Changes

== Single Variant Score Tests ==

== Single Variant Score Tests ==

Our single variant association test is the score test using linear mixed model, treating single variants as fixed effects. The alternative model is:

 

We used the following model for the trait:

<math> \mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\gamma_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} </math>.

<math> \mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\gamma_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} </math>.

In this model, the scalar parameter <math>\gamma_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:\gamma_i=0</math> is:

 

In this model, <math>\gamma_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:\gamma_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>