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## Unbalanced Variance Components

I need some help with an unbalanced Gauge Repeatability and Reproducibility (GRR) calculation.

Suppose I have a reading of zero for operator A and readings of 2 and 4 for operator B.  All readings are taken on the same unit.  I wish to estimate the variance within operator and the variance from operator to operator, using the Expected Mean Square (EMS) method.

Here’s what I think I do understand correctly:

SSE = MSE = (0-0)^2 + (2-3)^2 + (4-3)^2 = 2

Because the data vector is (0,2,4) and the operator means are (0,3,3).

SS operators = MS operators = (0-2)^2 + (3-2)^2 + (3-2)^2 = 6

Because the operator means are (0,3,3) and grand mean is (2,2,2).

SS total = 2 * MS total = (0-2)^2 + (2-2)^2 + (4-2)^2 = 8

Because the vector is (0,2,4) and the grand means are (2,2,2).

Here’s what I cannot figure out.    The JMP output gives me the expected values of the mean squares:

E(MSE) = variance(within operators)

E(MS operators) = variance(within operators)  +  ( 4/3 ) * variance(operator to operator)

How can I derive that factor of (4/3)?    Where does that come from?

Perhaps some linear model matrix theory, that I am just missing now?

I tried reviewing the appropriate passages in Searle's Variance Components book, but I cannot follow it.

Note:   I am not at all asking if it is appropriate to estimate these two variance components out of these three readings.  I feel that almost surely, it is not.  Instead, the three reading example is just that: a simple example from which I hope to figure out where that factor of (4/3) comes.

Regards, Joel Dobson