Multivariate analyses

Multivariate analysis is a set of techniques used for analysis of data sets that contain more than one variable, and the techniques are especially valuable when working with correlated variables. The techniques provide an empirical method for information extraction, regression, or classification some of these techniques have been developed quite recently because they require the computational capacity of modem computers. 

As a trivial example, say that you wanted to use impedance measurements to assess the quality or texture of the inside matter of an apple. One approach would be to develop an electrical model for the apple and figure out how texture differences depend on things such as cell stmcture and water content. Based on these assumptions, you could try to postulate how differences in texture will influence the impedance spectrum and then seek to have this confirmed by experiments. Referring to Section 9.1.2 this would be the explanatory approach, whereas a more descriptive approach could be the use of multivariate analysis. 

To use a multivariate regression technique, you would need another way of measuring or assessing the apple s texture as a golden standard to calibrate your model against. The measured impedance values will in this case often be called X-variables and the measurements from the calibration instmment is called Y-variables. 

There may exist a mechanical instmment for measuring the softness of the apple material or you may simply use a taste panel to score each apple on a scale from 1 to 10. What the multivariate regression software program then will do for you is basically to produce a model that you can use to calculate the softness of each apple (Y-variable) from the measured impedance data (X-variables) or from any data derived from the measured data, such as Cole parameters. 

The main purpose of multivariate methods would be information extraction. The simplest form of information extraction and data reduction is the PCA technique. The history of PCA can be traced to an article by Pearson (1901). It is a statistical method that can be performed in a wide variety of mathematical, statistical, or dedicated computer software such as Matlab (The MathWorks, Inc.), SPSS (SPSS, Inc.), or The Unscrambler (Camo, Inc.). We will here give a short nonmathematical introduction to this method, and we refer the reader to one of the many available text books on this topic for a more in-depth, formal presentation. 

A graphical representation is less easy for three variables and no longer possible for four or more it is here that computer analysis is particularly valuable in finding patterns and relationships. Matrix algebra is needed in order to describe the methods of multivariate analysis fully. No attempt will be made to do this here. The aim is to give an appreciation of the purpose and power of multivariate methods. Simple data sets will be used to illustrate the methods and some practical applications will be described. 

In this section we examine a few representative samples of multilinear techniques. 

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J. C. Gower. Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53 325, 1966. 

Correlations between structure and mass spectra were established on the basis of multivariate analysis of the spectra, database searching, or the development of knowledge-based systems, some including explicit management of chemical reactions. 

Chatfield C and A J CoHns 1980. Introduction to Multivariate Analysis. London, Chapman Hall. Desiraju G R 1997. Crystal Gazing Structure Prediction and Polymorphism. Sdence 278 404-405. Everitt B.S. 1993 Cluster Analysis. Chichester, John Wiley Sons. 

I J, J C Cole, J P M Lommerse, R S Rowland, R Taylor and M L Verdonk 1997. Isostar A Libraij )f Information about Nonbonded Interactions. Journal of Computer-Aided Molecular Design 11 525-531. g G, W C Guida and W C Still 1989. An Internal Coordinate Monte Carlo Method for Searching lonformational Space. Journal of the American Chemical Scociety 111 4379-4386. leld C and A J Collins 1980. Introduction to Multivariate Analysis. London, Chapman Hall, ig C-W, R M Cooke, A E I Proudfoot and T N C Wells 1995. The Three-dimensional Structure of 1 ANTES. Biochemistry 34 9307-9314. 

Jurs P C1990. Chemometrics and Multivariate Analysis in Analytical Chemistry. In Lipkowitz K B and D B Boyd (Editors) Reviews in Computational Chemistry Volume 1. New York, VCH Publishers, pp. 169-212. 

In multivariate least squares analysis, the dependent variable is a function of two or more independent variables. Because matrices are so conveniently handled by computer and because the mathematical formalism is simpler, multivariate analysis will be developed as a topic in matrix algebra rather than conventional algebra. 

For Multivariate Analysis, see McCuen, Reference 23, or other statistical texts. 

It should be noted that in this example the performance of only one variable, the three analysts, is investigated and thus this technique is called a one-way ANOVA. If two variables, e.g. the three analysts with four different titration methods, were to be studied, this would require the use of a two-way ANOVA. Details of suitable texts that provide a solution for this type of problem and methods for multivariate analysis are to be found in the Bibliography, page 156. 

Lorius et al. (1990) performed a simple multivariate analysis in which they correlate the temperature changes of the past 160 kyr (as recorded in the Vostok SD record) with changes in five forcings atmospheric CO2 plus CH4, ice volume, aerosol loading (dust and sepa-... 

