Empirically Corrected Rescaled Statistics for SEM with Small N and Large p

In many applications, the structural equation models with varying coefficients and changing quantiles are urgently needed. The estimation of the varying coefficients at different quantile levels brings difficulties and challenges especially in complex structural relations. In our article, we propose two kinds of quantile-based dynamic structural equation models: a quantile-based structural equation model with varying coefficients and a composite-quantile-based structural equation model with varying coefficients. Based on the above two new models, the paper first proposes a B-spline estimation algorithm based on the quantile regression estimation and then develops its extension algorithm based on the composite quantile regression estimation. The former algorithm allows the varying coefficients to be estimated at different quantiles, and the latter one simultaneously considers multiple quantile levels. A set of simulation studies is carried out to investigate the performances of our proposed algorithms and the paper finally applies the proposed models and algorithms to a real data example.

BackgroundLogistic regression is often used for mediation analysis with a dichotomous outcome. However, previous studies showed that the indirect effect and proportion mediated are often affected by a change of scales in logistic regression models. To circumvent this, standardization has been proposed. The aim of this study was to show the relative performance of the unstandardized and standardized estimates of the indirect effect and proportion mediated based on multiple regression, structural equation modeling, and the potential outcomes framework for mediation models with a dichotomous outcome.MethodsWe compared the performance of the effect estimates yielded by the three methods using a simulation study and two real-life data examples from an observational cohort study (n = 360).ResultsLowest bias and highest efficiency were observed for the estimates from the potential outcomes framework and for the crude indirect effect ab and the proportion mediated ab/(ab + c’) based on multiple regression and SEM.ConclusionsWe advise the use of either the potential outcomes framework estimates or the ab estimate of the indirect effect and the ab/(ab + c’) estimate of the proportion mediated based on multiple regression and SEM when mediation analysis is based on logistic regression. Standardization of the coefficients prior to estimating the indirect effect and the proportion mediated may not increase the performance of these estimates.

Joint analysis with clustering and structural equation modeling is one of the most popular approaches to analyzing heterogeneous data. The methods involved in this approach estimate a path diagram of the same shape for each cluster and interpret the clusters according to the magnitude of the coefficients. However, these methods have problems with difficulty in interpreting the coefficients when the number of clusters and/or paths increases and are unable to deal with any situation where the path diagram for each cluster is different. To tackle these problems, we propose two methods for simplifying the path structure and facilitating interpretation by estimating a different form of path diagram for each cluster using sparse estimation. The proposed methods and related methods are compared using numerical simulation and real data examples. The proposed methods are superior to the existing methods in terms of both fitting and interpretation.

SummaryGeneralized linear latent variable models (GLLVMs), as defined by Bartholomew and Knott, enable modelling of relationships between manifest and latent variables. They extend structural equation modelling techniques, which are powerful tools in the social sciences. However, because of the complexity of the log-likelihood function of a GLLVM, an approximation such as numerical integration must be used for inference. This can limit drastically the number of variables in the model and can lead to biased estimators. We propose a new estimator for the parameters of a GLLVM, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The new estimator can be viewed as an M-estimator, leading to readily available asymptotic properties and correct inference. A simulation study shows its excellent finite sample properties, in particular when compared with a well-established approach such as LISREL. A real data example on the measurement of wealth for the computation of multidimensional inequality is analysed to highlight the importance of the methodology.

We consider the problem of variable selection using Bayesian method in high-dimensional linear models, where the number of explanatory variables K n is possibly much larger than the sample size n. In particular, the true regression coefficients vector β* is required sparsity in the sense of , which describes a general situation when all explanatory variables are relevant, but most of them have very small effects. According to Bayesian inference of Wasserman (1998), given a proper prior to propose joint densities f(y, x) of response y and explanatory variables vector x, the posterior-proposed densities are often close to the true density f 0(y, x), for large n. It's the so-called “density consistency”. We aim to prove that in linear models, density consistency can induce regression function consistency, which ensures good performance of variable selection. Especially in the special case of , when some regression coefficients are bounded away from zero while the rest are exactly zero, our method would identify the underlying true model by giving consistent estimate of true regression coefficients vector. We also provide simulation studies and a real data example to demonstrate satisfactory finite-sample performance of our method.

