Testing normality of latent variables in the polychoric correlation
Keywords:Bayesian encompassing, partial observability, nonparametric specification test, discretization
This paper explores the feasibility of simultaneously facing three sources of complexity in Bayesian testing, namely (i) testing a parametric against a non-parametric alternative (ii) adjusting for partial observability (iii) developing a test under a Bayesian encompassing principle. Testing the normality of latent variables in the polychoric correlation model is taken as a case study. This paper starts from the specification of the model defining the polychoric correlation in the framework of manifest ordinal variables viewed as discretizations of underlying latent variables. Taking advantage of the fact that in this model, the marginal distributions of the latent variables are not identified, we use the approach of copula. Some identification issues are analysed. Next, we develop a Bayesian encompassing specification test for testing the Gaussianity of the underlying copula and consider the discretization model as a case of partial observability. The computational feasibility, the numerical stability and the discriminating power of the procedure are checked through a simulation experiment. An application completes the paper by illustrating the working of the procedure on a meta-analysis of clinical trials on acute migraine. The final section proposes, in the form of conclusions, an evaluation of the actual achievements of the paper.
A. AGRESTI (1984). Analysis of ordinal categorical data. Wiley, New York.
C. ALMEIDA (2007). Testing specifications on partial observability models: a Bayesian encompassing approach. Ph.D. thesis, Institut de statistique, Université catholique de Louvain, Louvain-la-Neuve.
C. ALMEIDA, M. MOUCHART (2005). Bayesian encompassing test under partial observability. Discussion Paper 05-14, Institut de statistique, Université catholique de Lou vain, Louvain-la-Neuve. http://www.stat.ucl.ac.be/ISpub/dp/2005/dp0514.pdf - http://www.stat.ucl.ac.be/ISpub/dp/2005/dp0514.pdf , submitted.
C. ALMEIDA, M. MOUCHART (2007a). Bayesian encompassing specification test under not completely known partial observability. Bayesian Analysis, 2, pp. 303–318.
C. ALMEIDA, M. MOUCHART (2007b). Bayesian encompassing specification test under not completely known partial observability (summary of Almeida and Mouchart (2007a)). In J.-M. BERNARDO, M.-J. BAYARRI, J.-O. BERGER, A.-P. DAWID, D. HECKERMAN, A.-F.-M. SMITH, M.WEST (eds.), Bayesian Statistics 8. Oxford:Oxford University Press, pp. 567–572.
J.-M. BERNARDO, A. F. M. SMITH (1994). Bayesian theory. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Ltd., Chichester.
P. DAMIEN, J. WAKEFIELD, S. WALKER (1999). Gibbs sampling for Bayesian nonconjugate and hierarchical models by using auxiliary variables. J. Roy. Statist. Soc. Ser. B, 61, no. 2, pp. 331–344.
J.-P. FLORENS, M.MOUCHART (1993). Bayesian testing and testing Bayesians. In Econometrics, North-Holland, Amsterdam, vol. 11 of Handbook of Statist., pp. 303–334.
J.-P. FLORENS, M. MOUCHART, J.-M. ROLIN (1990). Elements of Bayesian statistics, vol. 134 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York.
J.-P. FLORENS, J.-F. RICHARD, J.-M. ROLIN (2003). Bayesian encompassing specification test of a parametric model against a nonparametric alternative. Unpublished mimeo, revised version of Discussion Paper 96-08, Institut de statistique, Université catholique de Louvain, Louvain-la-Neuve.
L. A. GOODMAN (1981). Association models and the bivariate normal for contingency tables with ordered categories. Biometrika, 68, no. 2, pp. 347–355.
N. L. JOHNSON, S. KOTZ (1972). Distributions in statistics: continuous multivariate distributions. John Wiley & Sons Inc., New York. Wiley Series in Probability and Mathematical Statistics.
K. JÖRESKOG (2002). Structural equation modeling with ordinal variables sudsing LISREL. Tech. rep., technical report.
K. JÖRESKOG, D. SÖRBOM, S. DU TOIT, M. DU TOIT (2002). LISREL8: New statistical features. SSI Scientific International Inc. B. MUTHÉN (1983). Latent variable structural equation modeling with categorical data. J. Econometrics, 22, no. 1-2, pp. 43–65.
B. MUTHÉN (1984). A general structural equation model with dichotomous, ordered categorical variables, and continuous latent variables indicators. Psychometrika, 49, pp. 115–132.
B. MUTHÉN, C. HOFACKER (1988). Testing the assumptions underlying tetrachoric correlations. Psychometrika, 83, pp. 563 – 578.
R. B. NELSEN (1999). An introduction to copulas, vol. 139 of Lecture Notes in Statistics. Springer-Verlag, New York.
U.OLSSON (1979). Maximul likelihood estmation of the polychoric correlation coefficient. Psychometrika, 44, pp. 443-460.
K. PEARSON (1900). Mathematical contribution to the theory of evolution vii: On the correlation of characters not quantitatively measurable. Phil. Trans. R. Soc. Lond., 195A, pp. 1 – 47.
K. PEARSON, E. S. PEARSON (1922). On polychoric coefficients of correlation. Biometrika, 52, pp. 127 – 156.
G. M. TALLIS (1962). The maximum likelihood estimation of correlation from contingency tables. Biometrics, 18, pp. 342–353.
F. VANDENHENDE (2003). Copula models for the analysis of longitudinal ordinal responses in clinical trials on acute migraine. Ph.D. thesis, Institut de statistique, Université catholique de Louvain, Louvain-la-Neuve.
Q.WANG, S. R. KULKARNI, S. VERDÚ (2005). Divergence estimation of continuous distributions based on data-dependent partitions. IEEE Trans. Inform. Theory, 51, no. 9, pp. 3064–3074.