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Multivariate Normal Integrals and cOntingency Tables with Ordered Categories

Published online by Cambridge University Press:  01 January 2025

Yuchung J. Wang
Yuchung J. Wang*
Affiliation:
Rutgers University
*
Requests for reprints should be sent to Yuchung J. Wang, Department of Mathematical Sciences, Rutgers University, Camden, New Jersey 08102.
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Abstract

A 1k-dimensional multivariate normal distribution is made discrete by partitioning the k-dimensional Euclidean space with rectangular grids. The collection of probability integrals over the partitioned cubes is a k-dimensional contingency table with ordered categories. It is shown that loglinear model with main effects plus two-way interactions provides an accurate approximation for the k-dimensional table. The complete multivariate normal integral table is computed via the iterative proportional fitting algorithm from bivariate normal integral tables. This approach imposes no restriction on the correlation matrix. Comparisons with other numerical integration algorithms are reported. The approximation suggests association models for discretized multivariate normal distributions and contingency tables with ordered categories.

Keywords

association modelloglinear modeliterative proportional fitting algorithmlocal odds ratiosmultivariate dependence functions
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 2 , June 1997 , pp. 267 - 284
Copyright
Copyright © 1997 The Psychometric Society

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Footnotes

The contingency-table approach occurred to me while I was collaborating with Paul Holland of the Educational Testing Service in 1985 on bivariate dependence functions. Holland maintains a belief that “the continuous” can learn from “the discrete.” This work is a reassertion of his claim.

This research was sponsored by the National Science Council, Republic of China.

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