scholarly journals Tail Probabilities for the Null Distribution of Scanning Statistics

Bernoulli ◽  
2000 ◽  
Vol 6 (2) ◽  
pp. 191 ◽  
Author(s):  
David Siegmund ◽  
Benjamin Yakir
Keyword(s):  
Null Distribution ◽  
Tail Probabilities

scholarly journals FastSKAT: Sequence kernel association tests for very large sets of markers

2016 ◽  
Author(s):  
Thomas Lumley ◽  
Jennifer Brody ◽  
Gina Peloso ◽  
Alanna Morrison ◽  
Kenneth Rice
Keyword(s):  
Genetic Variants ◽  
Null Distribution ◽  
Singular Values ◽  
Tail Probabilities ◽  
Large Sets ◽  
Single Term ◽  
Major Bottleneck ◽  
Largest Eigenvalues ◽  
New Applications ◽  
Satterthwaite Approximation

AbstractThe Sequence Kernel Association Test (SKAT) is widely used to test for associations between a phenotype and a set of genetic variants, that are usually rare. Evaluating tail probabilities or quantiles of the null distribution for SKAT requires computing the eigenvalues of a matrix related to the genotype covariance between markers. Extracting the full set of eigenvalues of this matrix (an n × n matrix, for n subjects) has computational complexity proportional to n3. As SKAT is often used when n > 104, this step becomes a major bottleneck in its use in practice. We therefore propose fastSKAT, a new computationally-inexpensive but accurate approximations to the tail probabilities, in which the k largest eigenvalues of a weighted genotype covariance matrix or the largest singular values of a weighted genotype matrix are extracted, and a single term based on the Satterthwaite approximation is used for the remaining eigenval-ues. While the method is not particularly sensitive to the choice of k, we also describe how to choose its value, and show how fastSKAT can automatically alert users to the rare cases where the choice may affect results. As well as providing faster implementation of SKAT, the new method also enables entirely new applications of SKAT, that were not possible before; we give examples grouping variants by topologically assisted domains, and comparing chromosome-wide association by class of histone marker.

Bootstrap Analysis

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Goodness Of Fit ◽  
Null Distribution ◽  
Real Data ◽  
Confidence Regions ◽  
Confidence Bands ◽  
Cumulative Distribution ◽  
Attractive Alternative ◽  
Bootstrap Analysis ◽  
Parametric Bootstrapping ◽  
Anderson Darling

Parametric bootstrapping (BS) provides an attractive alternative, both theoretically and numerically, to asymptotic theory for estimating sampling distributions. This chapter summarizes its use not only for calculating confidence intervals for estimated parameters and functions of parameters, but also to obtain log-likelihood-based confidence regions from which confidence bands for cumulative distribution and regression functions can be obtained. All such BS calculations are very easy to implement. Details are also given for calculating critical values of EDF statistics used in goodness-of-fit (GoF) tests, such as the Anderson-Darling A2 statistic whose null distribution is otherwise difficult to obtain, as it varies with different null hypotheses. A simple proof is given showing that the parametric BS is probabilistically exact for location-scale models. A formal regression lack-of-fit test employing parametric BS is given that can be used even when the regression data has no replications. Two real data examples are given.

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