On the Estimation of Small Frequencies in Contingency Tables

Summary Suppose that we require estimates of the population frequencies corresponding to small entries in a large pure contingency table containing heterogeneity. It would be natural to lump rows or columns together with weights depending on the correlation coefficients. But if the rows, say, are nearly orthogonal, no lumping method will be reasonable. Moreover, if many of the entries in the table are missing, a lumping method may fail. A method is given here (which it may be possible to combine with a lumping method when appropriate) that depends on an assumption for the initial distribution of the “association factor” in each cell, i.e. the ratio of the population frequency in the cell to the product of the population frequencies of its row and column. If the logarithm of the association factor is assumed to have, initially, a normal distribution, then the final expectations and variances of the population frequencies and of their logarithms can be expressed in terms of the “after-effect” function which was originally tabulated by K. W. Wagner for an electrodynamical application. Instead of a log-normal distribution, a Pearson Type III distribution may be assumed for the association factor. This assumption may be less accurate but is easier to handle. The paper concludes with a list of properties of the after-effect function (and of some of its generalizations) that may be useful for further tabulation.