Multivariate reference values

Faber K, Kowalski BR (1997a) Improved prediction error estimates for multivariate calibration by correcting for the measurement error in the reference values. Appl Spectrosc 51 660... 

In order to construct a calibration model, the values of the parameters to be determined must be obtained by using a reference method. The optimum choice of reference method will be that providing the highest possible accuracy and precision. The quality of the results obtained with a multivariate calibration model can never exceed that of the method used to obtain the reference values, so the choice should be carefully made as the quality of the model will affect every subsequent prediction. The averaging of random errors inherent in regression methods can help construct models with a higher precision than the reference method. 

The elements of fhe vector y are the reference values of the response variable, used for building the model. The uncertainty on the coefficient estimation varies inversely with the determinant of the information matrix (X X) which, in the case of a unique predictor, corresponds to its variance. In multivariate cases, the determinant value depends on the variance of fhe predictors and on their intercorrelation a high correlation gives a small determinant of the information matrix, which means a big uncertainty on the coefficients, that is, unreliable regression results. 

The multivariate quantitative spectroscopic analysis of samples with complex matrices can be performed using inverse calibration methods, such as ILS, PCR and PLS. The term "inverse" means that the concentration of the analyte of interest is modelled as a function of the instrumental measurements, using an empirical relationship with no theoretical foundation (as the Lambert Bouguer-Beer s law was for the methods explained in the paragraphs above). Therefore, we can formulate our calibration like eqn (3.3) and, in contrast to the CLS model, it can be calculated without knowing the concentrations of all the constituents in the calibration set. The calibration step requires only the instrumental response and the reference value of the property of interest e.g. concentration) in the calibration samples. An important advantage of this approach is that unknown interferents may be present in the calibration samples. For this reason, inverse models are more suited than CLS for complex samples. 

In multivariate calibration, accuracy reports the closeness of agreement between the reference value and the value found by the calibration model and is generally expressed as the root mean square error of prediction (RMSEP, as described in section 4.5.6) for a set of validation samples ... 

M. J. Griffiths and S. L. R. Ellison, A simple numerical method of estimating the contribution of reference value uncertainties to sample-specific uncertainties in multivariate regression, Chemom. Intell. Lab. Syst., 83, 2006, 133-138. 

Fig. 16.4. Three methods of obtaining Raman-based estimates of biofluid concentrations in vivo, a Confocal isolation of a subsurface volume occupied by a blood vessel, enabling direct measurement of a blood spectrum, b Difference measurement between tissue in two states, one with more blood in the sampling volume (in this case, due to pressure modulation by the subject [6]). Computing the difference removes the bulk tissue contributions to the spectral measurement and emphasizes the contribution from blood, c Statistical correlation approach of measuring many volunteers tissue in a region where sufficient blood is present (e.g., the forearm as shown here) and obtaining a correlated reference value from a blood sample drawn at the same time. Multivariate calibration is then used to find correlations between the reference value and the spectral data vector. Unlike the previous two methods, this does not intrinsically isolate the blood chemicals Raman signatures from those of the surrounding tissue volume...

The body of this chapter discusses population-based univariate reference values and quantities derived from them. If, for example, we produce, treat, and use separate reference values for cholesterol and triglycerides in serum, we have two sets of univariate reference values. The term multivariate... 

The variable x in the preceding formulas denotes a quantity that varies. In our context, it signifies a reference value. If the variable by chance may take any one of a specified set of values, we use the term variate (i.e, a random variable). In this section, we consider distributions of single variates (i.e., univariate distributions). In a later section, we also discuss the joint distribution of two or more variates bivariate or multivariate distributions). 

The topic of the previous sections of this chapter has been univariate population-based reference values and quantities derived from them. Such values do not, however, fit the common clinical situation in which observed values of several different laboratory tests are available for interpretation and decision making. For example, the average number of individual clinical chemistry tests requested on one specimen received in the author s laboratory is 9.7. There are two models for interpretation by comparison in this situation. We can compare each observed value with the corresponding reference values or inteiwal (i.e., we perform multiple, univariate comparisons) or we can consider the set of observed values as a single multivariate observation and interpret it as such by a multivariate comparison. In this section, the relative merits of these two approaches are discussed, and methods for the latter type of comparison are presented. 

The use of multivariate reference regions usually requires the assistance of a computer program, which takes a set of results obtained by several laboratory tests on the same clinical specimen and calculates an index. The interpretation of a multivariate observation in relation to reference values is then the task of comparing the index with a critical value estimated from the reference values. This, obviously, is much simpler than comparing each result with its proper reference interval. 

The index is essentially a distance measure, Mahalanobis" squared distance (D ), which expresses the multivariate distance between the observation point and the common mean of the reference values, taldng into account the dispersion and correlation of the variables.More interpreta-tional guidance may be obtained from this distance by expressing it as a percentile analogous to the percentile presentation of univariate observed values. Also, the index of atypicality has a multivariate counterpart. ... 

Albert A, Heusghem C. Relating observed values to reference values The multivariate approach. In Grasbeck R, Alstrom T, eds. Reference values in laboratory medicine. Chichester, England John Wdey, 1981 289-96. 

To determine Sb in marine sediments by ETAAS, a direct method was developed based on quantitating the analyte in the liquid phase of the slurries (prepared directly in autosampler cups). The variables influencing the extraction of Sb into the liquid phase and the experimental setup were set after a literature search and a subsequent multivariate optimisation procedure. After the optimisation, a study was carried out to assess robustness. Six variables were considered at three levels each (see Table 2.13). In addition, two noise factors were set after observing that two ions, which are currently present into marine sediments, might interfere in the quantitations. In order to evaluate robustness, a certified reference material was used throughout, BCR-CRM 277 Estuarine Sediment (guide value for Sb 3.5 0.4pgg ). Table 2.13 depicts the experimental setup. 

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