Association Properties of KS Distributions: Consequences for

Association Properties of KS Distributions: Consequences
for Graphical Modelling
Astrid Heinicke
Institute for Statistics
Ludwigstr. 33
80539 Munich, Germany
[email protected]
1. Introduction
Most literature related to graphical models concerns so{called concentration graphs based
on Conditional Gaussian distributions, which include multivariate normal distributions and
multinomial distributions as special cases. Less attention has been paid to covariance graphs
which represent marginal independencies, but appropriate Markovian properties have been introduced by Cox and Wermuth (1993). Kauermann (1996) has shown that multivariate normal
distributions fulll the equivalence of these Markovian properties and analysed properties like
collapsibility for these models. Caputo (1998) dened graphical models on covariance graphs
for a family of distributions, so{called KS distributions, introduced by Koehler and Symanowski
(1995). One very nice property of KS distributions is that they allow for an arbitrary choice of
the univariate margins of the involved variables, but until now no information has been available on the kind of association structures captured by KS distributions and the consequences
for graphical modelling in this framework. In this talk a brief introduction on covariance models based on KS distributions as well as important characteristics concerning their association
structure is given. This is supplemented by the representation of the main results of a simulation study. Conclusions are drawn for the applicability of KS distributions to graphical
modelling.
2. The family of KS distributions
Let V = f1; : : : ; pg be an index set and V the set of all subsets of V . For all sets
I 2 I :=
P
+
+
fI 2 V with jI j 2g let I 2 IR0 and for all i 2 V let i 2 IR0 with i+ := i + I2I I > 0.
For any set of univariate marginal distributions Fi (xi), i 2 V , the joint cdf is
i2I
Y
(1)
F (x1 ; : : : ; xp ) =
i2V
"
1 Y XY
Fj (xj ) j+
I 2I i2I j 2I
j 6=i
Fi (xi )
Y
, (jI j , 1) Fi (xi)
i2I
1
i+
#I :
3. Association
Multivariate KS distributions fulll certain concepts of positive association (Joe, 1997). Among
others they are positively associated and their corresponding random variables are positive
dependent through the stochastic ordering as well as conditional increasing in sequence. Thus
the associations are monotone. Apart of special bivariate cases they are not multivariate totally
positive of order 2. Strength of association between pairs of variables depends on the relative
size of the association parameters in (1), and restricting certain parameters to zero leads to
exact marginal indepencies of the involved variables as well as certain parameter constellations
induce almost marginal independencies (quasi{independence) that do not go together with zero
restrictions of parameters.
4. Consequences
The fact that quasi{independencies cannot be related to zero{restrictions on parameters in (1) proofs to be a serious problem when it comes to applying graphical models with
KS distributions to real datasets. One further disadvantage of KS distributions is that they
have a limited range of dependence, i.e. they cannot capture all dierent patterns of positive
associatons among the components of a random vector.
REFERENCES
Caputo, A. (1998). Eine alternative Familie von Modellverteilungen fur Kovarianz{ und Konzentrationsgraphen. Dissertation. Ludwigs{Maximilians Universitat Munchen. Munchen.
Cox, D. R. and Wermuth, N. (1993). Linear dependencies represented by chain graphs (with
discussion). Statistical Science 8, 204{283.
Joe, H. (1997). Multivariate models and dependence concepts. Chapman and Hall. London.
Kauermann, G. (1996). On a dualization of graphical Gaussian models. Scandinavian Journal
of Statistics 23, 105{116.
Koehler, K. J. and Symanowski, J. T. (1995). Constructing multivariate distributions with
specic marginal distributions. Journal of Multivariate Analysis 55, 261{282.
RESUM
E
La plupart de la literature sur les modeles graphiques traite le cas de graphes de concentracion bases sur la distribution gaussienne conditionnelle. Beaucoup moins d'attention est pr^etee
aux graphes de covariance representants des independences marginales excepte l'introduction
des proprietes markoviennes appropriees par Cox et Wermuth (1993). De plus, Kauermann
(1996) demontre que la distribution gaussienne multivariee admet l'equivalence de ces proprietes markoviennes, et analyse le probleme de collapsibilite pour ces modeles. Recemment,
Caputo (1998) denit des modeles graphiques pour graphes de covariance bases sur une distribution multivariee proposee par Koehler et Symanowski (1995). La construction de celle{ci
permet un choix de distributions marginales quelleconques ce qui semble tres favorable. Mais
jusque'a present, aucune information est disponible sur la nature des associations qui peuvent ^etre modelees par cette distribution, et sur les consequences pour les modeles graphiques
correspondants. Cet expose donne une courte introduction aux modeles de covariance bases
sur la distribution de Koehler et Symanowski et adresse l'importante question de la structure
d'associations immanentes. Ceci est complete par les resultas centrals d'une simulation et
par des conclusions sur l'utilite de la distribution de Koehler et Symanowski pour les modeles
graphiques.