Because replicate measurements are not included in Table 1 of Randich et al., we are unable to assess the existence of an interaction term between position and lot; such an interaction, if it exists, must be incorporated into the error term, which also includes simple measurement error. The parameters of the model (_{k}, ρ_{ik}, λ_{jk}) can also be estimated more robustly via median polish (Ref. 2), which uses medians rather than means and thus provides more robust estimates, particularly when the data include a few outliers or extreme values that will adversely affect sample means (but not sample medians). This additive model was verified for each element by using Tukey’s diagnostic plot for twoway tables (Ref. 2, 3).
The conventional way to assess the signficance of the two factors is to compare the variance of the position effects, Var and the variance of the lot effects, Var scaled to the level of a single observation, with the variance of the estimated error term, Var(r_{ijk}). Under the null hypothesis that all ρ_{ik} are zero (position has no particular effect on the measurements, beyond the anticipated measurement error), the ratio of 28·Var to Var should follow an F distribution with two and 54 degrees of freedom; ratios that exceed 3.168 would be evidence that position affects measurements more than could be expected from mere measurement error.
Table G.1 below provides the results of the twoway analysis of variance with two factors, position and lot, for each element. The variances of the effects, scaled to the level of a single observation, are given in the column headed “Mean Sq”; the ratio of the mean squares is given under “F Value”; and the P value of
TABLE G.1 Analyses of Variance for Log(Measurement) Using Table 1 in Randich et al. (Ref. 1)
Sb 
Df 
Sum Sq 
Mean Sq 
F Value 
Pr (> F) 
MS (median polish) 
Position 
2 
0.001806 
0.000903 
2.9449 
0.06111 
0.004 
Lot 
27 
0.111378 
0.004125 
13.4514 
1.386e15 
0.0042 
Residuals 
54 
0.016560 
0.000307 



Sn 
Df 
Sum Sq 
Mean Sq 
F Value 
Pr (> F) 
MS (median polish) 
Position 
2 
2.701 
1.351 
7.5676 
0.001267 
0.2345 
Lot 
27 
147.703 
5.470 
30.6527 
<2.2e16 
6.0735 
Residuals 
54 
9.637 
0.178 



Cu 
Df 
Sum Sq 
Mean Sq 
F Value 
Pr (> F) 
MS (median polish) 
Position 
2 
0.006 
0.003 
0.1462 
0.8643 
0.00003 
Lot 
27 
102.395 
3.792 
176.9645 
<2e16 
4.1465 
Residuals 
54 
1.157 
0.021 


