Univariate linear calibration via replicated errors-in-variables model

Abstract

In this paper, we deal with the comparative calibration problem, i.e. with the situation when one instrument or measurement technique is calibrated against another, each of which is subject to the measurement error. We propose an approximate, small sample, calibration confidence interval of the unknown true value of the measured substance in units of the more precise instrument, given measurement in units of the less precise instrument. Here we deal with the simplest case—single-use linear univariate calibration, i.e. the case in which we assume linear relationship between the two measurement techniques (instruments), and, further, that the calibration procedure is conducted in order to obtain one value for an unknown, reported together with an interval estimate. The method for deriving the approximate confidence interval is based on estimation of the calibration line via the replicated errors-in-variables model. The model is locally linearized and the Wald-type F-statistic is constructed. An essential point in this approach is the use of the F-approximation of the distribution of the F-statistic suggested by Kenward and Roger [Kenward, M.G. and Roger, J.H., 1997, Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53, 983–997]. The statistical properties of the proposed interval estimator are verified by simulations for wide spectrum of experimental designs and compared with two standard univariate interval estimates—the classical approach for deriving the calibration interval was proposed by Eisenhart [Eisenhart, C., 1939, The interpretation of certain methods and their use in biological and industrial research. Annals of Mathematical Statistics, 10, 162–186] and the inverse method was proposed by Krutchkoff [Krutchkoff, R.G., 1967, Classical and inverse methods of calibration. Technometrics, 9, 425–439].