Ordinal data in social science statistics are often modeled as discretizations of a multivariate normal vector. In contrast to the continuous case, where SEM estimation is also consistent under non-normality, violation of underlying normality in ordinal SEM may lead to inconsistent estimation. In this article, we illustrate how underlying non-normality induces bias in polychoric estimates and their standard errors. This bias is strongly affected by how we discretize. It is therefore important to consider tests of underlying multivariate normality. In this study we propose a parametric bootstrap test for this purpose. Its performance relative to the test of Maydeu-Olivares is evaluated in a Monte Carlo study. At realistic sample sizes, the bootstrap exhibited substantively better Type I error control and power than the Maydeu-Olivares test in ordinal data with ten dimensions or higher. R code for the bootstrap test is provided.

Foldnes, Njål & Grønneberg, Steffen (2019)

On Identification and Non-normal Simulation in Ordinal Covariance and Item Response Models

A standard approach for handling ordinal data in covariance analysis such as structural equation modeling is to assume that the data were produced by discretizing a multivariate normal vector. Recently, concern has been raised that this approach may be less robust to violation of the normality assumption than previously reported. We propose a new perspective for studying the robustness toward distributional misspecification in ordinal models using a class of non-normal ordinal covariance models. We show how to simulate data from such models, and our simulation results indicate that standard methodology is sensitive to violation of normality. This emphasizes the importance of testing distributional assumptions in empirical studies. We include simulation results on the performance of such tests.

The assessment of model fit has received widespread interest by researchers in the structural equation modeling literature for many years. Various model fit test statistics have been suggested for conducting this assessment. Selecting an appropriate test statistic in order to evaluate model fit, however, can be difficult as the selection depends on the distributional characteristics of the sampled data, the magnitude of the sample size, and/or the proposed model features. The purpose of this paper is to present a selection procedure that can be used to algorithmically identify the best test statistic and simplify the whole assessment process. The procedure is illustrated using empirical data along with an easy to use computerized implementation.

Grønneberg, Steffen & Foldnes, Njål (2019)

A problem with discretizing Vale-Maurelli in simulation studies

Previous influential simulation studies investigate the effect of underlying non-normality in ordinal data using the Vale–Maurelli (VM) simulation method. We show that discretized data stemming from the VM method with a prescribed target covariance matrix are usually numerically equal to data stemming from discretizing a multivariate normal vector. This normal vector has, however, a different covariance matrix than the target. It follows that these simulation studies have in fact studied data stemming from normal data with a possibly misspecified covariance structure. This observation affects the interpretation of previous simulation studies.

We introduce and evaluate a new class of approximations to common test statistics in structural equation modeling. Such test statistics asymptotically follow the distribution of a weighted sum of i.i.d. chi-square variates, where the weights are eigenvalues of a certain matrix. The proposed eigenvalue block averaging (EBA) method involves creating blocks of these eigenvalues and replacing them within each block with the block average. The Satorra–Bentler scaling procedure is a special case of this framework, using one single block. The proposed procedure applies also to difference testing among nested models. We investigate the EBA procedure both theoretically in the asymptotic case, and with simulation studies for the finite-sample case, under both maximum likelihood and diagonally weighted least squares estimation. Comparison is made with 3 established approximations: Satorra–Bentler, the scaled and shifted, and the scaled F tests.

Foldnes, Njål & Grønneberg, Steffen (2017)

The asymptotic covariance matrix and its use in simulation studies

We propose a new and flexible simulation method for non-normal data with user-specified marginal distributions, covariance matrix and certain bivariate dependencies. The VITA (VIne To Anything) method is based on regular vines and generalizes the NORTA (NORmal To Anything) method. Fundamental theoretical properties of the VITA method are deduced. Two illustrations demonstrate the flexibility and usefulness of VITA in the context of structural equation models. R code for the implementation is provided.