Structural equation modeling (SEM) is a quantitative technique for evaluating the causal relations between and among a number of variables using a combination of statistical methods and assumptions. Although there have been recent developments in expanding SEM to include climatic changes, most applications have been restricted to the simulation of causal processes; this is especially true for the modelling of environmental changes. However, when SEM is applied as an exploratory technique in climate studies, the proposed model can illustrate and examine the relationships among several climatic elements in the environment. To investigate the relationship between temperature and precipitation, a combination of SEM, partial least squares (PLS) and GIS methods were employed in this study. A measurement model, a structural model and a spatial model for examining the relationships between temperature and precipitation were proposed. A SEM-PLS-GIS model consisting of three measurement sub-models was created on the basis of the concentration values of 14 climatic elements; the data were the monthly and seasonal values for the period between 1975 and 2012, and were obtained from 140 stations. The results of the new SEM-PLS-GIS model showed that seven temperature factors affected, directly or indirectly, the precipitation with minimum temperatures were being the most effective of all. It can be concluded that employing SEM-PLS-GIS model for the purpose of describing causal patterns of climate variations especially the variations of precipitation can be of great value.

Extension of SO-PLS to multi-way arrays: SO-N-PLS

One of the major challenges for epidemiologists is to understand causal relationships between risk factors and health outcomes by analysing data from observational studies. Epidemiologists know too well that correlations, such as those in regression analysis, rarely mean causation, and it would be very desirable if there is a methodology for observational studies, analogous to randomization in experimental studies, which can discover causes and effects amongst variables or at least confirm or refute the proposed causal relationships. Randomized controlled trials (RCTs) are certainly the ‘gold standard’ of establishing causes and effects, but quite often it is either unethical or unfeasible to conduct RCTs to test causal relations in epidemiological research. Epidemiologists need a methodology, which is sort of a combination of the directed acyclic graphs (DAGs) for conceptual construction of causal models and regression analysis for testing those models. It is therefore surprising that structural equation modelling (SEM) has not been so frequently used in epidemiology as in the social sciences, given that both epidemiologists and social scientists want to delineate causes and effects from observational data. The difference between DAGs and path diagrams in SEM is almost trivial to epidemiologists: the path between two variables can only have one direction in DAGs. 1,2 An individual path in SEM is tested in the same way the regression coefficient is in regression analysis, and model fit indices provided by SEM software packages help the analysts to assess the adequacy of the proposed causal model compared with the observed associations in the sample data. 3,4 Why then is SEM still under-utilized in epidemiology? This is a question posed by a commentary 5 in this journal a few years ago. The answers cited included unfamiliar terminology (SEM theory is formulated in Greek), restriction in the assumptions of variables (outcome variables need to be continuous), difficulties in testing interaction and non-linear relationship (it can be quite tedious to set up SEM models to do these), and equivalent models (two different causal models imply the same correlation structure and consequently, it is impossible to tell which is better). 6 Recent advances in SEM theory and software development has resolved some of these issues: new estimation methods do not require the strict assumption of multivariate normality, 7 and the outcome variables can be binary, ordinal or counts. 7–9 The basic rationale behind SEM is rather simple: multiple linear equations are used to specify causal relationship between variables some of which are manifest variables (i.e. observed and collected by the researchers), whilst others are latent variables (i.e. derived from the observed variables by specifying their relations using equations), such as those in factor analysis. The multiple equations in each causal model entail a certain structure of correlational relationships between observed variables which is usually given as a correlation (or covariance) matrix . The estimation procedure is to minimize the difference between and the observed correlation/covariance matrix S formulated by a likelihood function. The 2 -test is then used to evaluate the difference between these two matrices by taking into account the number of the estimated parameters in the proposed model. When the 2 -value is large (i.e. the difference between the two matrices is large) relative to the model’s degree of freedom, the proposed model is rejected, i.e. something in the causal relationships specified by the proposed model is not quite right and require a second thought. When the 2 -value is small, we fail to reject the model or tentatively accept the model as adequate. Because of the possibility of equivalent models, our model may still be wrong in terms of the causal relations amongst variables but happen to entail the same correlation structure (the same S) of the ‘true’ model. So SEM seems to be an endeavour to search for the truth by approximation, and in this aspect, doing SEM is quite similar to the process of ‘Conjectures and Refutations’ as described by the great philosopher

