遗传关联研究中MAX统计分布的分形特征
Fractal Characterizations of MAX Statistical Distribution in Genetic Association Studies
摘要：
为MAX统计量定义了两个非整数参数，它们是$d$简单检验统计量的最大值。第一个参数$d_{MAX}$是测试的分数，表示MAX中独立测试的等效数量。如果$d$测试是依赖性的，则为$d_{MAX} < d$。第二个参数是拟合MAX零分布的卡方分布$χ^2_k$的分数自由度$k$。这两个参数$d_{MAX}$和$k$可以独立地定义，并且即使$d_{MAX}$是整数，$k$也可以是非整数。我们使用遗传病例对照研究中MAX2和MAX 3统计量的例子来说明这两个参数。我们推测，$k$是有关的测试推断的模型的歧义量。在病例-对照遗传关联中，$k$低水平检测（例如$k=1$）能够提供关于疾病模型的明确信息，而$k$高水平检测（例如$k=2$）则完全不确定疾病模型。与海森堡的不确定性原理类似，推断疾病模型的能力和检测显著关联的能力可能无法同时得到优化，而$k$似乎可以衡量两者平衡的程度。
英文摘要：
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 $χ^2_k$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
