Optimal sample size planning for the Wilcoxon-Mann-Whitney test.

There are many different proposed procedures for sample size planning for the Wilcoxon-Mann-Whitney test at given type-I and type-II error rates α and ß, respectively. Most methods assume very specific models or types of data to simplify calculations (eg, ordered categorical or metric data, location shift alternatives, etc). We present a unified approach that covers metric data with and without ties, count data, ordered categorical data, and even dichotomous data. For that, we calculate the unknown theoretical quantities such as the variances under the null and relevant alternative hypothesis by considering the following "synthetic data" approach. We evaluate data whose empirical distribution functions match the theoretical distribution functions involved in the computations of the unknown theoretical quantities. Then, well-known relations for the ranks of the data are used for the calculations. In addition to computing the necessary sample size N for a fixed allocation proportion t = n1 /N, where n1 is the sample size in the first group and N = n1 + n2 is the total sample size, we provide an interval for the optimal allocation rate t, which minimizes the total sample size N. It turns out that, for certain distributions, a balanced design is optimal. We give a characterization of such distributions. Furthermore, we show that the optimal choice of t depends on the ratio of the two variances, which determine the variance of the Wilcoxon-Mann-Whitney statistic under the alternative. This is different from an optimal sample size allocation in case of the normal distribution model.

Inference for low- and high-dimensional multigroup repeated measures designs with unequal covariance matrices.

We propose tests for main and simple treatment effects, time effects, as well as treatment by time interactions in possibly high-dimensional multigroup repeated measures designs. The proposed inference procedures extend the work by Brunner et al. (2012) from two to several treatment groups and remain valid for unbalanced data and under unequal covariance matrices. In addition to showing consistency when sample size and dimension tend to infinity at the same rate, we provide finite sample approximations and evaluate their performance in a simulation study, demonstrating better maintenance of the nominal α-level than the popular Box-Greenhouse-Geisser and Huynh-Feldt methods, and a gain in power for informatively increasing dimension. Application is illustrated using electroencephalography (EEG) data from a neurological study involving patients with Alzheimer's disease and other cognitive impairments.

Floral Scents of a Deceptive Plant Are Hyperdiverse and Under Population-Specific Phenotypic Selection.

Floral scent is a key mediator in plant-pollinator interactions. However, little is known to what extent intraspecific scent variation is shaped by phenotypic selection, with no information yet in deceptive plants. In this study, we collected inflorescence scent and fruit set of the deceptive moth fly-pollinated Arum maculatum L. (Araceae) from six populations north vs. five populations south of the Alps, accumulating to 233 samples in total, and tested for differences in scent, fruit set, and phenotypic selection on scent across this geographic barrier. We recorded 289 scent compounds, the highest number so far reported in a single plant species. Most of the compounds occurred both north and south of the Alps; however, plants of the different regions emitted different absolute and relative amounts of scent. Fruit set was higher north than south of the Alps, and some, but not all differences in scent could be explained by differential phenotypic selection in northern vs. southern populations. This study is the first to provide evidence that floral scents of a deceptive plant are under phenotypic selection and that phenotypic selection is involved in shaping geographic patterns of floral scent in such plants. The hyperdiverse scent of A. maculatum might result from the imitation of various brood substrates of its pollinators.

High-dimensional repeated measures.

Recently, new tests for main and simple treatment effects, time effects, and treatment by time interactions in possibly high-dimensional multigroup repeated-measures designs with unequal covariance matrices have been proposed. Technical details for using more than one between-subject and more than one within-subject factor are presented in this article. Furthermore, application to electroencephalography (EEG) data of a neurological study with two whole-plot factors (diagnosis and sex) and two subplot factors (variable and region) is shown with the R package HRM (high-dimensional repeated measures).