Olav M. Kvalheim

Multivariate data analysis in pharmaceutics: A tutorial review

We provide an overview of latent variable methods used in pharmaceutics and integrated with advanced characterization techniques such as vibrational spectroscopy. The basics of the most common latent variable methods, principal component analysis (PCA), principal component regression (PCR) and partial least-squares (PLS) regression, are presented. Multiple linear regression (MLR) and methods for improved interpretation, variable selection, classification and validation are also briefly discussed. Extensive use of the methods is demonstrated by compilation of the recent literature.

Interpretation of latent-variable regression models

Abstract In this work, we show that the projections of the predictors on the normalized regression vectors represent a target rotation with the responses (concentration vectors) as targets. By means of this operation the predictive ability of a latent-variable (LV) regression model and the importance of each predictor for all the responses is obtained. The two features can be portrayed simultaneously and quantitatively in an LV regression BIPLOT display. This graph shows how modelled interferents influence prediction, information as important as the detection of and correction for unmodelled interferents when using a regression model for prediction. For samples characterized by whole digital profiles rather than a collection of peaks, graphs showing the covariances between the responses and the original or the reproduced predictor space appear to provide the most useful information for interpreting an LV regression model.

Maturity determination of organic matter in coals using the methylphenanthrene distribution

Abstract The distribution of phenanthrene and monomethylphenanthrenes in extracts of 15 different coals was determined by gas chromatography. The maturity as measured by vitrinite reflectance varied from 0.53% to 1.29% R 0 , covering the maturity range normally associated with the “oil window” ( 0.5–1.2% R 0 ). p ]Principal component analysis followed by factor rotation reveals that coal maturity correlates only with the distribution of the mono-methylphenanthrenes and not with the relative abundance of phenanthrene. This implies a methylphenanthrene distribution fraction (MPDF) linearly related to vitrinite reflectance for coal with maturity within the oil window.

Discriminating variable test and selectivity ratio plot: quantitative tools for interpretation and variable (biomarker) selection in complex spectral or chromatographic profiles.

The discriminating variable (DIVA) test and the selectivity ratio (SR) plot are developed as quantitative tools for revealing the variables in spectral or chromatographic profiles discriminating best between two groups of samples. The SR plot is visually similar to a spectrum or a chromatogram, but with the most intense regions corresponding to the most discriminating variables. Thus, the variables with highest SR represent the variables most important for interpretation of differences between groups. Regions with variables that are positively or negatively correlated to each other are displayed as corresponding negative and positive regions in the SR plot. The nonparametric DIVA test is designed for connecting SR to discriminatory ability of a variable quantified as probability for correct classification. A mean probability for a certain SR range is calculated as the mean correct classification rate (MCCR) for all variables in the same SR interval. The MCCR is thus similar to a mean sensitivity in each SR interval. In addition to the ranking of all variables according to their discriminatory ability provided by the SR plot, the DIVA test connects a probability measure to each SR interval. Thus, the DIVA test makes it possible to objectively define thresholds corresponding to mean probability levels in the SR plot and provides a quantitative means to select discriminating variables. In order to validate the approach, samples of untreated cerebrospinal fluid (CSF) and samples spiked with a multicomponent peptide standard were analyzed by matrix-assisted laser desorption ionization (MALDI) mass spectrometry. The differences in the multivariate spectral profiles of the two groups were revealed using partial least-squares discriminant analysis (PLS-DA) followed by target projection (TP). The most discriminating mass-to-charge (m/z) regions were revealed by calculating the ratio of explained to unexplained variance for each m/z number on the target-projected component and displaying this measure in SR plots with quantitative boundaries determined from the DIVA test. The results are compared to some established methods for variable selection.

