Classical test theory based and factor analytic structural models for analyzing change: A note on differences and similarities

This note is concerned with differences and similarities between structural models for analyzing change, which are conceptualized within two different modelling traditions: the one based on the classical test theory, and that within the factor-analytic approach. It is shown that these two possibilities lead to models for studying change, which are indistinguishable when using for data analytic purposes structural modeling programs, such as LISREL, EQS, COSAN, LISCOMP, RAMONA, EzPATH, SAS PROC CALIS. The reason for this data-analytic equivalence of the two conceptually different types of models is the confounding of their differences in the corresponding implied covariance matrix structures.     

Multilevel models for multiple-baseline data: modeling across-participant variation in autocorrelation and residual variance

Multilevel models (MLM) have been used as a method for analyzing multiple-baseline single-case data. However, some concerns can be raised because the models that have been used assume that the Level-1 error covariance matrix is the same for all participants. The purpose of this study was to extend the application of MLM of single-case data in order to accommodate across-participant variation in the Level-1 residual variance and autocorrelation. This more general model was then used in the analysis of single-case data sets to illustrate the method, to estimate the degree to which the autocorrelation and residual variances differed across participants, and to examine whether inferences about treatment effects were sensitive to whether or not the Level-1 error covariance matrix was allowed to vary across participants. The results from the analyses of five published studies showed that when the Level-1 error covariance matrix was allowed to vary across participants, some relatively large differences in autocorrelation estimates and error variance estimates emerged. The changes in modeling the variance structure did not change the conclusions about which fixed effects were statistically significant in most of the studies, but there was one exception. The fit indices did not consistently support selecting either the more complex covariance structure, which allowed the covariance parameters to vary across participants, or the simpler covariance structure. Given the uncertainty in model specification that may arise when modeling single-case data, researchers should consider conducting sensitivity analyses to examine the degree to which their conclusions are sensitive to modeling choices.     

Fitting direct covariance structures by the MSTRUCT modeling language of the CALIS procedure

This paper demonstrates the usefulness and flexibility of the general structural equation modelling (SEM) approach to fitting direct covariance patterns or structures (as opposed to fitting implied covariance structures from functional relationships among variables). In particular, the MSTRUCT modelling language (or syntax) of the CALIS procedure (SAS/STAT version 9.22 or later: SAS Institute, 2010) is used to illustrate the SEM approach. The MSTRUCT modelling language supports a direct covariance pattern specification of each covariance element. It also supports the input of additional independent and dependent parameters. Model tests, fit statistics, estimates, and their standard errors are then produced under the general SEM framework. By using numerical and computational examples, the following tests of basic covariance patterns are illustrated: sphericity, compound symmetry, and multiple‐group covariance patterns. Specification and testing of two complex correlation structures, the circumplex pattern and the composite direct product models with or without composite errors and scales, are also illustrated by the MSTRUCT syntax. It is concluded that the SEM approach offers a general and flexible modelling of direct covariance and correlation patterns. In conjunction with the use of SAS macros, the MSTRUCT syntax provides an easy‐to‐use interface for specifying and fitting complex covariance and correlation structures, even when the number of variables or parameters becomes large.     

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.     

Evaluating the robustness of repeated measures analyses: The case of small sample sizes and nonnormal data

Repeated measures analyses of variance are the method of choice in many studies from experimental psychology and the neurosciences. Data from these fields are often characterized by small sample sizes, high numbers of factor levels of the within-subjects factor(s), and nonnormally distributed response variables such as response times. For a design with a single within-subjects factor, we investigated Type I error control in univariate tests with corrected degrees of freedom, the multivariate approach, and a mixed-model (multilevel) approach (SAS PROC MIXED) with Kenward–Roger’s adjusted degrees of freedom. We simulated multivariate normal and nonnormal distributions with varied population variance–covariance structures (spherical and nonspherical), sample sizes (N), and numbers of factor levels (K). For normally distributed data, as expected, the univariate approach with Huynh–Feldt correction controlled the Type I error rate with only very few exceptions, even if samples sizes as low as three were combined with high numbers of factor levels. The multivariate approach also controlled the Type I error rate, but it requires N ≥ K. PROC MIXED often showed acceptable control of the Type I error rate for normal data, but it also produced several liberal or conservative results. For nonnormal data, all of the procedures showed clear deviations from the nominal Type I error rate in many conditions, even for sample sizes greater than 50. Thus, none of these approaches can be considered robust if the response variable is nonnormally distributed. The results indicate that both the variance heterogeneity and covariance heterogeneity of the population covariance matrices affect the error rates.     

有调节的多层中介效应分析

多层中介和多层调节效应分析在社科领域已常有应用,但如果将多层中介和调节整合在一起,可以产生2（多层中介类型）×2（调节变量的层次）×3（调节的中介路径）共12种有调节的多层中介模型。面对有调节的多层中介效应分析,研究者往往束手无策。详述了基于多层线性模型的12种有调节的多层中介的分析方法和基于多层结构方程模型的4类有调节的多层中介分析方法,包括正交分割法,随机系数预测法,潜调节结构方程法和贝叶斯合理值法。这四类方法的核心议题在于如何处理潜调节项。当样本量足够大时,建议选择潜调节结构方程法;当样本量不足时,建议选择贝叶斯合理值法。随后用一个实际例子演示如何进行有调节的多层中介效应分析并有相应的Mplus程序。最后展望了有调节的多层中介效应分析研究的拓展方向。     

Reciprocal influences in children's and families' adaptation to early childhood wheezing.

