Difference between revisions of "RAREMETALWORKER METHOD"
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| + | [[Category:RAREMETALWORKER]] | ||
| + | ==Useful Links== | ||
| + | |||
| + | Here are some useful links to key pages: | ||
| + | * The [[RAREMETALWORKER | '''RAREMETALWORKER documentation''']] | ||
| + | * The [[RAREMETALWORKER_command_reference | '''RAREMETALWORKER command reference''']] | ||
| + | * The [[RAREMETALWORKER_SPECIAL_TOPICS | '''RAREMETALWORKER special topics''']] | ||
| + | * The [[Tutorial:_RAREMETAL | '''RAREMETALWORKER quick start tutorial''']] | ||
| + | * The [[RAREMETAL_method | '''RAREMETAL method''']] | ||
| + | * The [[RAREMETAL_FAQ | '''FAQ''']] | ||
| + | |||
== Brief Introduction== | == Brief Introduction== | ||
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== Key Statistics for Analysis of Single Study == | == Key Statistics for Analysis of Single Study == | ||
| + | |||
| + | ===NOTATIONS=== | ||
We use the following notations to describe our methods: | We use the following notations to describe our methods: | ||
| − | <math>\mathbf{y}</math> is the observed | + | <math>\mathbf{y}</math> is the vector of observed quantitative trait |
<math>\mathbf{X}</math> is the design matrix | <math>\mathbf{X}</math> is the design matrix | ||
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<math>\boldsymbol{\varepsilon}</math> is the non-shared environmental effects | <math>\boldsymbol{\varepsilon}</math> is the non-shared environmental effects | ||
| − | == | + | <math> \hat{\boldsymbol{\Omega}} </math> is the estimated covariance matrix of <math>\mathbf{y}</math> |
| + | |||
| + | <math>\mathbf{K}</math> is the kinship matrix | ||
| + | |||
| + | <math>\mathbf{K_X}</math> is the kinship matrix of Chromosome X | ||
| + | |||
| + | <math> \sigma_g^2 </math> is the genetic component | ||
| + | |||
| + | <math> {{\sigma_g}_X}^2 </math> is the genetic component for markers on chromosome X | ||
| + | |||
| + | <math>\sigma_e^2 </math> is the non-shared-environment component. | ||
| + | |||
| + | ===SINGLE VARIANT SCORE TEST=== | ||
We used the following model for the trait: | We used the following model for the trait: | ||
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<math> \mathbf{y}=\mathbf{X}\boldsymbol{\beta_c}+\beta_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} </math>. | <math> \mathbf{y}=\mathbf{X}\boldsymbol{\beta_c}+\beta_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} </math>. | ||
| − | Here, | + | Here, the quantitive trait for an individual is a sum of covariate effects, additive genetic effect from the <math> i^{th} </math> variant and the polygenic background effects together with non-shared environmental effect. |
| − | In this model, <math>\beta_i</math> is to measure the additive genetic effect of the <math>i^{th}</math> variant. As usual, the score statistic for testing <math>H_0:\ | + | In this model, <math>\beta_i</math> is to measure the additive genetic effect of the <math>i^{th}</math> variant. As usual, the score statistic for testing <math>H_0:\beta_i=0</math> is: |
<math> U_i=(\mathbf{G_i}-\mathbf{\bar{G_i}} )^T \hat{\boldsymbol{\Omega}}^{-1}(\mathbf{y}-\mathbf{X}\boldsymbol{\beta}) </math> | <math> U_i=(\mathbf{G_i}-\mathbf{\bar{G_i}} )^T \hat{\boldsymbol{\Omega}}^{-1}(\mathbf{y}-\mathbf{X}\boldsymbol{\beta}) </math> | ||
| Line 40: | Line 65: | ||
<math> \mathbf{V}=(\mathbf{G}-\bar{\mathbf{G}})^T (\hat{\boldsymbol{\Omega}}^{-1}-\hat{\boldsymbol{\Omega}}^{-1} \mathbf{X}(\mathbf{X^T}\hat{\boldsymbol{\Omega}}^{-1}\mathbf{X})^{-1} \mathbf{X^T} \hat{\boldsymbol{\Omega}}^{-1})(\mathbf{G}-\bar{\mathbf{G}}) </math>. | <math> \mathbf{V}=(\mathbf{G}-\bar{\mathbf{G}})^T (\hat{\boldsymbol{\Omega}}^{-1}-\hat{\boldsymbol{\Omega}}^{-1} \mathbf{X}(\mathbf{X^T}\hat{\boldsymbol{\Omega}}^{-1}\mathbf{X})^{-1} \mathbf{X^T} \hat{\boldsymbol{\Omega}}^{-1})(\mathbf{G}-\bar{\mathbf{G}}) </math>. | ||
| − | The score test statistic, <math>T_i=(U_i^2)/V_{ii}</math>, is asymptotically distributed as chi-squared with one degree of freedom. | + | The score test statistic, <math>T_i=(U_i^2)/V_{ii}</math>, is asymptotically distributed as chi-squared with one degree of freedom. The score test p-value is reported in RAREMETALWORKER. |
| − | == | + | ===SUMMARY STATISTICS AND COVARIANCE MATRICES=== |
