Quality & Quantity 37: 111–134, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
The Structure of the Log Odds-Ratios in
Non-Independence and Symmetry Diagonal
Models for Square Contingency Tables
H. BAYO LAWAL
Department of Statistics, St. Cloud State University, St. Cloud, MN 56301, U.S.A.
Abstract. This paper examines the structure of the interaction term when expressed as the log of
in the log-linear model formulation of non-independence and symmetry diagonal
models applied to an I × I contingency table having ordinal classiﬁcatory variables. The class of
principal diagonal, diagonal band, full diagonal and diagonal-parameters symmetry models are
examined in the study. The general structure of
is examined for each class of diagonal models.
The 5 × 5 British Social Mobility data (Glass, 1954) will be employed as an example in this paper.
The structure of the log odds-ratios are formulated in terms of the parameter estimates obtained from
the application of SAS PROC GENMOD. Goodman in several of his papers on the subject has also
derived some of these log odds ratios but our focus here is the structure of these odds, when viewed
from the non-standard log-linear perspective (Lawal, 2001, 2002).
Key words: symmetry, diagonals models, log odds ratios
In analyzing a two-way contingency table in which the classiﬁcatory variables are
ordinal, we are often concerned in studying the association in such a table. Such
associations are usually measured in normal distribution theories by the correlation
. This measure will however be unsuitable for discrete data in con-
tingency tables where observations are counts and as such, such a single measure
will be inadequate for contingency table analyses as it values depend on the assign-
ment of scores to each category of the variables. The local odds ratio is the choice
measure of association in this case and model interpretations are often made in
terms of the odds-ratios. Among its several properties of the odds-ratios (Bishop et
al., 1975) is the fact that the odds-ratio is independent of the marginal distributions
of the table. It is therefore important that we be able estimate these odds ratios
under any model. Our approach in this paper is to formulate the structure of the
odds ratios in terms of the parameters of the models under consideration, rather
than from computation from expected cell frequencies under the same model.