The Maximum Likelihood Estimation of Correlation from Contingency Tables

It is sometimes desirable in practice to estimate the degree of correlation existing in 2 X 2 or 3 X 3 contingency tables. In some instances an underlying bivariate normal distribution can be assumed and the problem reduces to estimating p from the observed frequencies in the table. In the case of the 2 X 2 tables, this may be accomplished with the aid of tetrachoric functions, although this technique does not seem to have been extended to p X q tables in general. It is the purpose of the present paper to show how Maximum Likelihood (M.L.) estimates of p, p, may be obtained from such tables when, in fact, the parent distribution is bivariate normal. A somewhat similar problem has been considered by Mosteller 1946 who investigated the efficiency of estimating p from punch card data. Mosteller considered the case when the cards are sorted with respect to the two co-ordinates, x and y, in a particular manner, and derives the M.L. estimate of p from an order statistics argument. However, his results are not generally applicable to the problem considered here because of the special sorting model which he employs. This topic is part of a wider attempt to develop more satisfactory methods of analysing discrete and continuous data arising in some fields of quantitative genetics. For instance, in the study of heritability and repeatability of birth records of domestic animals, the required correlations are usually calculated by the product-moment or intra-class correlation techniques and hence satisfactory tests of hypotheses are lacking. In the particular case of fertility, a reasonable assumption may be that the potential to produce offspring is normally distributed, but, that, necessarily, phenotypic expression is only possible at distinct threshold values of this potential. Thus, for all potentials below a certain threshold no offspring result; for potentials above this critical value but below a second threshold level one offspring is produced