Graphical Models and
Multigraph Representations
Numbers
in parentheses correspond to the numbered references in my publication
list.
A very useful technique for
analyzing and interpreting hierarchical loglinear models in a graphical
way was introduced by Darroch, Lauritzen, and Speed
(1980) in a landmark paper. The usefulness of this approach is principally
due to the simple graphical characterization of models that can be understood
purely in terms of conditional independence relationships. Beginning with
this paper, chordal graphs have emerged as an important type of model for
the statistical analysis of contingency tables. A limited survey of this
literature is given in (50).
An alternative approach to
that of Darroch et al. makes use of the generator multigraph. The multigraph
approach has several strategic advantages over the firstorder interaction
graph used by Darroch et al. An account of how to use the multigraph approach
in maximum likelihood estimation and in identifying conditional independencies
in hierarchical loglinear models, as well as examples, are given in (41).
The theoretical details can be found in (43).
One of the examples given
in (41) involves data
from the Dayton Area Drug Survey, a survey conducted in 1992 by the Wright
State University School of Medicine and the United Health Services of the
Dayton area (a United Way agency) of all Dayton area school children in
grades 6 through 12. The Statistical Consulting Center at Wright State
analyzed this very large, complex set of data. The multigraph approach
was used to interpret structural relationships among the variables. It
was gratifying to have the opportunity to use a statistical method developed
by Wright State researchers (43)
in analyzing data from a survey conducted cooperatively by Wright State
and a Dayton United Way agency. Results of the study enhanced understanding
and knowledge of drug and substance abuse by Dayton area youth.
In the SAGE monograph #167, a brief review of the loglinear model and a presentation of the association graph and the multigraph are given  see (82).
Reference
Darroch,
J.N., Lauritzen, S.L., and Speed, T.P. (1980). Markov fields and loglinear
interaction models for contingency tables. Annals of Statistics
8: 522539.
