Fresenius J Anal Chem (1997) 357: 789—795  Springer-Verlag 1997 ORIGINAL PAPER Klaas Faber · Bruce R. Kowalski Improved estimation of the limit of detection in multivariate calibration Received: 3 April 1996/Revised: 31 July 1996/Accepted: 2 August 1996 Abstract The limit of detection is one of the most important performance characteristics of an analytical procedure. This paper critically examines a recently introduced expression for estimating a multivariate limit of detection. The main result is that the proposed expression is inconsistent. An application from induc- tively coupled plasma — optical emission spectrometry (ICP-OES) shows that the true limit of detection may be overestimated by more than 20%. In addition, the practical evaluation of the proposed expression amounts to an iterative procedure, which is unattrac- tive from an application point of view. A modification is derived that solves both problems. The relevant ex- pressions are discussed with respect to their interpreta- bility in terms of experimental design. Introduction The limit of detection is a quantitative measure for the ability of an analytical procedure to handle trace amounts of analyte. It is thereby one of the most important figures of merit. In a groundbreaking paper Currie [1] presented a literature survey that revealed numerous definitions for limit of detection. He demon- strated that their practical evaluation could lead to results that span three orders of magnitude. The large number of contradicting definitions in the analytical chemistry literature has led to considerable confusion on the subject. This confusion is the reason why, before addressing the main issue of this paper, i.e. examining a recently introduced multivariate limit of detection K. Faber · B.R. Kowalski (¥) Center for Process Analytical Chemistry, University of Washington, Box 351700, Seattle, WA 98195, U.S.A. Present address: Netherlands Forensic Science Institute, Volmerlaan 17, 2288 GD Rijswijk, The Netherlands estimator, we will briefly review the most important concepts. We will adopt the nomenclature introduced by Currie [1], which has recently been recommended by the Union of Pure and Applied Chemistry (IUPAC) [2]. Three limiting values are distinguished for the instru- ment response and corresponding analyte concentra- tion, i.e. the limit of decision, limit of detection and limit of determination. The limit of decision, ¸  , is the c  riti- cal level that allows for the a posteriori decision about the presence of an analyte, i.e. after the measurement is made. It is given by ¸  "k  ·   (1) where k  is a multiplier that leads to a probability  of an error of the first kind (false positive) and   is the standard deviation in the measurement or concentra- tion estimate resulting from a blank sample. The sub- script ‘‘0’’ indicates that this expression evaluates the tail of a distribution under the null-hypothesis H  , i.e. the analyte is not present. The limit of decision largely rules out the possibility of false positive decisions, but has the serious weakness that a constituent will not be detected with 50% probability if its true signal or concentration is equal to ¸  . In other words it allows for a large error of the second kind (false negative). The limit of detection, ¸  , is defined as the level that allows one to a priori assess the d  etectability of a com- ponent, i.e. before the measurement is made. It is given by ¸  "¸  #k  ·   (2) where k  is a multiplier that leads to a probability  of an error of the second kind and   is the standard deviation in the measurement or concentration esti- mate resulting from a sample that contains the analyte at the level of the limit of detection. The subscript ‘‘A’’ indicates that this expression evaluates the tail of a dis- tribution under the alternative hypothesis H  , i.e. the analyte is present at a ‘‘detectable’’ amount.