Changes

== Summary Statistics and Covariance Matrices==

== Summary Statistics and Covariance Matrices==

RAREMETALWORKER generates the score statistics for each marker ( <math> U_i </math>) together with quality information of that marker, including HWE p-value, call rate, and allele counts.  

RAREMETALWORKER automatically stores the score statistics for each marker ( <math> U_i </math>) together with quality information of that marker, including HWE p-value, call rate, and allele counts.  

RAREMETALWORKER also generates the covariance matrices (<math> \mathbf{V} </math>) of the score statistics of markers within a window.

RAREMETALWORKER also stores the covariance matrices (<math> \mathbf{V} </math>) of the score statistics of markers within a window.

==Chromosome X==

==Chromosome X==

To analyze markers on chromosome X, we fit an extra variance components <math> {{\sigma_g}_X}^2 </math>, to model the variance explained by chromosome X. A kinship for chromosome X, <math> \boldsymbol{K_X} </math>, can be estimated either from a pedigree, or from genotypes of marker from chromosome X. Then the estimated covariance matrix can be written as <math>\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+2\hat{{\sigma_g}_X^2}\mathbf{K_X}+\hat{\sigma_e^2}\mathbf{I}</math>.

To analyze markers on chromosome X, we fit an extra variance components <math> {{\sigma_g}_X}^2 </math>, to model the variance explained by chromosome X. A kinship for chromosome X, <math> \boldsymbol{K_X} </math>, can be estimated either from a pedigree, or from genotypes of marker from chromosome X. Then the estimated covariance matrix can be written as <math>\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+2\hat{{\sigma_g}_X^2}\mathbf{K_X}+\hat{\sigma_e^2}\mathbf{I}</math>.