Two-way ORDANOVA: Analyzing ordinal variation in a cross-balanced design

Variability assessment of qualitative data has an important role in diverse areas such as sociology, quality engineering, healthcare, decision making, genetics, metrology in chemistry and many others. To test the homogeneity hypothesis, we use two main categorical factors. Using a cross-balanced design, we provide a decomposition theorem of the sample total-variation into ‘intra’(within) component and ‘inter’(between) component for any scale. Building on this, we provide a way for decomposing the ‘inter’component itself into variation components contributed by the two factors and their interaction (when we consider more than one replicated test in the cross-design). Moreover, focusing on an ordinal scale, we offer segregation power indices for testing the significate effects of the two factors and their interaction. A two-way ORDANOVA calculator is provided to display the empirical distributions of the segregation power indices and to find critical points of the test statistics distributions at the given confidence level. Some examples are given to demonstrate the proposed method.