Uncertainty estimates for calibration measurements

The volume of liquid in a process tank is typically determined from a calibration function that relates tank volume to some measure of liquid height. This calibration equation is determined from data acquired during one or more calibration runs. Solutions are proposed for two statistical problems that arise when the calibration function is derived and used: that of computing plausible variance estimates if the data exhibit statistically significant run-to-run differences and that of computing confidence intervals that hold simultaneously for an arbitrary number of volume determinations. When statistically significant run-to-run differences are ignored and standard statistical procedures are applied to data, resulting variance estimates can seriously underestimate actual measurement variability. In the proposed procedure, the variance of a predicted new response is estimated as the sum of two components: one due to run-to-run differences and one due to the fitted model. Depending upon the nature of the model(s) required to fit the data, estimates of the response variance components take one of three forms. For linear functions, estimates of component variances are derived from sums of squares that are computed during an analysis of covariance. The estimated calibration function is used to determine the volumes that correspond to observed liquidmore » heights. Confidence intervals are required that hold simultaneously for arbitrarily many volume determinations. A procedure for computing simultaneous discrimination intervals employs variance estimates computed as described above in Scheffe-type simultaneous prediction intervals. These intervals are combined with tolerance intervals on the response variable to obtain equations that can be solved to obtain the desired discrimination intervals for volume determinations.« less