Abstract
<p>In this article we develop the Inter-battery Factor Analysis (IBA) byusing PLS (Partial Least Squares) methods. As the PLS methods are algorithms that iterate until convergence, an adequate intervention in some of their stages provides a solution to problems such as missing data. Specifically, we take the iterative stage of the PLS regression and implement the “available data” principle from the NIPALS (Non-linear estimation by Iterative Partial Least Squares) algorithm to allow the algorithmic development of the IBA with missing data. We provide the basic elements to correctly analyse and interpret the results. This new algorithm for IBA, developedunder the R programming environment, fundamentally executes iterative convergent sequences of orthogonal projections of vectors coupled with the available data, and works adequately in bases with or without missing data. To present the basic concepts of the IBA and to cross-reference the results derived from the algorithmic application, we use the complete Linnerud database for the classical analysis; then we contaminate this database with a random sample that represents approximately 7% of the non-available (NA) data for the analysis with missing data. We ascertain that the results obtained from the algorithm running with complete data are exactly the same as those obtained from the classic method for IBA, and that the results withmissing data are similar. However, this might not always be the case, as it depends on how much the ‘original’ factorial covariance structure is affected by the absence of information. As such, the interpretation is only valid in relation to the available data.</p>
Highlights
Among the PLS methods created by Wold (1985), the most important are NIPALS, PLS-Regression (PLS-R) and PLS-Path Modeling (PLS-PM), which were designed for the treatment of one, two and k quantitative data matrices, respectively
The development of PLS algorithms that replace classical methods like Inter-battery Factor Analysis (IBA) is important, as it happened with NIPALS (Wold 1966) for the Principal Component Analysis (PCA) or GNM-NIPALS (Aluja & González 2014) for the treatment of a mixed data matrix
The application of this algorithm through the fAIBna(Y,X) function to the complete linnerud database, formed by the X and Y subgroups, leads to the same results as those obtained by applying the classical IBA method (Tenenhaus 1998)
Summary
Among the PLS methods created by Wold (1985), the most important are NIPALS, PLS-Regression (PLS-R) and PLS-Path Modeling (PLS-PM), which were designed for the treatment of one, two and k quantitative data matrices, respectively. PLS-R studies the relationship between two groups of variables X and Y even in the presence of multicollinearity, and has been applied with great success in fields such as Chemometrics, Sensometrics, Genetics, Medical Imaging (Pérez & González 2013), among others These PLS methods are convergent algorithms, and, as such, they allow intervention in some of their stages or phases in order to optimally handle missing data problems, mixed data, etc. For this reason, the development of PLS algorithms that replace classical methods like IBA is important (that being the main focus of this article), as it happened with NIPALS (Wold 1966) for the Principal Component Analysis (PCA) or GNM-NIPALS (Aluja & González 2014) for the treatment of a mixed data matrix. We highlight future investigations oriented towards IBA with missing data and mixed data that optimally quantify the qualitative variables from a k-dimensional function starting point, according to the GNM-NIPALS method (Aluja & González 2014)
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