K-Sample Anderson–Darling Tests

Abstract

Two k-sample versions of an Anderson–Darling rank statistic are proposed for testing the homogeneity of samples. Their asymptotic null distributions are derived for the continuous as well as the discrete case. In the continuous case the asymptotic distributions coincide with the (k – 1)-fold convolution of the asymptotic distribution for the Anderson–Darling one-sample statistic. The quality of this large sample approximation is investigated for small samples through Monte Carlo simulation. This is done for both versions of the statistic under various degrees of data rounding and sample size imbalances. Tables for carrying out these tests are provided, and their usage in combining independent one- or k-sample Anderson–Darling tests is pointed out. The test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample. One weighting adjusts for the possibly different sample sizes, and the other is inside the integration placing more weight on tail differences of the compared distributions. The two versions differ mainly in the definition of the empirical distribution function. These tests are consistent against all alternatives. The use of these tests is two-fold: (a) in a one-way analysis of variance to establish differences in the sampled populations without making any restrictive parametric assumptions or (b) to justify the pooling of separate samples for increased sample size and power in further analyses. Exact finite sample mean and variance formulas for one of the two statistics are derived in the continuous case. It appears that the asymptotic standardized percentiles serve well as approximate critical points of the appropriately standardized statistics for individual sample sizes as low as 5. The application of the tests is illustrated with an example. Because of the convolution nature of the asymptotic distribution, a further use of these critical points is possible in combining independent Anderson–Darling tests by simply adding their test statistics.