Testing Categorized Bivariate Normality with Two-Stage Polychoric Correlation Estimates

@inproceedings{MaydeuOlivares2003TestingCB,
  title={Testing Categorized Bivariate Normality with Two-Stage Polychoric Correlation Estimates},
  author={Albert Maydeu-Olivares},
  year={2003}
}
We show that when the thresholds and the polychoric correlation are estimated in two stages, neither Pearson's X^2 nor the likelihood ratio G^2 goodness of fit test statistics are asymptotically chi-square. We propose a new test statistic, Mn, that is asymptotically chi-square in this situation. Mn, may have a wide range of applications beyond the one considered here as it is asymptotically chi-square for a broad class of consistent and asymptotically normal estimators. Mn equals X^2 with an… Expand

Tables from this paper

Pernicious Polychorics: The Impact and Detection of Underlying Non-normality
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 underExpand
Contributions to Estimation of Polychoric Correlations
  • S. Monroe
  • Mathematics, Medicine
  • Multivariate behavioral research
  • 2018
TLDR
The proposed Monte Carlo estimator for the asymptotic covariance matrix (ACM) of the polychoric correlation estimates can be more efficient than its sample-based counterpart, and this leads to better calibration for established test statistics, in particular with small samples. Expand
Testing the assumption of multivariate normality.
Methods of assessing the degree to which multivariate data deviate from multinormality are discussed. The best known of these methods, Mardia’s tests of multivariate skewness and kurtosis, allow oneExpand
What to Do About Zero Frequency Cells When Estimating Polychoric Correlations
Categorical structural equation modeling (SEM) methods that fit the model to estimated polychoric correlations have become popular in the social sciences. When population thresholds are high inExpand
A recommendation for applied researchers to substantiate the claim that ordinal variables are the product of underlying bivariate normal distributions
TLDR
It is recommended that in applications of the polychoric correlation coefficient, data is tested with goodness-of-fit of the BND, that such statistic is reported in published applications, and that the polyChoric correlation is not applied when the test is significant. Expand
Collapsing Categories is Often More Advantageous than Modeling Sparse Data: Investigations in the CFA Framework
TLDR
Collapsing categories were advantageous for ULSMV and WLSVV, yielding higher convergence rates, more accurate estimation of parameters and standard errors, and chi-square test rejection rates close to the nominal level. Expand
Comparing the Fit of Item Response Theory and Factor Analysis Models
Linear factor analysis (FA) models can be reliably tested using test statistics based on residual covariances. We show that the same statistics can be used to reliably test the fit of item responseExpand
The Use of Discrete Data in PCA: Theory, Simulations, and Applications to Socioeconomic Indices
TLDR
The simulation results show that the currently used method of running PCA on a set of dummy variables as proposed by Filmer & Pritchett (2001) is inferior to other methods for analyzing discrete data, both simple such as using ordinal variables, and more sophisticatedsuch as using the polychoric correlations. Expand
Goodness-of-Fit Assessment of Item Response Theory Models
The article provides an overview of goodness-of-fit assessment methods for item response theory (IRT) models. It is now possible to obtain accurate p-values of the overall fit of the model ifExpand
Why Ordinal Variables Can (Almost) Always Be Treated as Continuous Variables: Clarifying Assumptions of Robust Continuous and Ordinal Factor Analysis Estimation Methods
The analysis of factor structures is one of the most critical psychometric applications. Frequently, variables (i.e., items or indicators) resulting from questionnaires using ordinal items with 2–7Expand
...
1
2
3
...

References

SHOWING 1-10 OF 41 REFERENCES
Limited Information Goodness-of-Fit Testing in Multidimensional Contingency Tables
We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables of arbitrary dimensions. These statistics are quadratic forms inExpand
Limited-and Full-Information Estimation and Goodness-ofFit Testing in 2 n Contingency Tables : A Unified Framework
High-dimensional contingency tables tend to be sparse, and standard goodness-of-fit statistics such as X2 cannot be used without pooling categories. As an improvement on arbitrary pooling, forExpand
An Empirical Investigation of Goodness-of-Fit Statistics for Sparse Multinomials
Abstract Traditional discussions of goodness-of-fit tests for multinomial data consider asymptotic chi-squared properties under the assumption that all expected cell frequencies become large. ThisExpand
A two-stage estimation of structural equation models with continuous and polytomous variables.
TLDR
A partition maximum likelihood approach is used to obtain the first stage estimates of the thresholds and the polyserial and polychoric correlations in the underlying correlation matrix, and a generalized least squares approach is employed to estimate the structural parameters in the correlation structure. Expand
Maximum likelihood estimation of the polychoric correlation coefficient
The polychoric correlation is discussed as a generalization of the tetrachoric correlation coefficient to more than two classes. Two estimation methods are discussed: Maximum likelihood estimation,Expand
Structural equation models with continuous and polytomous variables
A two-stage procedure is developed for analyzing structural equation models with continuous and polytomous variables. At the first stage, the maximum likelihood estimates of the thresholds,Expand
On the estimation of polychoric correlations and their asymptotic covariance matrix
A general theory for parametric inference in contingency tables is outlined. Estimation of polychoric correlations is seen as a special case of this theory. The asymptotic covariance matrix of theExpand
Technical aspects of Muthén's liscomp approach to estimation of latent variable relations with a comprehensive measurement model
Muthén (1984) formulated a general model and estimation procedure for structural equation modeling with a mixture of dichotomous, ordered categorical, and continuous measures of latent variables. AExpand
Testing the assumptions underlying tetrachoric correlations
A method is proposed for empirically testing the appropriateness of using tetrachoric correlations for a set of dichotomous variables. Trivariate marginal information is used to get a set ofExpand
Structural equation modeling with ordinal variables
The statistical models used in structural equation modeling are described. The estimation theory for these models is reviewed for the case when all variables are continuous. Estimation theory for theExpand
...
1
2
3
4
5
...