  In several modern applications, ranging from genetics to genomics and
neuroimaging, there is a need to compare observations across different
populations, such as groups of healthy and diseased individuals. The interest
is in detecting a group effect. When the observations are vectorial,
real-valued and follow a multivariate Normal distribution, multivariate
analysis of variance (MANOVA) tests are routinely applied. However, such
traditional procedures are not suitable when dealing with more complex data
structures such as functional (e.g. curves) or graph-structured (e.g. trees and
networks) objects, where the required distributional assumptions may be
violated. In this paper we discuss a distance-based MANOVA-like approach, the
DBF test, for detecting differences between groups for a wider range of data
types. The test statistic, analogously to other distance-based statistics, only
relies on a suitably chosen distance measure that captures the pairwise
dissimilarity among all available samples. An approximate null probability
distribution of the DBF statistic is proposed thus allowing inferences to be
drawn without the need for costly permutation procedures. Through extensive
simulations we provide evidence that the proposed methodology works well for a
range of data types and distances, and generalizes the traditional MANOVA
tests. We also report on an application of the proposed methodology for the
analysis of a multi-locus genome-wide association study of Alzheimer's disease,
which has been carried out using several genetic distance measures.
