@article{Lang2013,
title = {Improved tests of independence in singly-ordered two-way contingency tables},
author = {Joseph B. Lang and Maria Iannario},
url = {http://www.sciencedirect.com/science/article/pii/S0167947313002260},
doi = {10.1016/j.csda.2013.06.014},
issn = {0167-9473},
year = {2013},
date = {2013-12-31},
journal = {Computational Statistics & Data Analysis},
volume = {68},
pages = {339-351},
publisher = {Elsevier B.V.},
abstract = {A new approach is described for improving statistical tests of independence between two categorical variables R and C, where C is ordinal and R may or may not be ordinal. Common tests of independence that exploit the ordinality of C use a restricted-alternative approach. A different, relaxed-null approach to improving tests of independence is considered. Specifically, the M-moment score test is introduced and shown to be an attractive alternative to well known restricted-alternative tests, such as the row-means Cochran–Mantel–Haenszel test, the Kruskal–Wallis test, and the likelihood-ratio test based on the cumulative-logit row-effects model or the log-linear row-effects model. Unlike these restricted-alternative tests, which are designed to detect location shifts, the M-moment score test is designed to be powerful for detecting shifts in any of the first M conditional moments of C across the values of R. Using multinomial–Poisson homogeneous modeling theory, the M-moment score tests are shown to be computationally and conceptually simple, with an attractive complement consistency property. Results of a simulation study compare the M-moment score test to several other commonly-used tests on the basis of their operating characteristics.},
keywords = {Cumulative-logit row-effects model, Exploiting ordinality, Kruskal–Wallis test, Latent distributions, Log-linear row-effects model, M-moment score tests, Multinomial–Poisson homogeneous models, Pearson’s chi-squared test, Row-means Cochran–Mantel–Haenszel test},
pubstate = {published},
tppubtype = {article}
}

A new approach is described for improving statistical tests of independence between two categorical variables R and C, where C is ordinal and R may or may not be ordinal. Common tests of independence that exploit the ordinality of C use a restricted-alternative approach. A different, relaxed-null approach to improving tests of independence is considered. Specifically, the M-moment score test is introduced and shown to be an attractive alternative to well known restricted-alternative tests, such as the row-means Cochran–Mantel–Haenszel test, the Kruskal–Wallis test, and the likelihood-ratio test based on the cumulative-logit row-effects model or the log-linear row-effects model. Unlike these restricted-alternative tests, which are designed to detect location shifts, the M-moment score test is designed to be powerful for detecting shifts in any of the first M conditional moments of C across the values of R. Using multinomial–Poisson homogeneous modeling theory, the M-moment score tests are shown to be computationally and conceptually simple, with an attractive complement consistency property. Results of a simulation study compare the M-moment score test to several other commonly-used tests on the basis of their operating characteristics.