Abstract: We develop an efficient test for the homogeneity of the mean directions of several independent circular populations (ANOMED) that can be universally implemented. Current tests for ANOMED are available only for highly concentrated and/or large groups. Thus, we fill the gap for a usable test under highly dispersed and/or small to medium-sized groups. Focusing on the popular von Mises distribution, a simple and elegant test statistic is derived under homogeneous concentrations across groups. The hurdle of the non-location-scale nuisance parameter κ is overcome by adopting a new approach based on the integrated likelihood ratio test (ILRT). Furthermore, a second-order-accurate asymptotic chi-square distribution is established for the ILRT. Notably, the test outperforms existing tests for small to moderate-size and highly dispersed (small κ) groups, which is precisely the parametric region of prime concern, where previous tests were either unusable or unsatisfactory. The test also outperforms the popular Watson-Williams test for highly concentrated small groups, and shows competitive performance compared with that of its best competitors and, hence, can be universally used in all situations. The ILRT extends naturally under heterogeneous concentrations, and is amenable to elegant generalizations to a rich variety of circular populations and to higher dimensions (i.e., to distributions on the sphere and hypersphere). Lastly, the test is illustrated using three real-life data sets.

Key words and phrases: Batschelet distribution, circular ANOVA, circular normal distribution, generalized von Mises distribution, integrated likelihood ratio tests, Watson-Williams test.