Get access

New heterogeneous test statistics for the unbalanced fixed-effect nested design


Correspondence should be addressed to Dr Wei-Ming Luh, Institute of Education, National Cheng Kung University, 1 University Road, Tainan City 701, Taiwan (e-mail:


When the underlying variances are unknown or/and unequal, using the conventional F test is problematic in the two-factor hierarchical data structure. Prompted by the approximate test statistics (Welch and Alexander–Govern methods), the authors develop four new heterogeneous test statistics to test factor A and factor B nested within A for the unbalanced fixed-effect two-stage nested design under variance heterogeneity. The actual significance levels and statistical power of the test statistics were compared in a simulation study. The results show that the proposed procedures maintain better Type I error rate control and have greater statistical power than those obtained by the conventional F test in various conditions. Therefore, the proposed test statistics are recommended in terms of robustness and easy implementation.