Methods and formulas for random effect predictions in Fit Mixed Effects Model

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Best linear unbiased predictors (BLUP)

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.

Standard error of BLUP

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

where

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

Degrees of freedom for BLUP

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

where

Notation

TermDescription
a vector with value 1 at the row and 0 elsewhere with dimension
Wthe asymptotic variance-covariance matrix of the variance component estimates

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

Confidence intervals for BLUP

The 100 × (1 – α)% confidence limits for the conditional expectation of the component of μ given response values have the following form:

Notation

TermDescription
the BLUP
the percentile from the t distribution with the given degrees of freedom
1 − confidence level
dfthe degrees of freedom for the BLUP

t-value for BLUP

p-value for BLUP

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

Notation

TermDescription
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.
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