J> < 0.01) and also more cost-effective, mainly because of the higher number of hospital admissions in the TCA group. This study had limitations in that patients prescribed TCAs were not randomly selected, a quarter of the patients in the TCA group failed to receive an effective dose, and objective measurements of outcome were not employed. Multivariate analysis suggested that despite the methodological limitations of the study, the differences in cost were due to the treatment received, and not to differences in patient characteristics. This study provides the first, albeit tentative, evidence of superior cost-effectiveness for SSRIs over TCAs in the UK. 

Selective serotonin reuptake inhibitor antidepressant selection and anxiolytic and sedative hypnotic prescribing a multivariate analysis./ Clin Outcomes Manage 4, 16—22. 

Hylan TR, Crown WH, Meneades L, et al (1998). SSRI and TCA antidepressant selection and health care costs a multivariate analysis. /... 

Adams, R.P. 1986. Geographic yariationm Juniperus silicicola and J. virginiana of the southeastern United States multivariate analysis of morphology and terpenoids. Taxon 35 61-75. 

To detect adulteration of wine. Bums et al. (2002) found that the ratios of acetylated to p-coumaroylated conjugates of nine characteristic anthocyanins served as useful parameters to determine grape cultivars for a type of wine. Our laboratory utilized mid-infrared spectroscopy combined with multivariate analysis to provide spectral signature profiles that allowed the chemically based classification of antho-cyanin-containing fruits juices and produced distinctive and reproducible chemical fingerprints, making it possible to discriminate different juices. " This new application of ATR-FTIR to detect adulteration in anthocyanin-containing juices and foods may be an effective and efficient method for manufacturers to assure product quality and authenticity. 

A multivariate analysis (Table XXV) shows the increased blood-lead level caused by the RSR smelter contribution and the traffic contribution to be 5.5 and 1.0, respectively. ... 

Mannhold, R., Crudani, G., Dross, K., Rekker, R. F. Multivariate analysis of experimental and calculative descriptors for molecular lipophilicity. J. Comput.-Aided Mol. Design 1998, 12, 573-581. 

One of the air of multivariate analysis is to reveal patterns in the data, whether they are in the form of a measurement table or in that of a contingency table. In this chapter we will refer to both of them by the more algebraic term matrix . In what follows we describe the basic properties of matrices and of operations that can be applied to them. In many cases we will not provide proofs of the theorems that underlie these properties, as these proofs can be found in textbooks on matrix algebra (e.g. Gantmacher [2]). The algebraic part of this section is also treated more extensively in textbooks on multivariate analysis (e.g. Dillon and Goldstein [1], Giri [3], Cliff [4], Harris [5], Chatfield and Collins [6], Srivastana and Carter [7], Anderson [8]). 

Usually, the raw data in a matrix are preprocessed before being submitted to multivariate analysis. A common operation is reduction by the mean or centering. Centering is a standard transformation of the data which is applied in principal components analysis (Section 31.3). Subtraction of the column-means from the elements in the corresponding columns of an nxp matrix X produces the matrix of... 

W.R. Dillon and M. Goldstein, Multivariate Analysis, Methods and Applications. Wiley, New York, 1984. 

C. Chatfield and A.J. Collins, Introduction to Multivariate Analysis. Chapman and Hall, London, 1980. 

P. E. Green and J.D. Carroll, Mathematical Tools for Applied Multivariate Analysis. Academic Press, New York, 1976. 

A. Gifi, Non-linear Multivariate Analysis. Wiley, Chichester, UK, 1990. 

Multivariate analysis of these different types of measurements (heterogeneous, homogeneous, compositional, ordered) may require special approaches for each of them. For example, compositional tables that are closed with respect to the rows, require a different type of analysis than heterogeneous tables where the columns are defined with different units. The basic approach of principal components... 

We consider an nxn table D of distances between the n row-items of an nxp data table X. Distances can be derived from the data by means of various functions, depending upon the nature of the data and the objective of the analysis. Each of these functions defines a particular metric (or yardstick), and the graphical result of a multivariate analysis may largely depend on the particular choice of distance function. 

J.P. van de Geer, Multivariate Analysis of Categorical Data Applications. Sage Publications, Newbury Park, CA, 1993. 

J.C. Gower, Adding a point to vector diagrams in multivariate analysis. Biometrika, 55 (1968) 582-585. 

P.N. Nyambi, J. Nkengasong, P. Lewi, K. Andries, W. Janssens, K. Fransen, L. Heyndrickx, P. Piot and G. van derGroen, Multivariate analysis of human immunodeficiency virus type 1 neutralization data. J. Virol., 70 (1996) 6235-6243. 

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