OBJECTIVE: The objective of the present study was to demonstrate the reciprocal relationships between family adaptation to illness and children's medication use over time among children who presented with wheezing illness in infancy but have varying illness outcomes by age 4. DESIGN: A longitudinal design and latent growth curve models (LGM) were used to predict change in family and caregiver adaptation to illness and children's medication use over three years among 140 infants with wheezing, among families from low socioeconomic, multi-ethnic backgrounds. MAIN OUTCOME MEASURES: One LGM predicted level and change (slope) of family adaptation to illness from children's baseline medication use. The second LGM predicted level and change (slope) of children's medication use from baseline family adjustment to illness. In both models, illness severity, caregivers' psychological resources, and emergency department use were covaried with the independent variable. RESULTS: Two latent growth models were found to adequately fit the data and demonstrate full reciprocal relations between family adaptation to illness and children's medication use while accounting for baseline variables. Baseline measures of caregiver psychological functioning and illness severity were also significant predictors of family adaptation and children's medication use over time. The two models were not statistically different for children with and without active asthma at 4 years of age. CONCLUSION: Findings support the reciprocal effects model of child and family influences on pediatric illness and underscore the importance of early indicators of individual and family functioning.     

Mixed-model pairwise multiple comparisons of repeated measures means

One approach to the analysis of repeated measures data allows researchers to model the covariance structure of the data rather than presume a certain structure, as is the case with conventional univariate and multivariate test statistics. This mixed-model approach was evaluated for testing all possible pairwise differences among repeated measures marginal means in a Between-Subjects x Within-Subjects design. Specifically, the authors investigated Type I error and power rates for a number of simultaneous and stepwise multiple comparison procedures using SAS (1999) PROC MIXED in unbalanced designs when normality and covariance homogeneity assumptions did not hold. J. P. Shaffer's (1986) sequentially rejective step-down and Y. Hochberg's (1988) sequentially acceptive step-up Bonferroni procedures, based on an unstructured covariance structure, had superior Type I error control and power to detect true pairwise differences across the investigated conditions.     

The challenges of fitting an item response theory model to the Social Anhedonia Scale

This study explored the application of latent variable measurement models to the Social Anhedonia Scale (SAS; Eckblad, Chapman, Chapman, & Mishlove, 1982), a widely used and influential measure in schizophrenia-related research. Specifically, we applied unidimensional and bifactor item response theory (IRT) models to data from a community sample of young adults (n = 2,227). Ordinal factor analyses revealed that identifying a coherent latent structure in the 40-item SAS data was challenging due to (a) the presence of multiple small content clusters (e.g., doublets); (b) modest relations between those clusters, which, in turn, implies a general factor of only modest strength; (c) items that shared little variance with the majority of items; and (d) cross-loadings in bifactor solutions. Consequently, we conclude that SAS responses cannot be modeled accurately by either unidimensional or bifactor IRT models. Although the application of a bifactor model to a reduced 17-item set met with better success, significant psychometric and substantive problems remained. Results highlight the challenges of applying latent variable models to scales that were not originally designed to fit these models.     

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.     

The Thorny Relation Between Measurement Quality and Fit Index Cutoffs in Latent Variable Models

Latent variable modeling is a popular and flexible statistical framework. Concomitant with fitting latent variable models is assessment of how well the theoretical model fits the observed data. Although firm cutoffs for these fit indexes are often cited, recent statistical proofs and simulations have shown that these fit indexes are highly susceptible to measurement quality. For instance, a root mean square error of approximation (RMSEA) value of 0.06 (conventionally thought to indicate good fit) can actually indicate poor fit with poor measurement quality (e.g., standardized factors loadings of around 0.40). Conversely, an RMSEA value of 0.20 (conventionally thought to indicate very poor fit) can indicate acceptable fit with very high measurement quality (standardized factor loadings around 0.90). Despite the wide-ranging effect on applications of latent variable models, the high level of technical detail involved with this phenomenon has curtailed the exposure of these important findings to empirical researchers who are employing these methods. This article briefly reviews these methodological studies in minimal technical detail and provides a demonstration to easily quantify the large influence measurement quality has on fit index values and how greatly the cutoffs would change if they were derived under an alternative level of measurement quality. Recommendations for best practice are also discussed.     

A Latent Growth Curve Modeling Approach to Pooled Interrupted Time Series Analyses

This paper presents a latent variable approach for the estimation of treatment effects within a pooled interrupted time series (ITS) design. Although considered quasi-experimental, the ITS design has been noted as representing one of the strongest alternatives to the randomized experiment, making it highly appropriate for use in documenting the presence of effects that might warrant further evaluation in a large-scale randomized study. Results suggest that the latent variable growth modeling (LGM) is capable of detecting simultaneous differences in both level and slope, and provides tests of significance for these two necessary indicators of an ITS intervention effect. As shown in the analyses, the LGM framework provides a comprehensive and flexible approach to research design and data analysis, making available to a wide audience of researchers an analytical framework for a variety of analyses of growth and developmental processes.     

