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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     

Community Trekker

Re: Unbalanced Variance Components

A friend suggested that I consult the book Design of Experiments A Realistic Approach from 1974 by Virgil L Anderson and Robert A. McLean. 

After spending several hours reading most of this book, I found what I needed in Appendix 12 of the book.  Using the method from Appendix 12, I was able to write out the expected value of the sums of squares.   My Expected Mean Squares results match to what JMP software gave me when using the EMS method in an unbalanced one-way random effects model.    To see the derivation, please review the attached PDF file.  

Let me know if this was helpful.

If nothing else, I know I learned something from this exercise.

Regards, Joel Dobson