The analysis of directional data: Performance comparisons of randomization and parametric methods

The analysis of directional data is an area of statistics concerned with observations collected initially in the form of unit vectors, or with vector-valued observations that can be reduced to that form through some type of preliminary normalization. Although there have been advances in the analysis of directional data in recent years, the collection of methods available is still extremely limited. In the present study, several Monte Carlo simulations are carried out to assess the performance of several parametric and nonparametric randomization methods for the analysis of directional data, with particular emphasis on the classic task of evaluating whether population mean directional differences exist over M independent samples of directions. To overcome the limited availability in many analysis paradigms, the present study investigates the randomization approach based on a measure of proximity between directions, which provides a mechanism for generalizing the randomization strategy rather directly to a wide range of different analysis problems. The latter generalization requires rephrasing whatever question is of interest to one defined in terms of the type of structure that should be present in the similarities between directions, as measured by the cosine of the angle between two vectors. The evaluation of the degree to which the conjectured structure is present is carried out by comparing an observed index of the degree to which the structure fits the data to what would be expected if no structure were present, i.e., under an appropriate randomization conjecture. Generally, the parametric ANAVA (Analysis of Angular Variance) test performs adequately unless the population is very dispersed, and the performance for skewed distributions is rather poorer than that for symmetrical distributions. However, the parametric methods examined in this study are influenced by dispersion more than by skewness. The ANAVA performs well under heterogeneous sample sizes, but is sensitive to heterogeneous sample dispersion especially when a very dispersed sample is included. In addition, the ANAVA shows poorer performances when the number of dimensions increases. The randomization methods are generally very adequate and are recommended for populations with a broad range of dispersion, skewness, and number of dimensions, but they are very sensitive to dispersion heterogeneity. When heterogeneous population distributions are expected for independent samples of directional data, this study suggests the possible use of a new test statistic, which is defined through the cosines of unit group mean vectors in an attempt to separate the effect of differences in dispersion from mean vector differences. This latter test statistic appears to be somewhat less sensitive to dispersion heterogeneity.U of I OnlyETDs are only available to UIUC Users without author permissio