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
