Fan Yang-Wallentin

Confirmatory Factor Analysis of Ordinal Variables With Misspecified Models

Ordinal variables are common in many empirical investigations in the social and behavioral sciences. Researchers often apply the maximum likelihood method to fit structural equation models to ordinal data. This assumes that the observed measures have normal distributions, which is not the case when the variables are ordinal. A better approach is to use polychoric correlations and fit the models using methods such as unweighted least squares (ULS), maximum likelihood (ML), weighted least squares (WLS), or diagonally weighted least squares (DWLS). In this simulation evaluation we study the behavior of these methods in combination with polychoric correlations when the models are misspecified. We also study the effect of model size and number of categories on the parameter estimates, their standard errors, and the common chi-square measures of fit when the models are both correct and misspecified. When used routinely, these methods give consistent parameter estimates but ULS, ML, and DWLS give incorrect standard errors. Correct standard errors can be obtained for these methods by robustification using an estimate of the asymptotic covariance matrix W of the polychoric correlations. When used in this way the methods are here called RULS, RML, and RDWLS.

Pairwise likelihood estimation for factor analysis models with ordinal data

A pairwise maximum likelihood (PML) estimation method is developed for factor analysis models with ordinal data and fitted both in an exploratory and confirmatory set-up. The performance of the method is studied via simulations and comparisons with full information maximum likelihood (FIML) and three-stage limited information estimation methods, namely the robust unweighted least squares (3S-RULS) and robust diagonally weighted least squares (3S-RDWLS). The advantage of PML over FIML is mainly computational. Unlike PML estimation, the computational complexity of FIML estimation increases either with the number of factors or with the number of observed variables depending on the model formulation. Contrary to 3S-RULS and 3S-RDWLS estimation, PML estimates of all model parameters are obtained simultaneously and the PML method does not require the estimation of a weight matrix for the computation of correct standard errors. The simulation study on the performance of PML estimates and estimated asymptotic standard errors investigates the effect of different model and sample sizes. The bias and mean squared error of PML estimates and their standard errors are found to be small in all experimental conditions and decreasing with increasing sample size. Moreover, the PML estimates and their standard errors are found to be very close to those of FIML.

ADHD symptoms, academic achievement, self-perception of academic competence and future orientation: A longitudinal study

In the investigation of the effect of attention deficit hyperactivity disorder (ADHD) symptoms on school careers there is a need to study the role of adolescent and childhood ADHD symptoms and academic achievement, and to incorporate measures that include the individuals perspective. Our aim was to gain an overview of the long-term development of school careers in relation to ADHD symptoms. We studied associations between ADHD symptoms and academic achievement at different time-points and future orientation at the end of high school, and assessed the role of self-perceptions of academic competence in these associations. Participants were 192 children (47% girls) with a range of ADHD symptoms taken from a community sample. Collecting data at three time points, in 6th, 11th and 12th grade we tested a structural equation model. Results showed that ADHD symptoms in 6th grade negatively affected academic achievement concurrently and longitudinally. ADHD symptoms in 11th grade negatively affected concurrent academic achievement and academic self-perception and future orientation in 12th grade. Academic achievement had a positive influence on academic self-perception and future orientation. Given the other factors, self-perception of academic competence did not contribute to outcomes. We concluded that early ADHD symptoms may cast long shadows on young peoples academic progress. This happens mainly by way of stability in symptoms and relations to early low academic achievement.

ML Versus MI for Missing Data With Violation of Distribution Conditions

Normal-distribution-based maximum likelihood (ML) and multiple imputation (MI) are the two major procedures for missing data analysis. This article compares the two procedures with respects to bias and efficiency of parameter estimates. It also compares formula-based standard errors (SEs) for each procedure against the corresponding empirical SEs. The results indicate that parameter estimates by MI tend to be less efficient than those by ML; and the estimates of variance -covariance parameters by MI are also more biased. In particular, when the population for the observed variables possesses heavy tails, estimates of variance -covariance parameters by MI may contain severe bias even at relative large sample sizes. Although performing a lot better, ML parameter estimates may also contain substantial bias at smaller sample sizes. The results also indicate that, when the underlying population is close to normally distributed, SEs based on the sandwich-type covariance matrix and those based on the observed information matrix are very comparable to empirical SEs with either ML or MI. When the underlying distribution has heavier tails, SEs based on the sandwich-type covariance matrix for ML estimates are more reliable than those based on the observed information matrix. Both empirical results and analysis show that neither SEs based on the observed information matrix nor those based on the sandwich-type covariance matrix can provide consistent SEs in MI. Thus, ML is preferable to MI in practice, although parameter estimates by MI might still be consistent.

