# Improving accuracy of total variance through variance components?

I'm following a measurement guidance that specifies a technique to improve the estimate of total variance by extracting individual variance components and performing a sum of squares. I'm not a statistician, so I'm looking for somebody to validate the methodology defined by the guidance. This is best explained by an example:

We sell a product that needs to meet particular performance requirements and is composed of two separate pieces which may potentially be interchanged outside of our control. As such, we cannot test every possible combination. To show that the product meets the requirements, we perform a study that samples the performance of pairings in a matrix, with two measurements on each pair to capture measurement error. Here's a data set for this:

The guidance states that the sample variance under-estimates the total variance and the solution is to extract the individual variance components and then combine them separately. The calculations, as prescribed by the guidance, are below:

Most of this technique actually makes a lot of sense to me. I see the value in applying the components separately to account for the different sample sizes, but there are also a few aspects of this that I don't fully understand. In particular:

1. The calculation of the measurement variance (Eq 3) is just taking the average of the variance at each pairing. Is this the proper technique to extract measurement variance?
2. Why do the component variances (Eqs 4 & 5) subtract out the variance in the measurement mean? This is particularly troublesome where the component variance is less than the subtracted variance in the measurement mean (as with Eq 5). Should we just set that component to zero in those cases?
3. In the calculation of the total variance of the mean (Eq 7), shouldn't the divisor for $S^2_{meas}$ actually be pq, not pqr?

Aside from this guidance, I've been unable to find any other examples that use this exact method. The data setup looks like a two-way crossed ANOVA but the analysis is different. For reference, the guidance refers to the book by Poduri S.R.S Rau, "Variance Components Estimation: Mixed Models, Methodologies, and Applications" [Chapman & Hall, 1997, pp 12-38] as the source of the technique. I'd appreciate it if anybody can validate this technique and help me understand the questions I posed.