Inference methods for multivariate and high-dimensional data

In light of greatly advanced computational resources, the scope of statistical data analysis now accommodates pressing new areas of application. This is particularly the case for the development of analysis tools for data sets containing several measured quantities. Those data are often complex due to their dimensionality, or structure, and it is especially challenging to derive appropriate analysis tools for those situations where classical and strongly simplifying model assumptions are not tenable. These, in practice rather common situations, are the target of the project, which proposes to develop methods for the analysis of complex, high-dimensional data. The methods shall be applicable in general situations, and also yield valid results when the application of currently existing procedures is not appropriate. Specifically, we are dealing with situations in which 1. data cannot be described by a normal distribution, 2. the response variables cannot be measured on a metric scale, but only by greater-smaller relations (ordinal data), 3. the number of measurements collected per person is larger than the number of persons (high dimensional data), or 4. not all persons under study could be observed long enough to obtain an accurate measurement, for example of the length of time leading up to an event (censored data). Using methods from mathematical statistics, we propose to develop procedures that meet the mentioned criteria. To this end, rank based methods as well as resampling techniques will be taken into account. The validity of each procedure will be shown by means of theoretical considerations concerning the performance of the tests for increasing sample sizes. These examinations will be complemented by extensive simulation studies in different designs of interest. Finally, all positively evaluated procedures will be made available in the form of statistical software packages for the free software environment R. The uniqueness and innovation of the project are that currently there exists no generally valid method for the complete analysis of complex data sets with many measured quantities. In practice, methods are used that are too simplistic, or procedures that are not appropriate due to strong underlying requirements, thus leading to unreasonable research results. The project aims to remedy this situation and to provide an effective methodology for the analysis of complex data. The expected results have wide potential applications and expand the role of statistical data analysis by allowing the treatment of new problems with meaningful and effective methods. Overall, we expect the results from the project to have a significant and lasting impact on modern "Data Science".

Nowadays, it is possible to generate bigger and bigger data sets with continuously decreasing technological effort. How does one make sense of these data? Classical statistical procedures are usually only applicable if certain restrictive assumptions are met. Many of the standard procedures are not valid if, for example, the data are high-dimensional, if the measurements cannot be represented as numbers, if observations in different groups exhibit differing variability, or if they are incompletely observed. We have developed and validated statistical methods that can still be applied sensibly in numerous challenging situations of big data or messy data. In order to make it easier for other researchers to actually apply these new methods, we have also devised corresponding free open source software packages and made them publicly available.