A variety of biological, statistical, and social science data come in the form of cross-classified tables of counts commonly known as contingency tables. Scaling the cell entries of such multidimensional matrices involves both mathematically and statistically well-posed problems of practical interest. In this chapter we first describe several situations where scaling can be useful. We then prove a very general theorem that demonstrates the existence of scaling factors. We also describe a natural scaling algorithm in the problem of scaling a nonnegative matrix to obtain prescribed row and column sums. In order to study the convergence properties of the algorithm it is convenient to work in terms of Hilbert's projective metric. Certain related concepts such as the contraction ratio of Birkhoff and the oscillation ratio of Hopf are introduced. In the last section we consider the problem of maximum likelihood estimation in contingency tables. This area of statistics, which forms part of discrete multivariate analysis, is of considerable interest to research workers at present.
Practical examples of scaling problems
Before we take up the mathematical problem of scaling, we illustrate some practical situations where scaling is useful.
Budget allocation problem
The Air Force, the Army, and the Navy have received their budget for the next fiscal year measured in some units to be allocated among technical, administrative, and research categories.