A Unified Latent Growth Curve Model

Applying item response theory models to repeated observations has demonstrated great promise in developmental research. By allowing the researcher to take account of the characteristics of both item response and measurement error in longitudinal trajectory analysis, it improves the reliability and validity of latent growth curve analysis. This has enabled the study, to differentially weigh individual items and examine developmental stability and change over time, to propose a comprehensive modeling framework, combining a measurement model with a structural model. Despite a large number of components requiring attention, this study focuses on model formulation, evaluates the performance of the estimators of model parameters, incorporates prior knowledge from Bayesian analysis, and applies the model using an illustrative example. It is hoped that this fundamental study can demonstrate the breadth of this unified latent growth curve model.     

定时截尾Pareto分布最小后验风险的Bayes推断

在定时截尾试验下,利用决策理论,充分利用产品的先验信息.考虑试验费用和试验时间,并结合损失函数,对分布制定了最小后验风险决策方案。     

Detecting Unobserved Heterogeneity in Latent Growth Curve Models

Growth mixture models combine latent growth curve models and finite mixture models to examine the existence of latent classes that follow distinct developmental patterns. Analyses based on these models are becoming quite common in social and behavioral science research because of recent advances in computing, the availability of specialized statistical programs, and the ease of programming. In this article, we show how mixture models can be fit to examine the presence of multiple latent classes by algorithmically grouping or clustering individuals who follow the same estimated growth trajectory based on an evaluation of individual case residuals. The approach is illustrated using empirical longitudinal data along with an easy to use computerized implementation.     

Comparison of Inverse Wishart and Separation-Strategy Priors for Bayesian Estimation of Covariance Parameter Matrix in Growth Curve Analysis

Growth curve modeling provides a general framework for analyzing longitudinal data from social, behavioral, and educational sciences. Bayesian methods have been used to estimate growth curve models, in which priors need to be specified for unknown parameters. For the covariance parameter matrix, the inverse Wishart prior is most commonly used due to its proper and conjugate properties. However, many researchers have pointed out that the inverse Wishart prior might not work as expected. The purpose of this study is to investigate the influence of the inverse Wishart prior and compare it with a class of separation-strategy priors on the parameter estimates of growth curve models. In this article, we illustrate the use of different types of priors with 2 real data analyses, and then conduct simulation studies to evaluate and compare these priors in estimating both linear and nonlinear growth curve models. For the linear model, the simulation study shows that both the inverse Wishart and the separation-strategy priors work well for the fixed effects parameters. For the Level 1 residual variance estimate, the separation-strategy prior performs better than the inverse Wishart prior. For the covariance matrix, the results are mixed. Overall, the inverse Wishart prior is suggested if the population correlation coefficient and at least 1 of the 2 marginal variances are large. Otherwise, the separation-strategy prior is preferred. For the nonlinear growth curve model, the separation-strategy priors work better than the inverse Wishart prior.     

Fitting Nonlinear Latent Growth Curve Models With Individually Varying Time Points

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

The Effect of Sampling-Time Variation on Latent Growth Curve Models

Longitudinal data are often collected in waves in which a participant’s data can be collected at different times within each wave, resulting in sampling-time variation that is unaccounted for when waves are treated as single time points. Little research has been reported on the effects of this temporal imprecision on longitudinal growth-curve modeling. This article describes the results of a simulation study into the effect of sampling-time variation on parameter estimation, model fit, and model comparison with an empirical validation of the model fit and comparison results.     

On the Correspondence between the Latent Growth Curve and Latent Change Score Models

There has been a great deal of work in the literature on the equivalence between the mixed-effects modeling and structural equation modeling (SEM) frameworks in specifying growth models (Willett &; Sayer, 1994). However, there has been little work on the correspondence between the latent growth curve model (LGM) and the latent change score model (see Grimm, Zhang, Hamagami, &; Mazzocco, 2013 Grimm, K. J., Zhang, Z., Hamagami, F., &; Mazzocco, M. M. (2013). Modeling nonlinear change via latent change and latent acceleration frameworks: Examining velocity and acceleration of growth trajectories. Multivariate Behavioral Research, 48, 117–143.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We demonstrate that four popular variants of the latent change score model – the no change, constant change, proportional change, and dual change models – have LGM equivalents. We provide equations that allow the translation of parameters from one approach to the other and vice versa. We then illustrate this equivalence using mathematics achievement data from the National Longitudinal Survey of Youth.     