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 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 stores 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, size of which can be specified through command line. |
| − | == | + | === MODELING RELATEDNESS === |
| − | + | We use a variance component model to handle familial relationships. We estimate the variance components under the null model: | |
<math>\mathbf{y}=\mathbf{X}\boldsymbol{\beta} +\mathbf{g}+ \boldsymbol{\varepsilon}</math> | <math>\mathbf{y}=\mathbf{X}\boldsymbol{\beta} +\mathbf{g}+ \boldsymbol{\varepsilon}</math> | ||
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We assume that genetic effects are normally distributed, with mean <math>\mathbf{0}</math> and covariance <math>\mathbf{K}\sigma_g^2</math> where the matrix <math>\mathbf{K}</math> summarizes kinship coefficients between sampled individuals and <math>\sigma_g^2</math> is a positive scalar describing the genetic contribution to the overall variance. We assume that non-shared environmental effects are normally distributed with mean <math>\mathbf{0}</math> and covariance <math>\mathbf{I}\sigma_e^2</math>, where <math>\mathbf{I}</math> is the identity matrix. | We assume that genetic effects are normally distributed, with mean <math>\mathbf{0}</math> and covariance <math>\mathbf{K}\sigma_g^2</math> where the matrix <math>\mathbf{K}</math> summarizes kinship coefficients between sampled individuals and <math>\sigma_g^2</math> is a positive scalar describing the genetic contribution to the overall variance. We assume that non-shared environmental effects are normally distributed with mean <math>\mathbf{0}</math> and covariance <math>\mathbf{I}\sigma_e^2</math>, where <math>\mathbf{I}</math> is the identity matrix. | ||
| − | To estimate <math>\mathbf{K}</math>, we either use known pedigree structure to define <math>\mathbf{K}</math> or else use the empirical estimator <math>\mathbf{K}=\frac{1}{l}\sum_{i=1}^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)} </math>, | + | To estimate <math>\mathbf{K}</math>, we either use known pedigree structure to define <math>\mathbf{K}</math> or else use the empirical estimator |
| − | where <math>l</math> is the count of variants, <math>G_i</math> and <math>f_i</math> are the genotype vector and estimated allele frequency for the <math>i^{th}</math> variant, respectively. Each element in <math>G_i</math> encodes the minor allele count for one individual. Model parameters <math>\hat{\boldsymbol{\beta}}</math>, <math>\hat{\sigma_g^2}</math> and <math>\hat{\sigma_e^2}</math>, are estimated using maximum likelihood and the efficient algorithm described in Lippert et. al. For convenience, let the estimated covariance matrix of <math>\mathbf{y}</math> be <math>\hat{\boldsymbol{\Omega}}= | + | |
| + | <math>\mathbf{K}=\frac{1}{l}\sum_{i=1}^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)} </math>, | ||
| + | |||
| + | where <math>l</math> is the count of variants, <math>G_i</math> and <math>f_i</math> are the genotype vector and estimated allele frequency for the <math>i^{th}</math> variant, respectively. Each element in <math>G_i</math> encodes the minor allele count for one individual. Model parameters <math>\hat{\boldsymbol{\beta}}</math>, <math>\hat{\sigma_g^2}</math> and <math>\hat{\sigma_e^2}</math>, are estimated using maximum likelihood and the efficient algorithm described in [http://www.nature.com/nmeth/journal/v8/n10/full/nmeth.1681.html Lippert et. al]. For convenience, let the estimated covariance matrix of <math>\mathbf{y}</math> be <math>\hat{\boldsymbol{\Omega}}=\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}</math>. | ||
| − | == | + | ===ANALYZING MARKERS ON 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}}= | + | 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}}=\hat{\sigma_g^2}\mathbf{K}+\hat{{\sigma_g}_X^2}\mathbf{K_X}+\hat{\sigma_e^2}\mathbf{I}</math>. |
Latest revision as of 17:49, 16 March 2018
Useful Links
Here are some useful links to key pages:
Brief Introduction
RAREMETALWORKER generates single variant association test statistics for a single study prior to meta-analysis. This page provides a brief description of the statistics that RAREMETALWORKER calculates, together with key formulae.
