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== 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 to test <math>H_0:\gamma_i=0</math> is:
Our single variant association test is the score test using linear mixed model, treating single variants as fixed effects. The alternative model is:
<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
<math> U_i=(\mathbf{G_i}-\mathbf{\bar{G_i}} )^T \boldsymbol{\Omega}^(-1)(\mathbf{y}-\mathbf{X}\boldsymbol{\beta}) </math>
U_i=(G_i-G ̅_i )^T Ω ̂^(-1) (y-Xβ ̂)
And the variance-covariance matrix of these statistics is (see Appendix A for details):
And the variance-covariance matrix of these statistics is (see Appendix A for details):
