scholarly journals A Note on Likelihood Ratio Tests for Models with Latent Variables

Psychometrika ◽  
2020 ◽  
Author(s):  
Yunxiao Chen ◽  
Irini Moustaki ◽  
Haoran Zhang
Keyword(s):  
General Theory ◽  
Likelihood Ratio ◽  
Latent Variables ◽  
Latent Variable ◽  
Structural Equation ◽  
Degrees Of Freedom ◽  
Structural Equation Models ◽  
Regularity Conditions ◽  
Ratio Test ◽  
Nested Models

AbstractThe likelihood ratio test (LRT) is widely used for comparing the relative fit of nested latent variable models. Following Wilks’ theorem, the LRT is conducted by comparing the LRT statistic with its asymptotic distribution under the restricted model, a $$\chi ^2$$ χ 2 distribution with degrees of freedom equal to the difference in the number of free parameters between the two nested models under comparison. For models with latent variables such as factor analysis, structural equation models and random effects models, however, it is often found that the $$\chi ^2$$ χ 2 approximation does not hold. In this note, we show how the regularity conditions of Wilks’ theorem may be violated using three examples of models with latent variables. In addition, a more general theory for LRT is given that provides the correct asymptotic theory for these LRTs. This general theory was first established in Chernoff (J R Stat Soc Ser B (Methodol) 45:404–413, 1954) and discussed in both van der Vaart (Asymptotic statistics, Cambridge, Cambridge University Press, 2000) and Drton (Ann Stat 37:979–1012, 2009), but it does not seem to have received enough attention. We illustrate this general theory with the three examples.

scholarly journals Products of Variables in Structural Equation Models: Multiplicative Reticular Action Models

2019 ◽  
Author(s):  
Steven M. Boker ◽  
Timo von Oertzen ◽  
Andreas Markus Brandmaier
Keyword(s):  
Latent Variables ◽  
Latent Variable ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Equation Model ◽  
Random Coefficients ◽  
Action Model ◽  
Action Models ◽  
Special Case ◽  
General Method

A general method is introduced in which variables that are products of other variables in the context of a structural equation model (SEM) can be decomposed into the sources of variance due to the multiplicands. The result is a new category of SEM which we call a Multiplicative Reticular Action Model (XRAM). XRAM can include interactions between latent variables, multilevel random coefficients, latent variable moderators, and novel constructs such as factors of paths and twin genetic decomposition of multilevel random coefficients. The method relies on an assumption that all variance sources in a model can be decomposed into linear combinations of independent normal standardized variables. Although the distribution of a variable that is an outcome of multiplication between other variables is not normal, the assumption is that it can be decomposed into sources that are normal if one takes into account the non-normality induced by the multiplication. The method is applied to an example to show how in a special case it is equivalent to known unbiased and efficient estimators in the statistical literature. Two simulations are presented that demonstrate the precision of the approximation and implement the method to estimate parameters in a multilevel autoregressive framework.

scholarly journals Optimizing Detection of True Within-Person Effects for Intensive Measurement Designs: A Comparison of Multilevel SEM and Unit-Weighted Scale Scores

2020 ◽  
Author(s):  
Jonathan Rush ◽  
Philippe Rast ◽  
Scott Michael Hofer
Keyword(s):  
Latent Variables ◽  
Latent Variable ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Simulated Data ◽  
Variable Model ◽  
Alternative Approach ◽  
Scale Scores ◽  
Weighted Model ◽  
The Impact

