Methods and formulas for the random effect predictions for Stability Study for random batches

Select the method or formula of your choice.

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.

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

where

Term	Description
Z_i	the n x m_i matrix of known codings for the 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.