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Analytical figures of merit such as selectivity are performance characteristics of an analytical determination. They can be used to select between potentially useful methods and to evaluate or optimize a method that is already in use. The importance of analytical figures of merit is generally accepted.

A case in point is how the premium journal of the field rightly describes itself:

Analytical Chemistry is a peer-reviewed research journal that explores the latest concepts in analytical measurements and the best new ways to increase accuracy, selectivity, sensitivity, and reproducibility.

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This page is organized as follows:


Univariate data: official literature

Analytical figures of merit are well-defined for univariate (i.e. zeroth-order) calibration, where a single number is measured for each sample. See, for example:


Consider a linear calibration function that relates content (x) to signal (y) according to the classical model. An important figure of merit is the sensitivity, which in this case is given by the slope of the calibration function:

FOM1.gif

Figure FOM 1: Linear calibration function without intercept - the simplest classical model. The sensitivity (s) is inversely proportional to the amount of error propagation when predicting the true content from the noisy signal. It is therefore the combination of sufficiently low instrument noise (y) and high sensitivity that ensures small changes in content (x) to be detectable. In addition, these characteristics determine the number and design of samples required to construct a sufficiently predictive model.


Other figures of merit such as signal-to-noise ratio and selectivity have an equally straightforward and intuitive interpretation.


Generalization to multivariate data and beyond


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The situation is obviously more complicated when a more complex data structure is measured for a single chemical sample and accordingly more sophisticated calibration methods are used to predict analyte concentrations. The theory of analytical chemistry enables one to generalize the figures of merit agreed upon for univariate calibration to more complex instrumental data structures. We have contributed to a generalization that directly relates to the uncertainty in the determination, quantified as a standard error of prediction. Some properties of this generalization are:

  • The figures of merit are analyte-specific. In other words, each analyte can be characterized by a different sensitivity, selectivity or signal-to-noise ratio. An analyte-specific signal-to-noise ratio may seem counterintuitive, because this figure of merit is traditionally used to characterize the instrument. However, this tradition has a univariate origin. In the multivariate context, a low noise level is of little use if severe spectral overlap (ideally non-existent in the univariate setting) hinders the construction of a reasonable model.
  • The figures of merit are consistent in the sense that they can be used to compare instruments of different complexity. For example, they can be used to quantify the gain in selectivity when moving from GC-MS to GC-MS-MS.


For a comprehensive review until 2005, see:

  • A. Olivieri, N.M. Faber, J. Ferré, R. Boqué, J.H. Kalivas and H. Mark
    Guidelines for calibration in analytical chemistry
    Part 3. Uncertainty estimation and figures of merit for multivariate calibration
    Pure & Applied Chemistry, 78 (2006) 633-661
    Download (icon_pdf.gif=645 kB: © IUPAC 2006)


The example of multivariate selectivity


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The International Union for Pure and Applied Chemistry (IUPAC) recommends the use of the term 'selectivity' in favour of 'specificity', see:

  • J. Vessman, R.I. Stefan, J.F. van Staden, K. Danzer, W. Lindner, D.T. Burns, A. Fajgelj and H. Müller
    Selectivity in analytical chemistry
    Pure & Applied Chemistry, 73 (2001) 1381-1386


In this document, selectivity is verbally defined as:

Selectivity refers to the extent to which a method can be used to determine particular analytes in mixtures or matrices without interferences from other components of similar behavior.

In the final section (6. Problems still to be solved), Vessman et al. mention that practical ways of calculating or quantifying selectivity constitute a topic for further research.

We have reviewed the subject in:

  • N.M. Faber, J. Ferré, R. Boqué and J.H. Kalivas
    Quantifying selectivity in spectrophotometric multicomponent analysis
    Trends in Analytical Chemistry, 22 (2003) 352-361


In this article, we reason that a selectivity criterion should satisfy the following requirements:

  1. Injectivity: any change of a single element in the calibration matrix should result in changes of the criterion.
  2. Continuity: minor or major changes in the partial sensitivities should induce minor or major changes in the criterion, respectively.
  3. Distinct value: for example, the fuzzy result infinity for full selectivity is unacceptable.
  4. Prediction uncertainty: the criterion should relate to prediction uncertainty, which implies that it be analyte-specific.
  5. Overdetermined systems: the criterion should allow for spectra with more wavelengths than components.
  6. Multiway data: the criterion should allow for a consistent generalization to multiway data (e.g., hyphenated instruments).


Among many currently known definitions, the only one that fits the description has been independently proposed in:

  • A. Lorber
    Error propagation and figures of merit for quantitation by solving matrix equations
    Analytical Chemistry, 58 (1986) 1167-1172
  • G. Bergmann, B. von Oepen and P. Zinn
    Improvement in the definitions of sensitivity and selectivity
    Analytical Chemistry, 59 (1987) 2522-2526


For example, its consistent generalization to multiway data (requirement 6) is discussed in:

  • N.J. Messick, J.H. Kalivas and P.M. Lang
    Selectivity and related measures for nth-order data
    Analytical Chemistry, 68 (1996) 1572-1579
  • N.M. Faber, A. Lorber and B.R. Kowalski
    Analytical figures of merit for tensorial calibration
    Journal of Chemometrics, 11 (1997) 419-461
  • N.M. Faber, J. Ferré, R. Boqué and J.H. Kalivas
    Second-order bilinear calibration: the effect of vectorising the data matrices of the calibration set Chemometrics and Intelligent Laboratory Systems, 63 (2002) 107-116


It is emphasized that this definition simultaneously accounts for all interferences in the mixture, just like the model that it intends to characterize. Only very recently has larger focus been directed towards quantifying the impact of individual interferences on multivariate analyte predictions, leading to a definition of pairwise multivariate selectivity coefficients:

  • M.A. Arnold, G.W. Small, D. Xiang, J. Qui and D.W. Murhammer
    Pure component selectivity analysis of multivariate calibration models from near-infrared spectra
    Analytical Chemistry, 76 (2004) 2583-2590
  • C.D. Brown and T.D. Ridder
    Framework for multivariate selectivity analysis, Part I: Theoretical and practical merits
    Applied Spectroscopy, 59 (2005) 787-803
  • T.D. Ridder, C.D. Brown and B.J. Ver Steeg
    Framework for multivariate selectivity analysis, Part II: Experimental applications
    Applied Spectroscopy, 59 (2005) 804-815


The radically different standpoint taken in that work may lead to a critical reexamination of multivariate selectivity assessment.


References & further information


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Open blue.gif Open a list of references. This list is rather extensive because recent studies have revealed a close connection between the calculation of multivariate analytical figures of merit and data preprocessing methods such as orthogonal signal correction, see e.g.:

  • A.C. Olivieri
    A simple approach to uncertainty propagation in preprocessed multivariate calibration
    Journal of Chemometrics, 16 (2002) 207-217
  • J. Ferré and N.M. Faber
    Generalization of rank reduction problems with Wedderburn's formula
    Journal of Chemometrics, 17 (2003) 603-607


For further information, please contact Joan Ferré: Joan Ferre.jpg