Intensive repeated measurement designs are frequently used to investigate within-person variation over relatively brief intervals of time. The majority of research utilizing these designs rely on unit-weighted scale scores, which assume that the constructs are measured without error. An alternative approach makes use of multilevel structural equation models (MSEM), which permit the specification of latent variables at both within-person and between-person levels. These models disaggregate measurement error from systematic variance, which should result in less biased within-person estimates and larger effect sizes. Differences in power, precision, and bias between multilevel unit-weighted and MSEM models were compared through a series of Monte Carlo simulations. Results based on simulated data revealed that precision was consistently poorer in the MSEM models than the unit-weighted models, particularly when reliability was low. However, the degree of bias was considerably greater in the unit-weighted model than the latent variable model. Although the unit-weighted model consistently underestimated the effect of a covariate, it generally had similar power relative to the MSEM model due to the greater precision. Considerations for scale development and the impact of within-person reliability are highlighted.

scholarly journals A protocol for modelling generalised biological responses using latent variables in structural equation models

One Ecosystem ◽  
2021 ◽  
Vol 6 ◽  
Author(s):  
James Grace ◽  
Magdalena Steiner
Keyword(s):  
Latent Variables ◽  
General Property ◽  
Latent Variable ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Evaluation Process ◽  
Theoretical Development ◽  
Biological Responses ◽  
Internal Integration ◽  
Multiple Species

In this paper, we consider the problem of how to quantitatively characterise the degree to which a study object exhibits a generalised response. By generalised response, we mean a multivariate response where numerous individual properties change in concerted fashion due to some internal integration. In latent variable structural equation modelling (LVSEM), we would typically approach this situation using a latent variable to represent a general property of interest (e.g. performance) and multiple observed indicator variables that reflect the specific features associated with that general property. While ecologists have used LVSEM in a number of cases, there is substantial potential for its wider application. One obstacle is that LV models can be complex and easily over-specified, degrading their value as a means of generalisation. It can also be challenging to diagnose causes of misspecification and understand which model modifications are sensible. In this paper, we present a protocol, consisting of a series of questions, designed to guide the researchers through the evaluation process. These questions address: (1) theoretical development, (2) data requirements, (3) whether responses to perturbation are general, (4) unique reactions by individual measures and (5) how far generality can be extended. For this illustration, we reference a recent study considering the potential consequences of maintaining biodiversity as part of agricultural management on the overall quality of grapes used for wine-making. We extend our presentation to include the complexities that occur when there are multiple species with unique reactions.

scholarly journals Generalized ridge estimators adapted in structural equation models

2020 ◽  
Vol 43 ◽  
pp. e49929
Author(s):  
Gislene Araujo Pereira ◽  
Mariana Resende ◽  
Marcelo Ângelo Cirillo
Keyword(s):  
Latent Variables ◽  
Latent Variable ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Real Data ◽  
Alternative Methods ◽  
Estimation Methods ◽  
Specification Error ◽  
Maximum Likelihood Methods ◽  
Pls Method

Multicollinearity is detected via regression models, where independent variables are strongly correlated. Since they entail linear relations between observed or latent variables, the structural equation models (SEM) are subject to the multicollinearity effect, whose numerous consequences include the singularity between the inverse matrices used in estimation methods. Given to this behavior, it is natural to understand that the suitability of these estimators to structural equation models show the same features, either in the simulation results that validate the estimators in different multicollinearity degrees, or in their application to real data. Due to the multicollinearity overview arose from the fact that the matrices inversion is impracticable, the usage of numerical procedures demanded by the maximum likelihood methods leads to numerical singularity problems. An alternative could be the use of the Partial Least Squares (PLS) method, however, it is demanded that the observed variables are built by assuming a positive correlation with the latent variable. Thus, theoretically, it is expected that the load signals are positive, however, there are no restrictions to these signals in the algorithms used in the PLS method. This fact implies in corrective areas, such as the observed variables removal or new formulations of the theoretical model. In view of this problem, this paper aimed to propose adaptations of six generalized ridge estimators as alternative methods to estimate SEM parameters. The conclusion is that the evaluated estimators presented the same performance in terms of accuracy, precision while considering the scenarios represented by model without specification error and model with specification error, different levels of multicollinearity and sample sizes.

scholarly journals 10. Using Instrumental Variable Tests to Evaluate Model Specification in Latent Variable Structural Equation Models