Key Statistics for Analysis of Single Study
NOTATIONS
We use the following notations to describe our methods:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{y}} is the vector of observed quantitative trait
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{X}} is the design matrix
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{G_i}} is the genotype vector of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} variant
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{\mathbf{G_i}}} is the vector of average genotype of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} variant
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\beta_c}} is the vector of covariate effects
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta_i} is the scalar of fixed genetic effect of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} variant
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{g}} is the random genetic effects
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\varepsilon}} is the non-shared environmental effects
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\boldsymbol{\Omega}} } is the estimated covariance matrix of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{y}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{K}} is the kinship matrix
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{K_X}} is the kinship matrix of Chromosome X
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_g^2 } is the genetic component
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{\sigma_g}_X}^2 } is the genetic component for markers on chromosome X
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_e^2 } is the non-shared-environment component.
SINGLE VARIANT SCORE TEST
We used the following model for the trait:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{y}=\mathbf{X}\boldsymbol{\beta_c}+\beta_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} } .
Here, the quantitive trait for an individual is a sum of covariate effects, additive genetic effect from the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th} } variant and the polygenic background effects together with non-shared environmental effect.
In this model, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta_i} is to measure the additive genetic effect of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} variant. As usual, the score statistic for testing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0:\beta_i=0} is:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i=(\mathbf{G_i}-\mathbf{\bar{G_i}} )^T \hat{\boldsymbol{\Omega}}^{-1}(\mathbf{y}-\mathbf{X}\boldsymbol{\beta}) }
We further derive the variance-covariance matrix of these statistics as
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}=(\mathbf{G}-\bar{\mathbf{G}})^T (\hat{\boldsymbol{\Omega}}^{-1}-\hat{\boldsymbol{\Omega}}^{-1} \mathbf{X}(\mathbf{X^T}\hat{\boldsymbol{\Omega}}^{-1}\mathbf{X})^{-1} \mathbf{X^T} \hat{\boldsymbol{\Omega}}^{-1})(\mathbf{G}-\bar{\mathbf{G}}) } .
The score test statistic, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_i=(U_i^2)/V_{ii}} , is asymptotically distributed as chi-squared with one degree of freedom. The score test p-value is reported in RAREMETALWORKER.
SUMMARY STATISTICS AND COVARIANCE MATRICES
RAREMETALWORKER automatically stores the score statistics for each marker ( Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i } ) together with quality information of that marker, including HWE p-value, call rate, and allele counts.
RAREMETALWORKER also stores the covariance matrices (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V} } ) of the score statistics of markers within a window, size of which can be specified through command line.
MODELING RELATEDNESS
We use a variance component model to handle familial relationships. We estimate the variance components under the null model:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{y}=\mathbf{X}\boldsymbol{\beta} +\mathbf{g}+ \boldsymbol{\varepsilon}}
We assume that genetic effects are normally distributed, with mean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{0}}
and covariance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{K}\sigma_g^2}
where the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{K}}
summarizes kinship coefficients between sampled individuals and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_g^2}
is a positive scalar describing the genetic contribution to the overall variance. We assume that non-shared environmental effects are normally distributed with mean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{0}}
and covariance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{I}\sigma_e^2}
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{I}}
is the identity matrix.
To estimate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{K}} , we either use known pedigree structure to define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{K}} or else use the empirical estimator
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{K}=\frac{1}{l}\sum_{i=1}^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)} } ,
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} is the count of variants, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_i} are the genotype vector and estimated allele frequency for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} variant, respectively. Each element in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_i} encodes the minor allele count for one individual. Model parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\boldsymbol{\beta}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\sigma_g^2}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\sigma_e^2}} , are estimated using maximum likelihood and the efficient algorithm described in Lippert et. al. For convenience, let the estimated covariance matrix of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{y}} be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\boldsymbol{\Omega}}=\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}} .
ANALYZING MARKERS ON CHROMOSOME X
To analyze markers on chromosome X, we fit an extra variance components Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{\sigma_g}_X}^2 } , to model the variance explained by chromosome X. A kinship for chromosome X, , can be estimated either from a pedigree, or from genotypes of marker from chromosome X. Then the estimated covariance matrix can be written as .
