| Abstract: | A generalizing, analytic model is developed for 2-way cross-classifications with fixed class sizes The model is expressed in closed form as a simple operator formula, and is readily extended to n-way cross-classifications, and to such cases where one or more cells are vacuous or fixed. The model permits easy derivation, by means of simple differential operators, of the exact power moments, product moments, and factorial moments of the cell frequencies. From this, the moments of the exact sampling distribution of the conventional x2-statistic can be computed, which, in turn, leads to a reappraisal of the Chi-square approximation for sparse and isotropic contingency tables. Here, the Gamma distribution is considered, and numerical results are presented that would suggest preference of the Gamma approximation over the Chi-square in such cases. |