The latent variable approach as applied to transitive reasoning

We discuss the limitations of hypothesis testing using (quasi-) experiments in the study of cognitive development and suggest latent variable modeling as a viable alternative to experimentation. Latent variable models allow testing a theory as a whole, incorporating individual differences with respect to developmental processes or abilities in the model. Experiments, in contrast, aim at testing hypotheses that refer to a specific part of a theory; also they ignore individual differences or model the individual differences using age group as a proxy for developmental stage. Drawing on a sample of 409 5–13-year olds, we demonstrate the advantages of latent variable models in the area of transitive reasoning. A comparison of three models showed that the latent variable model that represented fuzzy trace theory had a better fit than the models representing Piaget's theory or linear ordering theory.     

Selecting a linear mixed model for longitudinal data: repeated measures analysis of variance, covariance pattern model, and growth curve approaches

With increasing popularity, growth curve modeling is more and more often considered as the 1st choice for analyzing longitudinal data. Although the growth curve approach is often a good choice, other modeling strategies may more directly answer questions of interest. It is common to see researchers fit growth curve models without considering alterative modeling strategies. In this article we compare 3 approaches for analyzing longitudinal data: repeated measures analysis of variance, covariance pattern models, and growth curve models. As all are members of the general linear mixed model family, they represent somewhat different assumptions about the way individuals change. These assumptions result in different patterns of covariation among the residuals around the fixed effects. In this article, we first indicate the kinds of data that are appropriately modeled by each and use real data examples to demonstrate possible problems associated with the blanket selection of the growth curve model. We then present a simulation that indicates the utility of Akaike information criterion and Bayesian information criterion in the selection of a proper residual covariance structure. The results cast doubt on the popular practice of automatically using growth curve modeling for longitudinal data without comparing the fit of different models. Finally, we provide some practical advice for assessing mean changes in the presence of correlated data.     

Toward a Model-Based Approach to the Clinical Assessment of Personality Psychopathology

Recent years have witnessed tremendous growth in the scope and sophistication of statistical methods available to explore the latent structure of psychopathology, involving continuous, discrete, and hybrid latent variables. The availability of such methods has fostered optimism that they can facilitate movement from classification primarily crafted through expert consensus to classification derived from empirically based models of psychopathological variation. The explication of diagnostic constructs with empirically supported structures can then facilitate the development of assessment tools that appropriately characterize these constructs. Our goal in this article is to illustrate how new statistical methods can inform conceptualization of personality psychopathology and therefore its assessment. We use magical thinking as an example, because both theory and earlier empirical work suggested the possibility of discrete aspects to the latent structure of personality psychopathology, particularly forms of psychopathology involving distortions of reality testing, yet other data suggest that personality psychopathology is generally continuous in nature. We directly compared the fit of a variety of latent variable models to magical thinking data from a sample enriched with clinically significant variation in psychotic symptomatology for explanatory purposes. Findings generally suggested a continuous latent variable model best represented magical thinking, but results varied somewhat depending on different indexes of model fit. We discuss the implications of the findings for classification and applied personality assessment. We also highlight some limitations of this type of approach that are illustrated by these data, including the importance of substantive interpretation, in addition to use of model fit indexes, when evaluating competing structural models.     

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.     

Bayesian Semiparametric Structural Equation Models with Latent Variables

Structural equation models (SEMs) with latent variables are widely useful for sparse covariance structure modeling and for inferring relationships among latent variables. Bayesian SEMs are appealing in allowing for the incorporation of prior information and in providing exact posterior distributions of unknowns, including the latent variables. In this article, we propose a broad class of semiparametric Bayesian SEMs, which allow mixed categorical and continuous manifest variables while also allowing the latent variables to have unknown distributions. In order to include typical identifiability restrictions on the latent variable distributions, we rely on centered Dirichlet process (CDP) and CDP mixture (CDPM) models. The CDP will induce a latent class model with an unknown number of classes, while the CDPM will induce a latent trait model with unknown densities for the latent traits. A simple and efficient Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using simulated examples, and several applications.     

Extending the actor–partner interdependence model for binary outcomes: A multilevel logistic approach

Relationship researchers face many challenges when samples consist of dyads. The actor–partner interdependence model provides a framework for the analysis of dyadic data (D. A. Kashy & D. A. Kenny, 2000). Binary variables like discrete health diagnoses present additional challenges that are not easily handled using existing models. The analysis was demonstrated using SAS PROC GLIMMIX and HLM6. An example is presented where couple's personality traits are used to predict discrete health outcomes. A series of Monte Carlo simulations were performed to compare PROC GLIMMIX to HLM. GLIMMIX performed acceptably compared to HLM for small dyad‐level sample sizes (100 or fewer), but HLM clearly outperformed in larger samples. Both had problems estimating the variance of the random intercept and its standard error.