FRACTAL CHARACTERIZATIONS OF MAX STATISTICAL DISTRIBUTION IN GENETIC ASSOCIATION STUDIES

Two noninteger parameters are defined for MAX statistics, which are maxima of d simpler test statistics. The first parameter, dMAX, is the fractional number of tests, representing the equivalent numbers of independent tests in MAX. If the d tests are dependent, dMAX < d. The second parameter is the fractional degrees of freedom k of the chi-square distribution $\chi_k^2$ that fits the MAX null distribution. These two parameters, dMAX and k, can be independently defined, and k can be noninteger even if dMAX is an integer. We illustrate these two parameters using the examples 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.