  In a typical Morris water maze experiment, a mouse is placed in a circular
water tank and allowed to swim freely until it finds a platform, triggering a
route of escape from the tank. For reference purposes, the tank is divided into
four quadrants: the target quadrant where the trigger to escape resides, the
opposite quadrant to the target, and two adjacent quadrants. Several response
variables can be measured: the amount of time that a mouse spends in different
quadrants of the water tank, the number of times the mouse crosses from one
quadrant to another, or how quickly a mouse triggers an escape from the tank.
When considering time within each quadrant, it is hypothesized that normal mice
will spend smaller amounts of time in quadrants that do not contain the escape
route, while mice with an acquired or induced mental deficiency will spend
equal time in all quadrants of the tank. Clearly, proportion of time in the
quadrants must sum to one and are therefore statistically dependent; however,
most analyses of data from this experiment treat time in quadrants as
statistically independent. A recent paper introduced a hypothesis testing
method that involves fitting such data to a Dirichlet distribution. While an
improvement over studies that ignore the compositional structure of the data,
we show that methodology is flawed. We introduce a two-sample test to detect
differences in proportion of components for two independent groups where both
groups are from either a Dirichlet or nested Dirichlet distribution. This new
test is used to reanalyze the data from a previous study and come to a
different conclusion.
