To estimate the standard errors of the variance components, Minitab begins with the observed Fisher information matrix. The matrix has c
+ 1 rows and columns. The variable c
is the number of random effect terms in the model and 1 represents the variance for the error term. For i
= 1, …, c
= 1, …, c
the following is the formula for the ijth
component of the observed Fisher information matrix:
The following formula is the component of the last row and the
= 1, …, c:
This component is also the value of the last column and the row by the symmetry property of the variance-covariance matrix.
The following formula is the component of the last row and the last column:
The asymptotic variance-covariance matrix for the variance components estimates is twice the inverse of the observed Fisher information matrix. The estimates of the standard errors are the square roots of the diagonal elements of the variance-covariance matrix. The first c diagonal elements are for the variance components of the random effect terms. The last diagonal element is for the error variance component.