statistical techniques, latent growth mixture modelling, provides a robust framework for examining heterogeneity in patterns of development. This paper illustrates the use of latent growth mixture modelling for examining heterogeneity in developmental trajectories of adolescent antisocial behaviour. Copyright © 2006 John Wiley & Sons, Ltd.     

Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.     

多阶段增长模型的方法比较

多阶段增长模型（Piecewise Growth Modeling,PGM）可以解决发展趋势中具有转折点的情形,并且相对其他复杂的曲线增长模型,解释更简单.已有的统计方法主要通过多层线性模型和潜变量增长模型对多阶段模型进行估计.通过模拟研究,用HLM6.0和Mplus6.0对上述两种模型分别进行估计,结果发现在参数估计的精度上,两种估计方法没有差异,只是在犯一类错误的概率上后者略小.进一步通过对错误模型的探讨发现,在样本量小（n=50）,斜率变化小（△b=0.2）时,用线性模型拟合数据而非PGM所犯错误概率较小,整体拟合更佳.但随着样本的增加和斜率变化的增加,错误模型的犯错概率明显增大.故在实际应用中,为了能更好拟合数据,研究者应根据数据本身的情况选择恰当的模型.     

Development of early vocational behavior: Parallel associations between career engagement and satisfaction

The present five-wave longitudinal study examined the parallel development of career engagement and satisfaction among young adults over an eight-year period starting from the last stages of their secondary education and ending after the transition to higher education or working life. The research questions were analyzed with parallel process latent growth curve (LGC) modeling and growth mixture modeling (GMM). The study is part of the ongoing longitudinal Finnish Educational Transitions (FinEdu) study, and followed 826 participants from ages 17 to 25. The developmental dynamics showed that career engagement and satisfaction developed parallel, each predicting the changes in the other. Towards the end of secondary education, career engagement increased and career satisfaction decreased on the mean level; however, later on, after the transition to higher education/work, both processes leveled off. The GMM results also revealed the existence of two latent trajectory groups, one representing a high transitional and the other a low increasing trajectory of career engagement and satisfaction.     

Predicting Response to Intensive Multimodal Inpatient Treatment: A Comparison of Single- and Multiple-Class Growth Modeling Approaches

In a modest body of research, personality functioning assessed via performance-based instruments has been found to validly predict treatment outcome and, to some extent, differential response to treatment. However, state-of-the-science longitudinal and mixture modeling techniques, which are common in many areas of clinical psychology, have rarely been used. In this article, we compare multilevel growth curve modeling (MLM) and latent class growth modeling (LCGM) approaches with the same data set to illustrate the different research questions that can be addressed by each method. Global Assessment of Functioning (GAF) scores collected at 6 points during the course of a long-term multimodal inpatient treatment of 58 severely and persistently mentally ill adults were used to model the trajectory of treatment outcome. Pretreatment Rorschach-based markers of personality functioning and other markers of psychiatric severity were examined as covariates in each modeling approach. The results of both modeling approaches generally indicated that more psychologically impaired clients responded less favorably to treatment. The LCGM approach revealed 2 unique trajectories of improvement (a persistently low group and a higher starting, improving group). Personality functioning and baseline psychiatric variables significantly predicted group membership and the rate of change within the groups. A side-by-side examination of these 2 methods was found to be useful in predicting differential treatment response with personality functioning variables.     

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.     

Implementing continuous non-normal skewed distributions in latent growth mixture modeling: An assessment of specification errors and class enumeration

Abstract

Recent advances have allowed for modeling mixture components within latent growth modeling using robust, skewed mixture distributions rather than normal distributions. This feature adds flexibility in handling non-normality in longitudinal data, through manifest or latent variables, by directly modeling skewed or heavy-tailed latent classes rather than assuming a mixture of normal distributions. The aim of this study was to assess through simulation the potential under- or over-extraction of latent classes in a growth mixture model when underlying data follow either normal, skewed-normal, or skewed-t distributions. In order to assess this, we implement skewed-t, skewed-normal, and conventional normal (i.e., not skewed) forms of the growth mixture model. The skewed-t and skewed-normal versions of this model have only recently been implemented, and relatively little is known about their performance. Model comparison, fit, and classification of correctly specified and mis-specified models were assessed through various indices. Findings suggest that the accuracy of model comparison and fit measures are dependent on the type of (mis)specification, as well as the amount of class separation between the latent classes. A secondary simulation exposed computation and accuracy difficulties under some skewed modeling contexts. Implications of findings, recommendations for applied researchers, and future directions are discussed; a motivating example is presented using education data.     

Fitting the Reduced RUM with Mplus: A Tutorial

The Reduced Reparameterized Unified Model (Reduced RUM) is a diagnostic classification model for educational assessment that has received considerable attention among psychometricians. However, the computational options for researchers and practitioners who wish to use the Reduced RUM in their work, but do not feel comfortable writing their own code, are still rather limited. One option is to use a commercial software package that offers an implementation of the expectation maximization (EM) algorithm for fitting (constrained) latent class models like Latent GOLD or Mplus. But using a latent class analysis routine as a vehicle for fitting the Reduced RUM requires that it be re-expressed as a logit model, with constraints imposed on the parameters of the logistic function. This tutorial demonstrates how to implement marginal maximum likelihood estimation using the EM algorithm in Mplus for fitting the Reduced RUM.     

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。     

Trajectories of Youthful Antisocial Behavior: Categories or Continua?

The purpose of this study was to determine whether qualitatively distinct trajectories of antisocial behavior could be identified in 1,708 children (843 boys, 865 girls) from the 1979 National Longitudinal Survey of Youth–Child Data (NLSY-C). Repeated ratings were made on the Behavior Problems Index (BPI: Peterson and Zill Journal of Marriage and the Family, 48, 295–307, 1986) antisocial scale by the mothers of these children when the children were 6, 8, 10, 12, and 14 years of age. Scores on three indicators constructed from the six BPI Antisocial items (callousness, aggression, noncompliance) were then analyzed longitudinally (by summing across the rating periods) and cross-sectionally (by testing each individual rating period) in the full sample as well as in subsamples of boys and girls. Results obtained with the mean above minus below a cut (MAMBAC), maximum covariance (MAXCOV), and latent mode factor analysis (L-Mode) taxometric procedures revealed consistent evidence of continuous latent structure despite the fact Growth Mixture Modeling (GMM) and Latent Class Growth Analysis (LCGA) identified between two and eight trajectories, depending on the stopping rule, in the three antisocial indicators. From these results, it is concluded that the structural model underlying these data is better represented as continuous rather than as categorical. The implications of these results for future research on developmental trajectories of antisocial behavior are discussed.     

分类精确性指数Entropy在潜剖面分析中的表现:一项蒙特卡罗模拟研究

本研究通过蒙特卡洛模拟考查了分类精确性指数Entropy及其变式受样本量、潜类别数目、类别距离和指标个数及其组合的影响情况。研究结果表明:(1)尽管Entropy值与分类精确性高相关,但其值随类别数、样本量和指标数的变化而变化,很难确定唯一的临界值;(2)其他条件不变的情况下,样本量越大,Entropy的值越小,分类精确性越差;(3)类别距离对分类精确性的影响具有跨样本量和跨类别数的一致性;(4)小样本(N=50~100)的情况下,指标数越多,Entropy的结果越好;(5)在各种条件下Entropy对分类错误率比其它变式更灵敏。