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The BLUP are the predicted values of the random terms in the model. Recall the general form of the mixed model:

The vector that produces the BLUP estimates is:

where

For further details on the notation, go to the Methods section.

The estimated standard deviations are the square roots of the diagonals of this matrix:

where

Term | Description |
---|---|

Z_{i} | the n x m matrix of known codings for the _{i}i^{th} random effect in the model |

Z' | the transpose of Z |

y | the vector of the response values |

X | the design matrix |

b | the estimated coefficients for the fixed effects |

the variance component of the i^{th} random factor | |

the variance component for error | |

I_{m} | the identity matrix with m rows and columns |

The degrees of freedom for the test of the BLUP component are:

where

Term | Description |
---|---|

a vector with value 1 at the row and 0 elsewhere with dimension | |

W | the asymptotic variance-covariance matrix of the variance component estimates |

For further details on the notation, go to the Methods section.

The two-sided p-value for the null hypothesis that a Best Linear Unbiased Predictor (BLUP) equals 0 is:

Term | Description |
---|---|

The probability that under the null hypothesis T is less than the absolute value of the calculated . Here, T follows a t-distribution with df degrees of freedom. | |

The t value for the BLUP. |