  Two non-integer parameters are defined for MAX statistics, which are maxima
of $d$ simpler test statistics. The first parameter, $d_{MAX}$, is the
fractional number of tests, representing the equivalent numbers of independent
tests in MAX. If the $d$ tests are dependent, $d_{MAX} < d$. The second
parameter is the fractional degrees of freedom $k$ of the chi-square
distribution $\chi^2_k$ that fits the MAX null distribution. These two
parameters, $d_{MAX}$ and $k$, can be independently defined, and $k$ can be
non-integer even if $d_{MAX}$ is an integer. We illustrate these two parameters
using the example of MAX2 and MAX3 statistics in genetic case-control studies.
We speculate that $k$ is related to the amount of ambiguity of the model
inferred by the test. In the case-control genetic association, tests with low
$k$ (e.g. $k=1$) are able to provide definitive information about the disease
model, as versus tests with high $k$ (e.g. $k=2$) that are completely uncertain
about the disease model. Similar to Heisenberg's uncertain principle, the
ability to infer disease model and the ability to detect significant
association may not be simultaneously optimized, and $k$ seems to measure the
level of their balance.
