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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:
<math> \mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\gamma_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} </math>.
This model is a refinement of equation (1) above, adding a scalar parameter γ_i to measure the additive genetic effect of the ith variant. As usual41, the score statistic for testing H_0:γ_i=0 is
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):
V=(G-G ̅ )^T (Ω ̂^(-1)-Ω ̂^(-1) X(X^T Ω ̂^(-1) X)^(-1) X^T Ω ̂^(-1) )(G-G ̅).
Under the null, test statistics T_i=(U_i^2)/V_ii is asymptotically distributed as chi-squared with one degree of freedom.
== Summary Statistics ==
== Summary Statistics ==
