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We introduce a unified approach to testing a variety of rather general null hypotheses that can be formulated in terms of covariances matrices. These include as special cases, for example, testing for equal variances, equal traces, or for elements of the covariance matrix taking certain values. The proposed method only requires very few assumptions and thus promises to be of broad practical use. Two test statistics are defined, and their asymptotic or approximate sampling distributions are derived. In order to improve particularly the small-sample behavior of the resulting tests, two bootstrap-based methods are developed and theoretically justified. Several simulations shed light on the performance of the proposed tests. The analysis of a real data set illustrates the application of the procedures., Comment: Submitted to Statistica Sinica

In applied research, it is often sensible to account for one or several covariates when testing for differences between multivariate means of several groups. However, the "classical" parametric multivariate analysis of covariance (MANCOVA) tests (e.g., Wilks' Lambda) are based on quite restrictive assumptions (homoscedasticity and normality of the errors), which might be difficult to justify in small sample size settings. Furthermore, existing potential remedies (e.g., heteroskedasticity-robust approaches) become inappropriate in cases where the covariance matrices are singular. Nevertheless, such scenarios are frequently encountered in the life sciences and other fields, when for example, in the context of standardized assessments, a summary performance measure as well as its corresponding subscales are analyzed. In the present manuscript, we consider a general MANCOVA model, allowing for potentially heteroskedastic and even singular covariance matrices as well as non-normal errors. We combine heteroskedasticity-consistent covariance matrix estimation methods with our proposed modified MANCOVA ANOVA-type statistic (MANCATS) and apply two different bootstrap approaches. We provide the proofs of the asymptotic validity of the respective testing procedures as well as the results from an extensive simulation study, which indicate that especially the parametric bootstrap version of the MANCATS outperforms its competitors in most scenarios, both in terms of type I error rates and power. These considerations are further illustrated and substantiated by examining real-life data from standardized achievement tests.

When testing for superiority in a parallel-group setting with a continuous outcome, adjusting for covariates (e.g., baseline measurements) is usually recommended, in order to reduce bias and increase power. For this purpose, the analysis of covariance (ANCOVA) is frequently used, and recently, several exact and approximate sample size calculation procedures have been proposed. However, in case of multiple covariates, the planning might pose some practical challenges and surprising pitfalls, which have not been recognized so far. Moreover, since a considerable number of parameters have to be specified in advance, the risk of making erroneous initial assumptions, leading to substantially over- or underpowered studies, is increased. Therefore, we propose a method, which allows for re-estimating the sample size at a prespecified time point during the course of the trial. Extensive simulations for a broad range of settings, including unbalanced designs, confirm that the proposed method provides reliable results in many practically relevant situations. An advantage of the reassessment procedure is that it does not require unblinding of the data. In order to facilitate the application of the proposed method, we provide some R code and discuss a real-life data example.

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 $\alpha$ and $\beta$, respectively. Most methods assume very specific models or types of data in order to simplify calculations (for example, 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 with 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 $t = n_1/N$, where $n_1$ is the sample size in the first group and $N = n_1 + n_2$ 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 these 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.

Rank-based inference methods are applied in various disciplines, typically when procedures relying on standard normal theory are not justifiable, for example when data are not symmetrically distributed, contain outliers, or responses are even measured on ordinal scales. Various specific rank-based methods have been developed for two and more samples, and also for general factorial designs (e.g., Kruskal-Wallis test, Jonckheere-Terpstra test). It is the aim of the present paper (1) to demonstrate that traditional rank-procedures for several samples or general factorial designs may lead to paradoxical results in case of unbalanced samples, (2) to explain why this is the case, and (3) to provide a way to overcome these disadvantages of traditional rankbased inference. Theoretical investigations show that the paradoxical results can be explained by carefully considering the non-centralities of the test statistics which may be non-zero for the traditional tests in unbalanced designs. These non-centralities may even become arbitrarily large for increasing sample sizes in the unbalanced case. A simple solution is the use of socalled pseudo-ranks instead of ranks. As a special case, we illustrate the effects in sub-group analyses which are often used when dealing with rare diseases., Comment: 19 pages, 0 figures

It is well known that the standard F test is severely affected by heteroskedasticity in unbalanced analysis of covariance (ANCOVA) models. Currently available potential remedies for such a scenario are based on heteroskedasticity-consistent covariance matrix estimation (HCCME). However, the HCCME approach tends to be liberal in small samples. Therefore, in the present manuscript, we propose a combination of HCCME and a wild bootstrap technique, with the aim of improving the small-sample performance. We precisely state a set of assumptions for the general ANCOVA model and discuss their practical interpretation in detail, since this issue may have been somewhat neglected in applied research so far. We prove that these assumptions are sufficient to ensure the asymptotic validity of the combined HCCME-wild bootstrap ANCOVA. The results of our simulation study indicate that our proposed test remedies the problems of the ANCOVA F test and its heteroskedasticity-consistent alternatives in small to moderate sample size scenarios. Our test only requires very mild conditions, thus being applicable in a broad range of real-life settings, as illustrated by the detailed discussion of a dataset from preclinical research on spinal cord injury. Our proposed method is ready-to-use and allows for valid hypothesis testing in frequently encountered settings (e.g., comparing group means while adjusting for baseline measurements in a randomized controlled clinical trial).

The incidence of Alzheimer's disease (AD) and other forms of dementia is increasing in most western countries. For a precise and early diagnosis, several examination modalities exist, among them single-photon emission computed tomography (SPECT) and the electroencephalogram (EEG). The latter is highly available, free of radiation hazards, and non-invasive. Thus, its diagnostic utility regarding different stages of dementia is of great interest in neurological research, along with the question of whether its utility depends on age or sex of the person being examined. However, SPECT or EEG measurements are intrinsically multivariate, and there has been a shortage of sufficiently general inferential techniques for the analysis of multivariate data in factorial designs when neither multivariate normality nor equality of covariance matrices across groups should be assumed. We adapt an asymptotic model based (parametric) bootstrap approach to this situation, demonstrate its ability, and use it for a truly multivariate analysis of the EEG and SPECT measurements, taking into account demographic factors such as age and sex. These multivariate results are supplemented by marginal effects bootstrap inference whose theoretical properties can be derived analogously to the multivariate methods. Both inference approaches can have advantages in particular situations, as illustrated in the data analysis.