Methods and formulas for the variance components for Stability Study for random batches

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In This Topic

Variance components
Standard errors of variance components
Confidence intervals for variance components
Z-value and p-value
Variance-covariance matrix

Variance components

The variance components require an iterative solution to estimate the parameter θ_i. Once you have the parameter, the variance components have explicit solutions. The formula for the variance component for error is:

where

The following are the variance components for the random effect terms:

For details on the estimation of θ_i, see [1].

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

References

Hemmerle, W. and Hartley, H. (1973). Computing Maximum Likelihood Estimates for the Mixed A.O.V. Model using the W transformation. Technometrics, 15(4):819–831.

Standard errors of variance components

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 and j = 1, …, c the following is the formula for the ij^th component of the observed Fisher information matrix:

where

The following formula is the component of the last row and the

column, j = 1, …, c:

where

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.

Notation

Term	Description
	the trace of matrix
	the sum of squares of all the elements in the matrix M

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

Confidence intervals for variance components

Minitab uses the delta method to construct Wald-type confidence limits for the natural log of the variance components, then exponentiates the confidence interval to get the confidence interval for the variance component. The formula is:

Notation

Term	Description
	the variance component for the i^th random factor
	the quantile from the standard normal distribution
	1 − confidence level

Z-value and p-value

The null and alternative hypotheses for the test are:

The hypotheses for the error variance are the same.

The test statistic assumes a standard normal distribution:

The p-value is the upper tail probability from the standard normal distribution under the null hypothesis:

Notation

Term	Description
Z	the value of the inverse cumulative distribution function for the standard normal distribution

Variance-covariance matrix

The asymptotic variance-covariance matrix is the inverse of 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 and j = 1, …, c the following is the formula for the