Description Details Author(s) References

A computationally-efficient leading-eigenvalue approximation to tail probabilities and quantiles of large quadratic forms, in particular for the Sequence Kernel Association Test (SKAT) used in genomics <doi:10.1002/gepi.22136>. Also provides stochastic singular value decomposition for dense or sparse matrices.

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This package computes tail probabilities for large quadratic forms, with
the motivation being the SKAT test used in DNA sequence association
studies.

The true distribution is a linear combination of 1-df chi-squared
distributions, where the coefficients are the non-zero eigenvalues of
the matrix `A`

defining the quadratic form *z^TAz*. The package uses an
approximation to the distribution consisting of the largest `neig`

terms in the
linear combination plus the Satterthwaite approximation to the rest of
the linear combination.

The main function is `pQF`

, which has options for how to
compute the leading eigenvalues (Lanczos-type algorithm or stochastic
SVD) and how to compute the linear combination (inverting the
characteristic function or a saddlepoint approximation). The Lanczos
algorithm is from the `svd`

package; the stochastic SVD can be
called directly via `ssvd`

or `seigen`

Given a square matrix, `pQF`

uses it as `A`

. If the input is a
non-square matrix `M`

, then `A`

is `crossprod(M)`

. The
function can also be used matrix-free, given an object containing
functions to compute the product and transpose-product by `M`

. This last option
is described in the `"matrix-free"`

vignette. The matrix-free
algorithm also uses a randomised estimator to estimate
the trace of `crossprod(A)`

. The function `sparse.matrixfree`

constructs a object for
matrix-free use of `pQF`

from a sparse Matrix object. The
algorithms are described in the Lumley et al (2018) reference.

Finally, there are functions specifically for the SKAT family of genomic
tests. These take a genotype matrix and an adjustment model as arguments
and produce an object that contains the test statistic in its
`Q`

component and which can be used as an argument to `pQF`

to
extract p-values: `SKAT.matrixfree`

and `famSKAT`

. The
vignette `"Checking pQF vs SKAT"`

compares `SKAT.matrixfree`

to the `SKAT`

package and illustrates how it can be used

Thomas Lumley

Maintainer: Thomas Lumley <t.lumley@auckland.ac.nz>

Tong Chen, Thomas Lumley (2019) Numerical evaluation of methods approximating the distribution of a large quadratic form in normal variables. Computational Statistics & Data Analysis. 139: 75-81,

Lumley et al. (2018) Sequence kernel association tests for large sets of markers: tail probabilities for large quadratic forms. Genet Epidemiol . 2018 Sep;42(6):516-527. doi: 10.1002/gepi.22136

Nathan Halko, Per-Gunnar Martinsson, Joel A. Tropp (2010) Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. https://arxiv.org/abs/0909.4061.

Lee, S., with contributions from Larisa Miropolsky, and Wu, M. (2015). SKAT: SNP-Set (Sequence) Kernel Association Test. R package version 1.1.2.

Lee, S., Wu, M. C., Cai, T., Li, Y., Boehnke, M., and Lin, X. (2011). Rare-variant association testing for sequencing data with the sequence kernel association test. American Journal of Human Genetics, 89:82-93.

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