Quantification and handling of sampling errors in instrumental measurements: a case study

Instrumental measurements are often used to represent a whole object even though only a small part of the object is actually measured. This can introduce an error due to the inhomogeneity of the product. Together with other errors resulting from the measuring process, such errors may have a serious impact on the results when the instrumental measurements are used for multivariate regression and prediction. This paper gives examples of how errors influencing the predictions obtained by a multivariate regression model can be quantified and handled. Only random errors are considered here, while in certain situations, the effect of systematic errors is also considerable. The relevant errors contributing to the prediction error are: error in instrumental measurements (x-error), error in reference measurements (y-error), error in the estimated calibration model (regression coefficient error) and model error (due to non-linearities, lack of x-information, etc.). The size of the four error contributions are estimated by an errors-in-variables approach, and the importance of the different errors is evaluated. An approach based on the theory of the measurement error models is used to show how the influence of errors in the instrumental measurements can be partly adjusted for. The approach can be used when the number of replicates used for calibration and prediction differs. With few replicates during prediction and when the error in the instrumental measurements has significant impact on the predictions, the approach seems to provide more accurate predictions than the naive approach. Predictions of water content of fish fillets from low-field NMR relaxations are used as examples to show the applicability of the methods.