Errors-in-variables system identification using structural equation modeling

Errors-in-variables (EIV) identification refers to the problem of consistently estimating linear dynamic systems whose output and input variables are affected by additive noise. Various solutions have been presented for identifying such systems. In this study, EIV identification using Structural Equation Modeling (SEM) is considered. Two schemes for how EIV Single-Input Single-Output (SISO) systems can be formulated as SEMs are presented. The proposed formulations allow for quick implementation using standard SEM software. By simulation examples, it is shown that compared to existing procedures, here represented by the covariance matching (CM) approach, SEM-based estimation provide parameter estimates of similar quality.

Asymptotic Robustness Study of the Polychoric Correlation Estimation

Asymptotic robustness against misspecification of the underlying distribution for the polychoric correlation estimation is studied. The asymptotic normality of the pseudo-maximum likelihood estimator is derived using the two-step estimation procedure. The t distribution assumption and the skew-normal distribution assumption are used as alternatives to the normal distribution assumption in a numerical study. The numerical results show that the underlying normal distribution can be substantially biased, even though skewness and kurtosis are not large. The skew-normal assumption generally produces a lower bias than the normal assumption. Thus, it is worth using a non-normal distributional assumption if the normal assumption is dubious.

A Simulation Study of Polychoric Instrumental Variable Estimation in Structural Equation Models

Data collected from questionnaires are often in ordinal scale. Unweighted least squares (ULS), diagonally weighted least squares (DWLS) and normal-theory maximum likelihood (ML) are commonly used methods to fit structural equation models. Consistency of these estimators demands no structural misspecification. In this article, we conduct a simulation study to compare the equation-by-equation polychoric instrumental variable (PIV) estimation with ULS, DWLS, and ML. Accuracy of PIV for the correctly specified model and robustness of PIV for misspecified models are investigated through a confirmatory factor analysis (CFA) model and a structural equation model with ordinal indicators. The effects of sample size and nonnormality of the underlying continuous variables are also examined. The simulation results show that PIV produces robust factor loading estimates in the CFA model and in structural equation models. PIV also produces robust path coefficient estimates in the model where valid instruments are used. However, robustness highly depends on the validity of instruments.

Asymptotic efficiency of the pseudo‐maximum likelihood estimator in multi‐group factor models with pooled data

A multi-group factor model is suitable for data originating from different strata. However, it often requires a relatively large sample size to avoid numerical issues such as non-convergence and non-positive definite covariance matrices. An alternative is to pool data from different groups in which a single-group factor model is fitted to the pooled data using maximum likelihood. In this paper, properties of pseudo-maximum likelihood (PML) estimators for pooled data are studied. The pooled data are assumed to be normally distributed from a single group. The resulting asymptotic efficiency of the PML estimators of factor loadings is compared with that of the multi-group maximum likelihood estimators. The effect of pooling is investigated through a two-group factor model. The variances of factor loadings for the pooled data are underestimated under the normal theory when error variances in the smaller group are larger. Underestimation is due to dependence between the pooled factors and pooled error terms. Small-sample properties of the PML estimators are also investigated using a Monte Carlo study.

Errors-in-variables identification using covariance matching and structural equation modeling

Two approaches for errors-in-variables identification are compared. Covariance matching (CM) is known to be a computationally efficient method with good performance. Structural equation modeling (SEM) has been used for many years for static problems, particularly for social science applications. It is shown here how the SEM approach can be applied also for dynamic (time-series) problems, and that the resulting method is closely related to the CM approach.

Approximated penalized maximum likelihood for exploratory factor analysis: an orthogonal case

The problem of penalized maximum likelihood (PML) for an exploratory factor analysis (EFA) model is studied in this paper. An EFA model is typically estimated using maximum likelihood and then the estimated loading matrix is rotated to obtain a sparse representation. Penalized maximum likelihood simultaneously fits the EFA model and produces a sparse loading matrix. To overcome some of the computational drawbacks of PML, an approximation to PML is proposed in this paper. It is further applied to an empirical dataset for illustration. A simulation study shows that the approximation naturally produces a sparse loading matrix and more accurately estimates the factor loadings and the covariance matrix, in the sense of having a lower mean squared error than factor rotations, under various conditions.