2009 ◽  
Vol 39 (1) ◽  
pp. 327-355 ◽  
Author(s):  
James B. Kirby ◽  
Kenneth A. Bollen
Keyword(s):  
Latent Variables ◽  
Latent Variable ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Model Specification ◽  
Equation Modeling ◽  
Monte Carlo Experiment ◽  
Full Information ◽  
Limited Information ◽  
Finite Sample

Structural equation modeling (SEM) with latent variables is a powerful tool for social and behavioral scientists, combining many of the strengths of psychometrics and econometrics into a single framework. The most common estimator for SEM is the full-information maximum likelihood (ML) estimator, but there is continuing interest in limited information estimators because of their distributional robustness and their greater resistance to structural specification errors. However, the literature discussing model fit for limited information estimators for latent variable models is sparse compared with that for full-information estimators. We address this shortcoming by providing several specification tests basedon the 2SLS estimator for latent variable structural equation models developed by Bollen (1996). We explain how these tests can be used not only to identify a misspecified model but to help diagnose the source of misspecification within a model. We present and discuss results from a Monte Carlo experiment designed to evaluate the finite sample properties of these tests. Our findings suggest that the 2SLS tests successfully identify most misspecified models, even those with modest misspecification, and that they provide researchers with information that can help diagnose the source of misspecification.

scholarly journals Bayesian multilevel structural equation modeling: An investigation into robust prior distributions.

2020 ◽  
Author(s):  
Sara van Erp ◽  
William Browne
Keyword(s):  
Random Effects ◽  
Latent Variables ◽  
Latent Variable ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Equation Modeling ◽  
Prior Distributions ◽  
Inverse Gamma ◽  
Variance Parameters ◽  
Multilevel Structural Equation

Bayesian estimation of multilevel structural equation models (MLSEMs) offers advantages in terms of sample size requirements and computational feasibility, but does require careful specification of the prior distribution especially for the random effects variance parameters. The traditional “non-informative” conjugate choice of an inverse- Gamma prior with small hyperparameters has been shown time and again to be problematic. In this paper, we investigate alternative, more robust prior distributions. In contrast to multilevel models without latent variables, MLSEMs have multiple random effects variance parameters, both for the multilevel structure and for the latent variable structure. It is therefore even more important to construct reasonable priors for these parameters. We find that, although the robust priors outperform the traditional inverse-Gamma prior, their hyperparameters do require careful consideration.

scholarly journals Interindividual Differences in Treatment Effects based on Structural Equation Models with Latent Variables: An EffectLiteR Tutorial

2019 ◽  
Author(s):  
Axel Mayer ◽  
Johannes Zimmermann ◽  
Jürgen Hoyer ◽  
Simone Salzer ◽  
Jörg Wiltink ◽  
...  
Keyword(s):  
Latent Variables ◽  
Latent Variable ◽  
Structural Equation ◽  
Treatment Effects ◽  
Structural Equation Models ◽  
Controlled Trial ◽  
Psychological Research ◽  
Individual Treatment ◽  
Interindividual Differences ◽  
Multicenter Randomized Controlled Trial

The investigation of interindividual differences in the effects of a treatment is challenging, because many constructs-of-interest in psychological research such as depression or anxiety are latent variables and modeling heterogeneity in treatment effects requires interactions and potentially nonlinear relationships. In this paper, we present a tutorial of the EffectLiteR approach (Mayer, Dietzfelbinger, Rosseel, & Steyer, 2016) that allows for estimating individual treatment effects based on latent variable models. We describe step by step how to apply the approach using the EffectLiteR software package with data from the multicenter randomized controlled trial of the Social Phobia Psychotherapy Network (SOPHO-NET) and provide guidelines and recommendations for researchers. The focus of the paper is on explaining the results of a comprehensive effect analysis in an accessible language and on highlighting the opportunities the EffectLiteR approach offers for analyzing interindividual differences in treatment effects.

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