Another approach to computing consensus estimates is to try to get an explicit estimate of the intermethod (intergroup) variance, and sum that with a pooled estimate of the within-method (within-group) variance, using the combination to weight the contributions from the different methods to form a consensus mean. An estimation equation approach for the determination of the between-group variance developed by Mandel and Paule [21] has often been used at NIST, particularly in the certification of standard reference materials. Vangel-Rukhin [22] showed that the Mandel-Paule solution can be interpreted as an approximation to maximum likelihood. While most useful when the number of contributing methods is large, we compute the MPVR estimates of the inter method variances here and add them to a standard pooled estimates of the within method variances, take the square root to get an estimate of the overall standard errors, and multiply by an expansion factor of 2. This method has the virtue of explicitly quantifying within-group and between-group variation, and being rooted in a broadly applicable important general method of mathematical statistics, namely maximum likelihood. The drawback is that the estimation equation solution and formulations of its variance are strictly speaking only asymptotically correct, so that MPVR estimates are better for data sets where many (e.g. >10) methods/groups are present.
Comparing the consensus means and uncertainties obtained across the comparable sets of RM/analyte data, one can observe the following. Consensus means agree very well across all sets. Naive and BOB must agree because they are both unweighted means of (group) means. But the weighting scheme of MPVR (maximum likelihood) in general either doesn't move the consensus away from the unweighted mean - at least to within the reportable precision - or perturbs it only very slightly. Comparing the consensus uncertainties as calculated here, one can observe the general pattern: U(naive) < U(BOB) < U(MPVR). U(naive)is actually computed from an incorrect, or overly naive, underlying statistical model, one that assumes that all the varying groups' data come from a single parent normal population (same mean, same variance). Consequently U(naive) will, typically, lower bound the other estimates, and represents an anticonservative, and probably incorrect, estimate of the overall variation. U(BOB), on the other hand, makes no such naive population assumption, and takes explicitly into account intermethod biases. The U(BOB) estimates here represent credible consensus uncertainties. U(MPVR) estimates calculated from formula [19] of Vangel-Rukhin [22], representing confidence intervals about the consensus mean, would in fact - experience suggests - agree rather closely with the BOB estimates. However, here we have elected to combine pooled-across-group within-variance in quadrature with the MPVR estimate of between-variance, and multiply by an expansion factor of 2, to get numbers that can be thought of as representing over-conservative confidence intervals, or less-than-conservative prediction intervals for the consensus means. Because of the discrepancies among the methods clearly visible in the various boxplot figures in this publication, we elect to use these slightly inflated MPVR uncertainties as the certification numbers for the respective SRM's and listed in Table 6.