Stephen E. Fienberg and Alessandro Rinaldo
The common view of the history of contingency tables is that it
begins in 1900 with the work of Pearson and Yule, but it extends back
at least into the 19th century. Moreover it remains an active area of
research today. In this paper we give an overview of this history
focussing on the development of log-linear models and their estimation
via the method of maximum likelihood. S. N. Roy played a crucial role
in this development with two papers co-authored with his students
S. K. Mitra and Marvin Kastenbaum, at roughly the mid-point temporally
in this development. Then we describe a problem that eluded Roy and
his students, that of the implications of sampling zeros for the
existence of maximum likelihood estimates for log-linear
models. Understanding the problem of on-existence is crucial to the
analysis of large sparse contingency tables. We introduce some
relevant results from the application of algebraic geometry to the study of this statistical problem.