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Experimental Design Models with Random Components

has been used to estimate the variance components, then one could logi cal ly proceed with a likelihood ratio test for the variance component which has an asymptotic chi-square distribution.

In some cases it is possible to construct exact confidence intervals on variance components. In generaÅ‚, the 100(1 - a)% confidence interval on cr² is

df E ^    ^    2 ^ df E ^

2 ^ 0^2 Xa/^_E    2,df_£

where

ct² =SS_E/df_E

firom the ANOVA table; SS_E and df_E being the sum of squares and degrees of freedom for error, respectively. Unfortunately, we cannot find simple expressions for confidence intervals such as this one for all variance components of interest, or functions of variance components. In these cases, approximate confidence intervals, based on the Satterthwaite approximation for ratios of Iinear combinations of mean squares can be used. If the maximum likelihood or MINQUE method has been used to obtain point estimates of the variance components, then the asymptotic variances of these estimators could be used to obtain approximate confidence intervals.

Example

Data from semiconductor manufacturing is used to illustrate some of the methods which we have described. Originally, the experiment was only described as a full-factorial experiment with two factors. The analysis we pnesent assumes a completely randomized design. However, one factor is day, with 7 levels, suggesting a randomization restriction might have been used. The other factor is site, with 5 levels. There are 4 replicates for a total of 140 measurements.

The analysis of varianoe and estimates of the variance components for a completely randomized design with random factors is presented in FigurÄ™ 1. The output provides the ANOVA estimates of the components from SAS PROC VARCOMP. The variance component for error suggests substantial noise in the data.

FigurÄ™ 2 provides the maximum likelihood estimates of the components from PROC VARCOMP for the same model. FigurÄ™ 3 provides the restricted maximum likelihood estimates from SAS. Notice that for this balanced study with nonnegative ANOYA estimates of the components, these estimates are identical to