Spectra of water in the near- and mid-infrared region

Abstract Spectra of water have been acquired in the mid-infrared (MIR) and the near-infrared (NIR) region in the temperature range 2–96°C and 4–52°C, respectively. Loading plots from partial-least-squares regression were used to locate isosbestic points in the spectra bands of water. By means of least-squares, the original spectral profiles have been resolved into two spectra, one increasing and the other decreasing with temperature. Concentration profiles of the two structurally different water associations, for the investigated temperature range, were obtained by use of evolutionary curve resolution and first-order differentiation of the MIR spectra. Utilising information from the concentration profiles obtained in the MIR, the NIR spectra were resolved. Relative concentrations were obtained using spectral intensities from the isosbestic points. The complexity in both the fundamental and overtone region of the spectra shows that both structures of water are involved in H-bonding. This result indicates a pseudo-first-order reaction in water, either between an open and a more dense state, or between a rigid, strongly H-bonded state and a more loosely H-bonded state. The cross-correlation pattern between the two regions can be ascribed to the temperature-induced variation in the concentration of the two associations.

Evolving factor analysis in the presence of heteroscedastic noise

Abstract Evolving factor analysis (EFA) is a promixing method for the analysis of multivariate data with an intrinsic order. When applying EFA for assessment of peak homogeneity in liquid chromatography, one has to be aware of instrumental and experimental difficulties. Heteroscedasticity is one of the most serious problems and leads to additional eigenvalues that may be misinterpreted as being due to an impurity. After appropriate data pretreatment, the fixed-size window EFA technique proved successful for peak purity control in liquid chromatography with photodiode-array detection. Less than 1% of a spectrally similar impurity could be detected for Rs values as low as 0.3.

White, grey and black multicomponent systems: a classification of mixture problems and methods for their quantitative analysis

Abstract “Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry … If mathematical analysis should ever hold a prominent place in chemistry — an aberration which is happily almost impossible — it would cause a rapid and widespread degeneration of that science.” Auguste Comte, Philosophie Positive (1830) Liang, Y.-Z., Kvalheim, O.M. and Manne, R., 1993. White, grey and black multicomponent systems. A classification of mixture problems and methods for their quantitative analysis. Chemometrics and Intelligent Laboratory Systems, 18: 235–250. Multivariate calibration and resolution methods for handling samples of chemical mixtures are examined from the point of view of the analytical chemist. The methods are classified concordant to three different kinds of analytical mixture systems; i.e. ‘white’, ‘grey’, and ‘black’ analytical systems. Advantages and limitations of available multivariate calibration and resolution methods are discussed with respect to the proposed classification of the analytical mixture problem.

Robust methods for multivariate analysis — a tutorial review

Abstract Robust methods developed in statistics and chemometrics for multivariate calibration and exploratory analysis are reviewed. Robust methods can be classified according to aim: (i) regression methods, (ii) methods for outlier detection (diagnostics), and (iii) methods for dimensionality reduction (exploratory analysis). Based on this taxonomy, some of the methods are described in detail and illustrated with examples.

Latent-structure decompositions (projections) of multivariate data

Abstract Several approaches to the decomposition of multivariate data arrays in terms of latent structure are developed within a common mathematical frame. The methods considered are (i) decomposition into principal components, also called singular-value decomposition, (ii) decomposition using the partial-least-squares approach and (iii) projection on to axes defined by selected variables or objects, so-called markers. Graphic display, being of major importance in interactive data exploration and classification, is discussed.

Quantitative analysis in diffuse reflectance spectrometry: A modified Kubelka-Munk equation

Abstract The behaviour of the Kubelka-Munk equation with particle size is analysed by using mono-disperse polystyrene particles. Based on the experimental results a model is proposed to explain the diffuse reflectance spectra of powdered samples in a quantitative manner. The new model explains quantitatively the behaviour of the diffuse reflectance spectra of mono-disperse polystyrene particles in KBr. Furthermore, the results show that the model can be used to explain the diffuse reflectance spectra of mixtures of mono-disperse polystyrene spheres in KBr and of a coal sample mixture containing a distribution of particle sizes and KBr. It appears that the proposed model can be used for dilute samples in a quantitative manner.