Fractal Characterizations of MAX Statistical Distribution in Genetic Association Studies

TL;DR

This paper introduces two fractional parameters, $d_{MAX}$ and $k$, to characterize the MAX statistical distribution in genetic studies, linking $k$ to model ambiguity and uncertainty in disease association detection.

Contribution

It defines and illustrates the use of fractional parameters $d_{MAX}$ and $k$ for MAX statistics, providing insights into model ambiguity and test uncertainty in genetic association studies.

Findings

$d_{MAX}$ represents the fractional number of tests, indicating test dependence.

$k$ is a fractional degrees of freedom parameter related to model ambiguity.

Low $k$ values correspond to more definitive disease models.

Abstract

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.