Research Article

Assessing interlaboratory comparison data adjustment procedures

Data Science Group, National Physical Laboratory, Teddington, Middlesex, UK

^* Corresponding author: kavya.jagan@npl.co.uk

Received: 4 December 2018
Accepted: 16 March 2019

Abstract

Interlaboratory comparisons (ILCs) are one of the key activities in metrology. Estimates x = (x₁,…, x_n) ^T of a measurand α along with their associated standard uncertainties u₀ = (u_0,1,…, u_0,n) ^T, u_0,j  = u₀ (x_j) are provided by each of n laboratories. Employing a model of the formx_j ∈ N(α, v_0,j),  j = 1,…,n,

v_0,j = u_0,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 v_j  ≥  v_0,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.