The Use of Models in the Analysis of Some Classes of Contingency Tables

In some types of biological experiments, it frequently happens that the results take the form of a k-dimensionial table, the cells giving the outcomes of independent multinomial trials. This would be the case, for instance, in an experiment to determiine the viability of a particular organism in different environments. The experimenter may be interested in testing the effects of r difflerent media and c temperature conditions on the survival rate of his test organism, at a particular age. FoJ this purpose the experimnenter may set up rc tubes in such a way that i, jth tube contains the ith medium, is held at the jth temperature level and holds a random sample of N,j of his organisms. After the elapse of a predetermlined period of time, he counts the number of organisms surviving in each tube and he is then interested in analysing the effects of temperature and media on the proportion of survivors. His data may be arranged in an r X c table recording the proportion of survivors, and this example is obtained as a special case from the more general situation by using k = 2 and using binomial trials. It is the purpose of this paper to suggest a method of analysing such data, but the development is mainly in terms of two-dimensional tables with independent binomial trials. Since extensions to nmore involved situations are easily effected, more complicated cases will receive no further attention here. The same methods of analysis may also be used in somewhat different circumstances and the necessary modifications for dealing with r X c contingency tables in the same way are indicated as an example.