Collapsibility and response variables in contingency tables.

*(English)*Zbl 0549.62041The collapsibility of a hierarchical loglinear model L for a multidimensional contingency N are given in two equivalent definitions: 1) for all \(p=p(i)\in L\), \(p(i_ a)\in L_ a\); 2) for all \(i_ a\), \(\hat p(i{}_ a)=\hat p_ a(i_ a)\), where a is a subset of classifying factors.

The authors give a sufficient and necessary condition for a hierarchical loglinear model \(L_ c\) to be collapsible onto a, that is, every connected component of a being contained in a generator of L. Furthermore, if L is a graphical model, the condition simply means that the boundary of every connected component is complete. So a graphical model is collapsible iff \(a_ 1,a_ 2\in a\), \(a_ 1\perp a_ 2| s\) implies \(a_ 1\perp a_ 2| s\cap a.\)

Since a loglinear model may carry an inappropriate model, and many natural relevant models are not loglinear, it is necessary to distinguish response variables and explanatory variables. This leads to the concept of response model. In this paper, a sufficient and necessary condition for a loglinear model to be a response model was given and vice versa. As expected, the conditions are stated in terms of collapsibility.

Finally, since in an appropriate model the marginal distribution is independent of the conditional one, S-sufficiency of marginal tables is used as another description of collapsibility.

The authors give a sufficient and necessary condition for a hierarchical loglinear model \(L_ c\) to be collapsible onto a, that is, every connected component of a being contained in a generator of L. Furthermore, if L is a graphical model, the condition simply means that the boundary of every connected component is complete. So a graphical model is collapsible iff \(a_ 1,a_ 2\in a\), \(a_ 1\perp a_ 2| s\) implies \(a_ 1\perp a_ 2| s\cap a.\)

Since a loglinear model may carry an inappropriate model, and many natural relevant models are not loglinear, it is necessary to distinguish response variables and explanatory variables. This leads to the concept of response model. In this paper, a sufficient and necessary condition for a loglinear model to be a response model was given and vice versa. As expected, the conditions are stated in terms of collapsibility.

Finally, since in an appropriate model the marginal distribution is independent of the conditional one, S-sufficiency of marginal tables is used as another description of collapsibility.

Reviewer: Y.Zhang