X.X-transpose for a standardized SnpMatrix
The input SnpMatrix is first standardized by subtracting the mean (or stratum mean) from each call and dividing by the expected standard deviation under Hardy-Weinberg equilibrium. It is then post-multiplied by its transpose. This is a preliminary step in the computation of principal components.
The input matrix, of type
This computation forms the first step of the calculation of principal
components for genome-wide SNP data. As pointed out by Price et al.
(2006), when the data matrix has more rows than columns it is
most efficient to calculate the eigenvectors of
X.X-transpose, where X is a
SnpMatrix whose columns have been standardized to zero mean and
unit variance. For autosomes, the genotypes are given codes 0, 1 or 2
after subtraction of the mean, 2p, are divided by the standard
sqrt(2p(1-p)) (p is the estimated allele
frequency). For SNPs on the X chromosome in male subjects,
genotypes are coded 0 or 2. Then
the mean is still 2p, but the standard deviation is
2sqrt(p(1-p)). If the
strata is supplied, a
stratum-specific estimate value for p is used for
Missing observations present some difficulty. Price et al. (2006) recommended replacing missing observations by their means, this being equivalent to replacement by zeros in the standardized matrix. However this results in a biased estimate of the complete data result. Optionally this bias can be corrected by inverse probability weighting. We assume that the probability that any one call is missing is small, and can be predicted by a multiplicative model with row (subject) and column (locus) effects. The estimated probability of a missing value in a given row and column is then given by m = RC/T, where R is the row total number of no-calls, C is the column total of no-calls, and T is the overall total number of no-calls. Non-missing contributions to X.X-transpose are then weighted by w=1/(1-m) for contributions to the diagonal elements, and products of the relevant pairs of weights for contributions to off–diagonal elements.
A square matrix containing either the complete X.X-transpose matrix, or just its lower triangle
The correction for missing observations can result in an output matrix which is not positive semi-definite. This should not matter in the application for which it is intended
In genome-wide studies, the SNP data will usually be held as a series of
"XSnpMatrix"), one per chromosome.
Note that the X.X-transpose matrices
produced by applying the
xxt function to each object in turn
can be added to yield the genome-wide result.
If the matrix is converted to a correlation matrix by pre- and post-multiplying by the sqrt of the inverse of its diagonal, then this is an unbiased estimate of twice the kinship matrix.
David Clayton firstname.lastname@example.org
Price et al. (2006) Principal components analysis corrects for stratification in genome-wide association studies. Nature Genetics, 38:904-9
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