Calibration and confidence interval: The minimum allowable concentration

The calibration curve is based on a deterministic relationship between Y , the observed response, f , the instrumental function, and x , the concentration of the analyte. When the instrumental function is linear the model is\documentclass{article}\pagestyle{empty}\begin{document}$$ Y = a_0 + a_1 x + \varepsilon $$\end{document} with a 0 the blank and a 1 the sensitivity. In fact calibratio is a two‐step procedure consisting of the ‘calibration’, itself (computation of the curve coefficients) and the ‘counter‐calibration’, when the curve is used to predict the concentration of an unknown sample. Least‐squares regression (LSR) can be simply used in order to solve the first step. However, an error may occur when the classical results of LSR are used to compute the detection limit. The detection limit can be defined as the smallest concentration that can be detected with certainty from the blank with a risk of error α and a probability of being right 1 − β. If we define the critical level (CL) as\documentclass{article}\pagestyle{empty}\begin{document}$$ y({\rm CL)} = a_0 + t_{(1 - \alpha)} {\rm}\sqrt {{\rm var(}a_0)} $$\end{document} the detection limit (DL) is then\documentclass{article}\pagestyle{empty}\begin{document}$$ y({\rm DL)} = y({\rm CL}) + t_{(1 - \beta)} {\rm}\sqrt {{\rm var(}a_0)} $$\end{document} Since the units used to express y (DL) are the same as the units of y , it must be divided by a 1 in order to be expressed as a concentration. This creates a contradiction however, because a 1 is a random variable. Therefore the detection limit cannot be computed according to its previous definition. If we use tolerance interval theory, it is possible to define a new criterion called the minimum allowable concentration which does not have such a drawback. This new concept is based on an improved approach to counter‐calibration.