Assessing interlaboratory comparison data adjustment procedures

Abstract

Interlaboratory comparisons (ILCs) are one of the key activities in metrology. Estimates x = (x1,…, xn) T of a measurand α along with their associated standard uncertainties u0 = (u0,1,…, u0,n) T, u0,j  = u0 (xj) are provided by each of n laboratories. Employing a model of the form xj ∈ N(α, v0,j),  j = 1,…,n, v0,j = u0,j2, we may wish to find a consensus value for α. A χ2 test can be used to assess the degree to which the spread of the estimates x are consistent with the stated uncertainties u0. If they are judged to be inconsistent, then an adjustment procedure can be applied to determine vj  ≥  v0,j, so that x and v represent consistency. The underlying assumption behind this approach is that some or all of the laboratories have underestimated or neglected some uncertainty contributions, sometimes referred to as ‘dark uncertainty’, and the adjusted v provides an estimate of this dark uncertainty derived from the complete set of laboratory results. There are many such adjustment procedures, including the Birge and Mandel–Paule (M-P) procedures. In implementing an adjustment procedure, a desirable objective is to make as minimal an adjustment as necessary in order to bring about the required degree of consistency. In this paper, we discuss the use of relative entropy, also known as the Kullback–Leibler divergence, as a measure of the degree of adjustment. We consider parameterising v = v (b) as a function of parameters b with the input v0 = v (b0) for some b0. We look to perturb b from b0 to bring about consistency in a way that minimises how far b is from b0 in terms of the relative entropy or Kullback–Leibler divergence.