In a paper published by Farlie [2] the efficiency of independence test using several test statistics (rank correlation coefficients, ordinary sample correlation coefficient) for a more special class of two-dimensional distributions was examined. The present paper generalizes these results for a wider class of two-dimensional distributions with finite contingency. In the theorems 3 and 4 sufficient and necessary cirteria are given for the cases in which the above statistics are as efficient in the sense of Pitman as the statistics most efficient in the sense of Pitman whose existence is confirmed under certain conditions of regularity but generally unknown. It is then possible to replace this unknown test statistics by the known statistics.