Correspondence factor analysis

Correspondence factor analysis (CFA) is most appropriate when the data represent counts of contingencies, or when there are numerous true zeroes in the table (i.e. when zero means complete absence of a contingency, rather than a small quantity which has been rounded to zero [47]). A detailed description of the method is found in Section 32.3.6. 

Data obtained from a screening test on rats, differentiating between morphinomimetic (opioid analgesic) and neuroleptic compounds [48]. The six observations are scored on a 6-point scale ranging from absent (0) to highly pronounced (6). Compounds 1 to 13 are morphinomimetics compounds 14 to 26 are neuroleptics. 

Prostration Temperat. Temperat. Pupil diam. Pupil diam. Palpebral Total increase decrease increase increase decrease 

---

Hence is the fraction of the total sum of squares (or inertia) c of the data X that is accounted for by v,. The sum of squares (or inertia) of the projections upon a certain axis is also proportional to the variance of these projections, when the mean value (or sum) of these projections is zero. In data analysis we can assign different masses (or weights) to individual points. This is the case in correspondence factor analysis which is explained in Chapter 32, but for the moment we assume that all masses are identical and equal to one. 

Fig. 31.11. Biplot of chromatographic retention times in Table 31.2, resulting from correspondence factor analysis, i.e. after double-closure of the data. The line segments have been added to emphasize contrasts in the same way as in Fig. 31.10.

In this respect, the weight coefficients are proportional to the column-sums. Distances of Chi-square form the basis of correspondence factor analysis (CFA) which is discussed in Chapter 32. 

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

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. 

The log-linear model (LLM) is closely related to correspondence factor analysis (CFA). Both methods pursue the same objective, i.e. the analysis of the association (or correspondence) between the rows and columns of a contingency table. In CFA this can be obtained by means of double-closure of the data in LLM this is achieved by means of double-centring of the logarithmic data. 

Fig. 37.5. Biplot obtained from correspondence factor analysis of the data in Table 37.8 [43], Circles refer to compounds. Squares relate to observations. Areas of circles and squares are proportional to the marginal sums of the rows and columns in the table. The horizontal and vertical components represent 40 and 31 %, respectively, of the interaction in the data.

Double centering centers both columns and rows it is the basic transformation for correspondence factor analysis—a method conceptually similar to PCA, and often applied to contingency tables (Greenacre 1992). 

An optimum projection of both data matrix aspects, namely lines and columns is provided by correspondence factor analysis which also offers a number of sophisticated and useful variance decompositions. For details see [GREENACRE, 1984 MELLIN-GER, 1987]. One of the first applications to environmental data has been reported by FEINBERG [1986],... 

Ounnar and co-workers [31,32] widely apply in their QSRR studies the approach called correspondence factor analysis (CFA). CFA is mathematically related to PCA, differing in the preprocessing and scaling of the data. Those authors often succeeded in assigning definite physical sense to abstract factors, e.g., they identified the Hammett constants of substituents in meta and para positions of 72 substituted /V-benzylideneanilines (NBA) in determining the first factorial axis resulting from the CFA analysis of retention data of NBA in diverse normal-phase HPLC systems. 

Multivariate methods of data analysis were first applied in chromatography for retention prediction purposes [7. More recently, principal component analysis (PCA), correspondence factor analysis (CFA) and spectral mapping analysis (SMA) have been employed to objectively cla.ssify. stationary phase materials according to the retention... 

Large data tables may hide information which is not easily detected by simple inspection of the various columns. Principal component analysis and some closely related techniques such as factor analysis (FA), correspondence factor analysis (CFA) and non-linear mapping (NLM), reduce a data matrix to new supervariables retaining a maximum of information or variance from the original data matrix. These new variables are called latent variables or principal components, and are orthogonal vectors composed of linear combinations of the original variables. This concept is shown schematically in Fig. 22.15. 

L.) collected by beekeepers in apparently polluted and nonpolluted environments was performed by using inductively coupled plasma atomic emission spectrometry (ICP-AES) to measure significant concentrations of Ag, Ca, Cr, Co, Cu, Fe, Li, Mg, Mn, Mo, P, S, Zn, Al, Cd, Hg, Ni, and Pb. Fortunately, Cd, Hg, Ni, and Pb were not detected in the analyzed samples. Conversely, Ag, Cu, Al, Zn, and S were found in some samples located near industrial areas. Because a high variability was found in the concentration profiles, correspondence factor analysis was used to rationalize the data and provide a typology of the honeys based on the concentration of these different elements in the honeys. The results were confirmed by means of principal component analysis and hierarchical cluster analysis. Finally, the usefulness of the acacia honey as a bioindicator of heavy metal contamination is discussed. 

Among the different linear multivariate methods that can be used to analyze Table 12.2, correspondence factor analysis (CFA) was selected because its yf metrics permits work on data prohles and the natural biplot representation of the variables and objects which greatly facilitates the interpretation of the graphical displays [26], In addition, CFA has been used successfully on similar data matrices for rationalizing (eco)toxicologi-cal information [27-30],... 

Devillers, J. and Karcher, W. (1990). Correspondence factor analysis as a tool in environmental SAR and QSAR stndies. In Practical Applications of Quantitative Structure-Activity Relationships (QSAR) in Environmental Chemistry and Toxicology (Karcher, W. and Devillers, J., Eds). Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 181-195. 

Boolean descriptors. These descriptors are easy to derive but their introduction in dummy regression analysis can yield statistical pitfalls when they are used for predictive purposes. To overcome this problem, it is possible to use the stochastic regression analysis. This method consists of performing, as a first step, a correspondence factor analysis (CFA) and then carrying out a regression analysis from the CFA factors. The stochastic regression analysis can also be used when the molecular descriptors are frequencies of occurrence. ... 

---