# Fractal Characterizations of MAX Statistical Distribution in Genetic Association Studies - Quantitative Biology > Quantitative Methods

Abstract: Two non-integer parameters are defined for MAX statistics, which are maximaof $d$ simpler test statistics. The first parameter, $d {MAX}$, is thefractional number of tests, representing the equivalent numbers of independenttests in MAX. If the $d$ tests are dependent, $d {MAX} < d$. The secondparameter is the fractional degrees of freedom $k$ of the chi-squaredistribution $\chi^2 k$ that fits the MAX null distribution. These twoparameters, $d {MAX}$ and $k$, can be independently defined, and $k$ can benon-integer even if $d {MAX}$ is an integer. We illustrate these two parametersusing 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 modelinferred 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 diseasemodel, as versus tests with high $k$ e.g. $k=2$ that are completely uncertainabout the disease model. Similar to Heisenberg-s uncertain principle, theability to infer disease model and the ability to detect significantassociation may not be simultaneously optimized, and $k$ seems to measure thelevel of their balance.

Author: Wentian Li, Yaning Yang

Source: https://arxiv.org/