Journal of Biometrics & Biostatistics

Testing for the homogeneity of density functions of circular random variables is useful in many settings including the study of wind patterns, paleocurrents trends, the seasonality in human-related events such as homicides and suicides and the seasonality in the appearance of diseases. In this paper, we considered that the density functions are members of the flexible family of circular distributions based on nonnegative trigonometric (Fourier) sums (series) developed by Fernandez-Duran [1]. We constructed a test based on the likelihood ratio and we applied the proposed test to simulated and real datasets.

Various effect sizes have been proposed. However, different effect size measures are suitable for different types of data, and the interpretations of effect sizes are generally arbitrary and remain problematic. In this article, the concepts of contrast variable, its standardized mean (SMCV) and c + -probability are explored to link together the commonly used effect sizes including the probabilistic index and ratios of mean difference to variability. A contrast variable can provide both a probabilistic meaning and an index of signal-to-noise ratio to interpret the strength of a comparison, which offers us a strong base to classify the strength of a comparison. Contrast variable, SMCV and c + -probability not only give interpretations to both Cohen’s and McLean’s criteria but also work effectively and consistently for either relationship or group comparison in either independent or correlated situations and in either two or more than 2 groups. Treatment effect, main effect, interaction effect, linear relationship, quadratic relationship and any other contrasts in factorial experiments can all be addressed consistently using contrast variables and the SMCV-based classifying rule, as demonstrated using examples in this article. Therefore, contrast variable, SMCV, c + -probability and associated classifying rules may have the potential to offer a consistent interpretation to effect sizes.