Factor analysis scores

Higher order differences derivatives Difference ratios Fourier coefficients Principal components scores Factor analysis scores Curve-fitting coefficients Partial least-squares scores... 

Fig. 32.6. (a) Generalized score plot derived by correspondence factor analysis (CFA) from Table 32.4. The figure shows the distance of Triazolam from the origin, and the distance between Triazolam and Lorazepam. (b) Generalized loading plot derived by CFA from Table 32.4. The figure shows the distance of epilepsy from the origin, and the distance between epilepsy and anxiety. 

In their fundamental paper on curve resolution of two-component systems, Lawton and Sylvestre [7] studied a data matrix of spectra recorded during the elution of two constituents. One can decide either to estimate the pure spectra (and derive from them the concentration profiles) or the pure elution profiles (and derive from them the spectra) by factor analysis. Curve resolution, as developed by Lawton and Sylvestre, is based on the evaluation of the scores in the PC-space. Because the scores of the spectra in the PC-space defined by the wavelengths have a clearer structure (e.g. a line or a curve) than the scores of the elution profiles in the PC-space defined by the elution times, curve resolution usually estimates pure spectra. Thereafter, the pure elution profiles are estimated from the estimated pure spectra. Because no information on the specific order of the spectra is used, curve resolution is also applicable when the sequence of the spectra is not in a specific order. 

Factor analysis with the extraction of two factors and varimax rotation can be carried out in R as described below. The factor scores are estimated with a regression method. The resulting score and loading plots can be used as in PCA. 

Principal Component Analysis (PCA) is the most popular technique of multivariate analysis used in environmental chemistry and toxicology [313-316]. Both PCA and factor analysis (FA) aim to reduce the dimensionality of a set of data but the approaches to do so are different for the two techniques. Each provides a different insight into the data structure, with PCA concentrating on explaining the diagonal elements of the covariance matrix, while FA the off-diagonal elements [313, 316-319]. Theoretically, PCA corresponds to a mathematical decomposition of the descriptor matrix,X, into means (xk), scores (fia), loadings (pak), and residuals (eik), which can be expressed as... 

After determining the underlying factors which affect local precipitation composition at an Individual site, an analysis of the slmlllarlty of factors between different sites can provide valuable Information about the regional character of precipitation and Its sources of variability over that spatial scale. SIMCA ( ) Is a classification method that performs principal component factor analysis for Individual classes (sites) and then classifies samples by calculating the distance from each sample to the PGA model that describes the precipitation character at each site. A score of percent samples which are correctly classified by the PGA models provides an Indication of the separability of the data by sites and, therefore, the uniqueness of the precipitation at a site as modeled by PGA. 

In some diseases a simple ordinal scale or a VAS scale cannot describe the full spectrum of the disease. There are many examples of this including depression and erectile dysfunction. Measurement in such circumstances involves the use of multiple ordinal rating scales, often termed items. A patient is scored on each item and the summation of the scores on the individual items represents an overall assessment of the severity of the patient s disease status at the time of measurement. Considerable amoimts of work have to be done to ensure the vahdity of these complex scales, including investigations of their reprodu-cibihty and sensitivity to measuring treatment effects. It may also be important in international trials to assess to what extent there is cross-cultural imiformity in the use and imderstand-ing of the scales. Complex statistical techniques such as principal components analysis and factor analysis are used as part of this process and one of the issues that need to be addressed is whether the individual items should be given equal weighting. 

One of the major uses of multivariate techniques has been the discrimination of samples based on sensory scores, which also has been found to provide information concerning the relative importance of sensory attributes. Techniques used for sensory discrimination include factor analysis, discriminant analysis, regression analysis, and multidimensional scaling (8, 10-15). 

Since the sensory data collected involved degree of sample difference from a reference, it was felt that the analytical data should be analyzed in a similar manner. In cases where some peaks making up a multicomponent mixture are known to be specific to that mixture, this is a relatively simple matter. In such cases, the peak areas of the known components can be compared to a reference and average percent difference calculated. However, if it is not possible to pick out peaks that are clearly specific to a single multicomponent mixture, a more sophisticated technique such as factor analysis is required. There are circumstances where all peaks are common to each multicomponent mixture, i.e. qualitatively similar but quantitatively different. Also there are cases where peaks are found only in one of the multicomponent mixtures, but it is not clear to which mixture they belong. In these cases factor analysis is required to extract patterns that are characteristic of the specific multicomponent mixtures. Analytical concentrations of each of the multicomponent mixtures are then calculated as a set of factor scores where each score is directly proportional to the actual concentration of each multicomponent mixture. 

Correlation of Analytical/Sensory Results. Sensory data was correlated with headspace data of tobacco volatiles by factor analysis (BMDP4M) and canonical correlation BMDP6M. Analytical data included factor scores and discriminant analyses scores sensory data included scores from the two MDS dimensions. Sorted rotated factor loadings of combined sensory/analytical data using factor analysis are shown in Table II. Factor one contained those variables from the analytical and sensory data which related to differences between bright (A), burley (B), and oriental (C) (Figure 10). These included dimension 1 in the... 

Figure 8. Kriged image map showing the distribution of Factor I scores from the principal components analysis (see the caption for Figure 5 for the results of the principal components analysts).

Fig. 15.5. Factor analysis results for the C-H stretching region (2800-3050 cm 1 region) in human skin and in cultured skin model (Epiderm ). Data from human skin (8 x 12 pixels) and cultured skin (7 x 12 pixels) have been concatenated. Pixels marked with x s were excluded from the analysis, a Factor loadings for the methylene stretching region. The dashed vertical line marks 2876 cm-1 and emphasizes the shift in frequency between factors 1 and 2. b Score plots for factor 1 are depicted for human skin in the left set of 8 X 12 pixels and for cultured skin in the right set of 7 x 12 pixels, c Score plots for factor 2 are depicted for human skin in the left set of 8 x 12 pixels and for cultured skin in the right set of 7 x 12 pixels...

It is, furthermore, possible to interpret the latent vectors t or u. The latent vectors have got scores for each object, as in factor analysis. These scores can be used to display the objects. Another possibility is to compute the correlation between original features and the latent vectors to assess the kind of interacting features for both data sets. 

Regression techniques. Principal components are sometimes called abstract factors, and are primarily mathematical entities. In multivariate calibration the aim is to convert these to compound concentrations. PCR uses regression (sometimes called transformation or rotation) to convert PC scores onto concentrations. This process is often loosely called factor analysis, although terminology differs according to author and discipline. 

Factor Analysis and Multiple Regression. Factor scores for the 26 and 30 point cases, generated using the factor score coefficients in Tjble IV, were correlated with inverse median survival time, t... 

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