Effect size is crucial for quantifying differences and a key concept behind Type I errors and power, but measures of effect size are seldom studied in structural equation modeling (SEM). While fit indices such as the root mean square error of approximation may address the severity of model misspecification, they are not a direct generalization of commonly used effect size measures such as Cohen’s d. Moreover, with violations of normality and when a test statistic does not follow a noncentral chi-square distribution, measures of misfit that are defined through the assumed distribution of the test statistic are no longer valid.In this study, two new classes of effect size measures for SEM are developed by generalizing Cohen’s d. The first class consists of definitions that are theoretically equivalent to , the population counterpart of the normal-distribution-based discrepancy function. The second class of effect size measures bears a stricter resemblance to Cohen’s d in its original form. Several versions of these generalizations are investigated to identify the one that is least affected by sample size and population distribution but most sensitive to model misspecification. Their performances under violated distributional assumptions, severity of model misspecification, and various sample sizes are examined using both normal maximum likelihood estimation and robust M-estimation. Monte Carlo results indicate that one measure in the first class of effect size measures is little affected by sample size and distribution while preserving sensitivity to model misspecification and thus is recommended for researchers to report in publications.

ABSTRACTA definite linear algorithm based on the constraint least squares solution for the structural equation model (SEM) is proposed. First, the modular constraint least squares solution is obtained. Then, the prescription constraint (the weight coefficients are non negative and their sum is one) is added, and the last solution of SEM is obtained. This definite linear algorithm is a new and reasonable algorithm compared to traditional algorithms like linear structural relationship (LISREL) and partial least squares (PLS). Finally, the application of SEM with the definite linear algorithm in fire risk evaluation is addressed, and the data example is calculated using software Data Analysis and Statistical Computation (DASC).

This paper presents a novel variable selection method in additive nonparametric regression model. This work is motivated by the need to select the number of nonparametric components and number of variables within each nonparametric component. The proposed method uses a combination of hard and soft shrinkages to separately control the number of additive components and the variables within each component. An efficient algorithm is developed to select the importance of variables and estimate the interaction network. Excellent performance is obtained in simulated and real data examples.

We propose a new procedure for sparse factor analysis (FA) such that each variable loads only one common factor. Thus, the loading matrix has a single nonzero element in each row and zeros elsewhere. Such a loading matrix is the sparsest possible for certain number of variables and common factors. For this reason, the proposed method is named sparsest FA (SSFA). It may also be called FA-based variable clustering, since the variables loading the same common factor can be classified into a cluster. In SSFA, all model parts of FA (common factors, their correlations, loadings, unique factors, and unique variances) are treated as fixed unknown parameter matrices and their least squares function is minimized through specific data matrix decomposition. A useful feature of the algorithm is that the matrix of common factor scores is re-parameterized using QR decomposition in order to efficiently estimate factor correlations. A simulation study shows that the proposed procedure can exactly identify the true sparsest models. Real data examples demonstrate the usefulness of the variable clustering performed by SSFA.

ABSTRACTThe paper presents 3D diffraction imaging based on the spectral decomposition of a different combination of selective or partial images. These images are obtained by the pre‐stack asymmetric migration procedure, which is weighted data summation. Spectral decomposition is done in the Fourier domain with respect to spatial dip and azimuth angles. Numerical examples with the application of different workflows for the synthetic and real data examples demonstrate detailed reliable reconstruction of the fractured zones and reliable reconstruction of fracture orientation on synthetic and real 3D data examples.

The importance of sample size, although widely discussed in the literature on structural equation modeling (SEM), has not been widely recognized among applied SEM researchers. To narrow this gap, we focus on second language testing and learning studies and examine the following: (a) Is the sample size sufficient in terms of precision and power of parameters in a model using Monte Carlo analysis? (b) How are the results from Monte Carlo sample size analysis comparable with those from the N ≥ 100 rule and from the N: q ≥ 10 (sample size–free parameter ratio) rule? Regarding (a), parameter bias, standard error bias, coverage, and power were overall satisfactory, suggesting that sample size for SEM models in second language testing and learning studies is generally appropriate. Regarding (b), both rules were often inconsistent with the Monte Carlo analysis, suggesting that they do not serve as guidelines for sample size. We encourage applied SEM researchers to perform Monte Carlo analyses to estimate the requisite sample size of a model.

Position effects may occur in both paper–pencil tests and computerized assessments when examinees respond to the same test items located in different positions on the test. To examine position effects in large-scale assessments, previous studies often used multilevel item response models within the generalized linear mixed modeling framework. Using the equivalence of the item response theory and binary factor analysis frameworks when modeling dichotomous item responses, this study introduces a structural equation modeling (SEM) approach that is capable of estimating various types of position effects. Using real data from a large-scale reading assessment, the SEM approach is demonstrated for investigating form, passage position, and item position effects for reading items. The results from a simulation study are also presented to evaluate the accuracy of the SEM approach in detecting item position effects. The implications of using the SEM approach are discussed in the context of large-scale assessments.