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Montgomery & Runger

Statistica! Inference in tbe Random Model

An important aspect of the random model is estimating the variance components. In fact, this is often the primary purpose for conducting the experiment. There are several ways to obtain point estimaies.

The classical approach is to equate the observed and expected mean squares and solve for the variance components. This is a moment estimator. If the expected value of the ith sum of squares is

E(SS0 = aiicxi² + a2cr₂^{2 +}•••+ a*CTk² + a«cr²    (8)

for i=0,l,2,...,k. NotÄ™ that there are k+1 variance components; thercfore, we must have k+1 sums of squares. Equaiing E(SSj) to the observed SSi yields

SS« = an^i² + ai2^₂² + ••• + auc^*² + aio^ ²

for i=0,l,2,k. In matrix notation this becomes

SS = Aa²

and the variance component estimates are <x² = A^-ISS

For balanced design models, the estimators are uniformly minimum variance unbiased.

Unfortunately, the moment estimation can yield negative estimates of sonie of the variance components, an embarrassing, if not disturbing, result. For this reason, authors have recommended restricted maximum likelihood estimators or minimum norm Ä…uadratic unbiased estimators (MINQUE). See Milliken and Johnson (1984) for detaiis of these methods. SAS version 6, PROC VARCOMP implements these algorithms, as well as the method of moments. Montgomery and Runger (1993b) illustrate the application of this program to a random model for a measurement capability study. Another example is presented below.

Hypothesis testing about variance components usually relies on using the sum of squares for the ANOVA to construct ratios of mean sÄ…uares that are distributed as F random variables, at least approximately. The analyst must find (or construct) a linear combination of mean squares for the numerator and denominator of the ratio such that the diflference in expected value between the numerator and denominator mean sÄ…uare is the variance component in question. This is usually called a Satterthwaite (1946) approximation. If the maximum likelihood method