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This paper critically reviews the problem of over-fitting in multivariate calibration and the conventional validation-based approach to avoid it. It proposes a randomization test that enables one to assess the statistical significance of each component that enters the model. This alternative is compared with cross-validation and independent test set validation for the calibration of a near-infrared spectral data set using partial least squares (PLS) regression. The results indicate that the alternative approach is more objective, since, unlike the validation-based approach, it does not require the use of 'soft' decision rules. The alternative approach therefore appears to be a useful addition to the chemometrician's toolbox.

A new approach is developed for estimating the limit of detection in second-order bilinear calibration with the generalized rank annihilation method (GRAM). The proposed estimator is based on recently derived expressions for prediction variance and bias. It follows the latest IUPAC recommendations in the sense that it concisely accounts for the probabilities of committing both types I and II errors, i.e. false positive and false negative declarations, respectively. The estimator has been extensively validated with simulated data, yielding promising results.

Four methods are compared for quantifying the effect of measurement errors on the uncertainty in bilinear model predictions. These methods amount to (1) evaluating an approximate expression for prediction variance, (2) bootstrapping residuals left after fitting the data matrices using a singular value decomposition, (3) adding noise from an appropriate distribution to the original data, and (4) jack-knifing rows and columns of the data matrices. The comparison is carried out for liquid chromatography/ultraviolet data obtained from Malinowski and the models are constructed using the generalized rank annihilation method. It is found that the first three methods give highly consistent results, whereas the jack-knife yields uncertainty estimates that have no clear interpretation.

This work presents a new approach for calculating multivariate detection limits for the commonly used classical or direct calibration models. The derived estimator, which is in accordance with latest IUPAC recommendations, accounts for the different sources of error related to the calibration and prediction steps. Since the multivariate detection limit for a given analyte is influenced by the presence of other components in the sample, a different detection limit is calculated for each analyte and analysed sample, at the chosen significance levels α and β . The proposed methodology has been experimentally validated by determining four pesticides in water using a FIA method with diode-array detection. The results compare favourably with the ones obtained using previously proposed estimators.

Pseudorank estimation is a ubiquitous problem in multivariate data analysis. Many pseudorank estimation methods currently in use are based on the size of the eigenvalues calculated in abstract factor analysis (AFA). The basic assumption behind these methods is that the eigenvalues, when ordered according to their size, form two distinct sets, namely the so-called primary eigenvalues, that explain the systematic variation in the data together with embedded error, and the so-called secondary eigenvalues, that consist only of noise. This paper shows that a strict separation of eigenvalues in a primary and secondary set can not be expected a priori if the noise in the data is heteroscedastic. The main conclusion is that proper data pre-treatment is required to facilitate AFA-based pseudorank estimation.

In the univariate context analytical figures of merit such as sensitivity play a key role in the selection, validation and optimisation of analytical determinations. In the past, numerous definitions of multivariate sensitivity have been proposed to achieve the same goal in the multivariate context. Two definitions of multivariate sensitivity particularly stand out since they can be shown to relate in a transparent way to other figures of merit such as limit of detection and signal-to-noise ratio: the one due to Lorber and the one due to Bergmann, von Oepen and Zinn (BOZ). Lorber's definition has been recently criticised for containing selectivity information: Lorber's sensitivity depends on the overlap between the spectra of the analyte and the interferents whereas the BOZ sensitivity is a function of the analyte spectrum only. This paper gives a convenient explanation for this counter-intuitive aspect of Lorber's definition and states a recommendation for its application. A practical data set obtained from Prof. Malinowski serves to illustrate theoretical ideas.

The estimation of the pseudorank of a matrix, i.e., the rank of a matrix in the absence of measurement error, is a major problem in multivariate data analysis. In the practice of analytical chemistry it is often even the only problem. An important example is the determination of the purity of a chromatographic peak. In this paper we discuss three pseudorank estimation methods that make use of prior knowledge about the size of the measurement error. The first method (Method A) is based on the standard errors in the diagonal elements of the row-echelon form of the matrix, the second method (Method B) is based on the eigenvalues of principal component analysis (PCA) and the third method (a t-test) is based on the singular values. Methods A and B are modifications of methods that are well known in analytical chemistry. However, these methods cannot provide significance levels for the estimated pseudorank. This holds for the original methods as well as the present modifications. The main reason for introducing these modifications is that in this way relationships are established between the t-test and methods that are already known. The aspects that are covered in this paper include the sampling distribution of the test statistic, the number of degrees of freedom to be used in the test, the adequacy of theoretical predictions and the bias that results from random measurement noise. The object of this paper is to demonstrate that using prior knowledge about the size of the measurement error may yield powerful pseudorank estimation methods. This is illustrated by comparing the significance levels obtained by the t-test and Malinowski's F-test. The t-test yields sharper significance levels for experimental data obtained from the literature as well as simulated data. This can be satisfactorily explained by the larger number of degrees of freedom that is employed in this test. The viability of the new t-test is supported by a thorough evaluation of the test data by a large number of conventional methods. As a remarkable by-product of the present investigation we find that a plot of the singular values yields a promising graphical pseudorank estimation method. Graphical methods have proved their use in the past in the case that the size of the measurement error is unknown. This new graphical method therefore provides a natural complement to the t-test.

The jack-knife is a resampling method that is increasingly used for assessing the uncertainty in regression coefficient estimates, even when the predictor variables (X) are designed. Application of the jack-knife to designed data, however, violates a basic assumption underlying all resampling methods, namely that the resampled units should constitute a random sample from some distribution; the idea is to 'resample the sample.' This paper advances the view that the jack-knife should not be applied to estimate the uncertainty in regression coefficient estimates obtained from designed data, since a sound alternative is available. A literature data set is re-analyzed to lend support to this view.