浅谈磷酸分布曲线的应用

本文根据磷酸的分布曲线推导出NaH2PO4和Na2HPO4水溶液氢离子浓度计算的最简公式．     

Fitting Structural Equation Model Trees and Latent Growth Curve Mixture Models in Longitudinal Designs: The Influence of Model Misspecification

When conducting longitudinal research, the investigation of between-individual differences in patterns of within-individual change can provide important insights. In this article, we use simulation methods to investigate the performance of a model-based exploratory data mining technique—structural equation model trees (SEM trees; Brandmaier, Oertzen, McArdle, & Lindenberger, 2013)—as a tool for detecting population heterogeneity. We use a latent-change score model as a data generation model and manipulate the precision of the information provided by a covariate about the true latent profile as well as other factors, including sample size, under the possible influences of model misspecifications. Simulation results show that, compared with latent growth curve mixture models, SEM trees might be very sensitive to model misspecification in estimating the number of classes. This can be attributed to the lower statistical power in identifying classes, resulting from smaller differences of parameters prescribed by the template model between classes.     

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.     

基于贝叶斯推断的多层软测量建模在丁醇发酵中的应用

针对丁醇生产过程中发酵产物品质参量难以实时测量,现有测量方法精度不高、测量结果受不确定因素影响较大的问题,提出一种基于贝叶斯推断和支持向量回归（Support vector machine regression,SVR）的多层软测量建模方法。首先应用贝叶斯推断计算后验概率、筛选偏置数据,并对偏置数据校准,建立第一层SVR模型;然后利用贝叶斯推断进行二次校准,建立第二层SVR模型,对第一层SVR模型输出进行修正,得到最终预测结果,克服干扰和偏差引起的模型不准确问题。将基于贝叶斯推断的多层支持向量回归（Bi-SVR）预测模型应用于丁醇发酵过程,仿真及实验结果表明,相较于传统SVR预测模型,系统在低干扰的情况下预测精度提高了4.52%,在高干扰时预测精度提高了5.37%。     

稳健统计推断在均值-方差模型中的应用

针对投资组合均值-方差模型,引入了Fast-MCD多变量稳健估计方法,稳健估计模型中股票期望收益和协方差矩阵,减小了离群值对投资组合决策的影响,并结合我国证券市场的特点,对沪市A股市场进行了实证分析,得到了证券投资组合的有效前沿.     

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.     

增长曲线模型中回归系数的线性估计的可容许性

对增长曲线模型,当协差阵是降秩的,在矩阵损失函数下,本文分别给出了回归系数的线性估计在非齐次线性估计类和全体估计类中是可容许估计的条件.     

Using Design-Based Latent Growth Curve Modeling With Cluster-Level Predictor to Address Dependency

The authors compared the effects of using the true Multilevel Latent Growth Curve Model (MLGCM) with single-level regular and design-based Latent Growth Curve Models (LGCM) with or without the higher-level predictor on various criterion variables for multilevel longitudinal data. They found that random effect estimates were biased when the higher-level predictor was not included and that standard errors of the regression coefficients from the higher-level were underestimated when a regular LGCM was used. Nevertheless, random effect estimates, regression coefficients, and standard error estimates were consistent with those from the true MLGCM when the design-based LGCM included the higher-level predictor. They discussed implication for the study with empirical data illustration.     

Weibull分布恒加试验的Bayes推断

利用自适应舍选抽样和Gibbs抽样对CE模型下Weibull分布场合恒定应力加速寿命试验进行了Bayes估计.最后通过模拟实例表明该估计是有效的.     

Nonlongitudinal Estimation of Growth Curve Parameters: The Cohort Growth Model

The cohort growth model (CGM) is a method for estimating the parameters of a latent growth model (LGM) based on cross-sectional data. The CGM models the interindividual differences in the growth rate, and it models how subjects’ growth rate is related to their initial status. We derive model identification for the CGM and illustrate, in a simulation study, that the CGM provides unbiased parameter estimates in most simulation conditions. Based on empirical data we compare the estimates of the CGM with the estimates of the LGM. The results were comparable for both models. Although the estimates of the (co)-variances were different, the estimates of both models led to similar conclusions on the developmental change. Finally, we discuss the advantages and limitations of the CGM, and we provide recommendations for its use in empirical research.     

Specification Search for Identifying the Correct Mean Trajectory in Polynomial Latent Growth Models

This study investigated the optimal strategy for model specification search under the latent growth modeling (LGM) framework, specifically on searching for the correct polynomial mean or average growth model when there is no a priori hypothesized model in the absence of theory. In this simulation study, the effectiveness of different starting models on the search of the true mean growth model was investigated in terms of the mean and within-subject variance-covariance (V-C) structure model. The results showed that specifying the most complex (i.e., unstructured) within-subject V-C structure with the use of LRT, ΔAIC, and ΔBIC achieved the highest recovery rate (>85%) of the true mean trajectory. Implications of the findings and limitations are discussed.