KEGG(Kyoto Encyclopedia of Genes and Genomes)〔和文〕 (特集 ゲノム医学の現在と未来--基礎と臨床) -- (データベース)

The Analysis of Variance.

The central goal of human genetics is to understand the inherited basis of human variation in phenotypes, elucidating human physiology, evolution and disease. Rare mutations have been found underlying two thousand mendelian diseases; more recently, it has become possible to assess systematically the contribution of common SNPs to complex disease. The known role of copy-number alterations in sporadic genomic disorders, combined with emerging information about inherited copy-number variation, indicate the importance of systematically assessing copy-number variants (CNVs), including common copy-number polymorphisms (CNPs), in disease. Here we discuss evidence that CNVs affect phenotypes, directions for basic knowledge to support clinical study of CNVs, the challenge of genotyping CNPs in clinical cohorts, the use of SNPs as markers for CNPs and statistical challenges in testing CNVs for association with disease. Critical needs are high-resolution maps of common CNPs and techniques that accurately determine the allelic state of affected individuals.

Copy number variation in association studies of human disease

How should one quantify the strength of association between two random variables without bias for relationships of a specific form? Despite its conceptual simplicity, this notion of statistical "equitability" has yet to receive a definitive mathematical formalization. Here we argue that equitability is properly formalized by a self-consistency condition closely related to Data Processing Inequality. Mutual information, a fundamental quantity in information theory, is shown to satisfy this equitability criterion. These findings are at odds with the recent work of Reshef et al. [Reshef DN, et al. (2011) Science 334(6062):1518-1524], which proposed an alternative definition of equitability and introduced a new statistic, the "maximal information coefficient" (MIC), said to satisfy equitability in contradistinction to mutual information. These conclusions, however, were supported only with limited simulation evidence, not with mathematical arguments. Upon revisiting these claims, we prove that the mathematical definition of equitability proposed by Reshef et al. cannot be satisfied by any (nontrivial) dependence measure. We also identify artifacts in the reported simulation evidence. When these artifacts are removed, estimates of mutual information are found to be more equitable than estimates of MIC. Mutual information is also observed to have consistently higher statistical power than MIC. We conclude that estimating mutual information provides a natural (and often practical) way to equitably quantify statistical associations in large datasets.

/pdf/equitability-mutual-information-and-the-maximal-information-3rcyuw3ll5.pdf

Equitability, mutual information, and the maximal information coefficient

We demonstrate that a popular class of nonparametric mutual information (MI) estimators based on k-nearest-neighbor graphs requires number of samples that scales exponentially with the true MI. Consequently, accurate estimation of MI between two strongly dependent variables is possible only for prohibitively large sample size. This important yet overlooked shortcoming of the existing estimators is due to their implicit reliance on local uniformity of the underlying joint distribution. We introduce a new estimator that is robust to local non-uniformity, works well with limited data, and is able to capture relationship strengths over many orders of magnitude. We demonstrate the superior performance of the proposed estimator on both synthetic and real-world data.

/pdf/efficient-estimation-of-mutual-information-for-strongly-1puypi63n3.pdf

Efficient Estimation of Mutual Information for Strongly Dependent Variables

SUMMARY We consider the problem of detecting associations between random vectors of any dimension. Few tests of independence exist that are consistent against all dependent alternatives. We propose a powerful test that is applicable in all dimensions and consistent against all alternatives. The test has a simple form, is easy to implement, and has good power.

/pdf/a-consistent-multivariate-test-of-association-based-on-ranks-csirf21ds6.pdf

A consistent multivariate test of association based on ranks of distances

A popular approach for testing if two univariate random variables are statistically independent consists of partitioning the sample space into bins, and evaluating a test statistic on the binned data. The partition size matters, and the optimal partition size is data dependent. While for detecting simple relationships coarse partitions may be best, for detecting complex relationships a great gain in power can be achieved by considering finer partitions. We suggest novel consistent distribution-free tests that are based on summation or maximization aggregation of scores over all partitions of a fixed size. We show that our test statistics based on summation can serve as good estimators of the mutual information. Moreover, we suggest regularized tests that aggregate over all partition sizes, and prove those are consistent too. We provide polynomial-time algorithms, which are critical for computing the suggested test statistics efficiently. We show that the power of the regularized tests is excellent compared to existing tests, and almost as powerful as the tests based on the optimal (yet unknown in practice) partition size, in simulations as well as on a real data example.

/pdf/consistent-distribution-free-k-sample-and-independence-tests-4n96lpso8r.pdf

Consistent distribution-free K-sample and independence tests for univariate random variables

Consistent distribution-free $K$-sample and independence tests for univariate random variables

http://www.math.tau.ac.il/~ruheller/Papers/draftHGH.pdf

A class of multivariate distribution-free tests of independence based on graphs

Bergsma and Dassios (2014) introduced an independence measure which is zero if and only if two random variables are independent. This measure can be naively calculated in $O(n^4)$. Weihs et al. (2015) showed that it can be calculated in $O(n^2 \log n)$. In this note we will show that using the methods described in Heller et al. (2016), the measure can easily be calculated in only $O(n^2)$.

Yair Heller

Papers

A consistent multivariate test of association based on ranks of distances

Consistent distribution-free K-sample and independence tests for univariate random variables

Consistent distribution-free $K$-sample and independence tests for univariate random variables

A class of multivariate distribution-free tests of independence based on graphs

Computing the Bergsma Dassios sign-covariance