The basic theory upon which several well-known tests of the independence of least squares regression disturbances are based is reviewed and then applied to formulate a testing procedure which is inexpensive when a large-capacity, high-speed electronic computer is employed. Investigations to date indicate that the significance points of the von Neumann ratio, which can be obtained by this procedure, are accurate to the order required in practical work even when the number of degrees of freedom is small, and the test is always conclusive. This is encouraging since the results from attempts to apply the Durbin-Watson or the Theil-Nagar tests frequently may be inconclusive or of doubtful accuracy. Consideration is also given to the question of what to do when the null hypothesis of residual independence is rejected.