Functional data analysis (FDA) is a branch of statistics that analyzes observations treated as functions, curves, or surfaces. To represent the data in such a way, one needs only to measure some variable over time or space, which is a scenario encountered in many fields, such as brain imaging data, medical measurements over time, biological development, etc. Then the discrete data observed at so-called design time points can be transformed into functional data. Such a representation allows us to avoid many problems of classical multivariate statistical methods, for example, the curse of dimensionality and missing data. Therefore, numerous methods have been developed for classification, clustering, dimension reduction, regression, and statistical hypothesis testing for functional data. The methods are based on different approaches, for example, dimension reduction (basis expansion, functional principal components), random projections on multivariate data, and aggregating pointwise statistics. During the talk, we present the different aspects and strategies of the functional data analysis methods with a special focus on the functional analysis of variance. The latter covers, in particular, one-way, multi-way, univariate, multivariate, independent observations, and repeated measurements.

In various scientific fields, functional data can be observed more and more frequently. This includes audiology, biology, ergonomy, meteorology, growth studies and environmentology (Zhang, 2013, p. 2), to name just a few examples.

Several tools for the analysis of functional data, including tests for complex factorial designs and functional ANOVA problems, have already been studied (e.g. Zhang, 2013). However, they mainly rely on homoscedasticity assumptions, which are often not justifiable in practice. Moreover, all these strategies are designed for global null hypotheses testing, e.g. for main and interaction effects, and do not directly allow a more in-depth analysis by testing several null hypotheses simultaneously.

To address the first problem, we obtain a test function that takes the heteroscedasticity of the functional data into account. Integrating over the test function yields a test statistic for general null hypotheses in factorial designs, which has a rather complicated limit null distribution. Therefore, we propose a resampling procedure to approximate the null distribution. In a next step, we explain how to use the described testing strategy and resample scheme to infer several local null hypotheses simultaneously. Hereby, we incorporate the asymptotic exact dependency structure between the local test statistics to avoid a significant power loss. The resulting multiple testing procedure is consonant and coherent as defined in Gabriel (1969). Moreover, the proposed multiple contrast tests control the level of significance of the global test as well as the family-wise type I error rate. The small sample performances of the proposed global and multiple testing procedures are analyzed in extensive simulations and finally illustrated by analyzing a real data example.

References:

K.R. Gabriel (1969). Simultaneous test procedures–some theory of multiple comparisons. The Annals of Mathematical Statistics, 40(1):224-250.

J.-T. Zhang (2013). Analysis of variance for Functional Data. Chapman & Hall/CRC.

In various fields, e.g., biology, ecology, medicine, or psychology, several outcome variables are of simultaneous interest leading to multivariate data. For example, an ecologist may study the aggression against predators and the relative reproductive success (fitness) of birds grouped by sex and colour morph. Other examples are psychological tests or different medical quantities, e.g., heart rate, blood pressure, weight, or height of a patient. As pointed out by Warne (2014) , multivariate analysis-of-variance (MANOVA) is “one of the most common multivariate statistical procedures in the social science literature”. However, classical MANOVA relies on restrictive assumptions as normality and homogeneity of covariances. But the “normality assumption becomes quasi impossible to justify when moving from univariate to multivariate observations” (Konietschke et al., 2015) and, similarly, homogeneity is often implausible. To overcome these difficulties there are less restrictive mean-based MANOVA concepts proposed for testing global hypothesis about multivariate expectations, e.g Friedrich and Pauly (2018). In case of outliers or distributions with larger tails, however, non-robust estimators like the mean can have some drawbacks. Despite the usage of quantiles is intuitive in that case and often applied in descriptive statistics, e.g. boxplots, quantiles "[appear] to be quite underused in medical research" (Beyerlein, 2014).

Therefore we developed a flexible quantile-based MANOVA method. The approach is adaptable to general factorial designs and has the advantage that it fits to median and other quantile-based statistical methods. To achieve this, we considered two quadratic-form type test statistics and three different strategies for estimating the covariance. The test statistics’ distribution is approximated via resampling. We prove that our method is valid in theory and even works in case of general heterogeneous or heteroscedastic data beyond normality. In a simulation study, we compare the novel procedures with state-of-the-art mean-based approaches and observe that the quantile-based approach produces more powerful tests in the case of heavy-tailed data. As an illustrative example we consider heavy-tailed data about the colour morphs of common buzzard chicks.

References
Beyerlein, A. (2014). Quantile Regression–Opportunities and Challenges From a User’s Perspective. American Journal of Epidemiology, 180(3):330–331.

Friedrich, S. and Pauly, M. (2018). MATS: Inference for potentially singular and heteroscedastic MANOVA. Journal of Multivariate Analysis, 165:166–179.

Konietschke, F., Bathke, A. C., Harrar, S. W., and Pauly, M. (2015). Parametric and nonparametric bootstrap methods for general MANOVA. Journal of Multivariate Analysis, 140:291–301.

Warne, R. (2014). A Primer on Multivariate Analysis of Variance (MANOVA) for Behavioral Scientists. Practical Assessment, Research & Evaluation, 19:17.

Covariance and correlation matrices are essential tools for investigating random vectors' dispersion and dependency structure. Especially
correlation matrices that are not affected by units allow investigating the dependency structures of random vectors or comparing them.
We introduce an approach for testing various null hypotheses that can be formulated based on the correlation matrix. Examples cover MANOVA-type hypothesis of equal correlation matrices as well as testing for special correlation structures such as, e.g., sphericity. Apart from existing fourth moments, our approach requires no other assumptions, allowing applications in various settings. To improve the small sample performance, a bootstrap technique is proposed and theoretically justified, as well as some Taylor-expansion-based approaches.