QSAR in grossly underdetermined systems: Opportunities and issues

Regression in grossly underdetermined systems has emerged as an important means for understanding molecular activity via comparative molecular field analysis (CoMFA) and other quantitative structure activity relationship (QSAR) studies. But this methodology has applications in much broader areas; for example, near-infrared spectroscopy, mutational enzyme activity studies including protein folding rates to determine which sites are important for determining conformation, and analyses of gene expression data from chip arrays. An error analysis which answers questions concerning the quality of the predictivity, the relative importance of each descriptor, the quality of the estimates of the contribution by each descriptor, and the number of independent components expressed by the associated data is indispensable in understanding whether some particular set of structure variables is important in defining the mechanisms driving the chemical or biological activities. This paper reviews opportunities for QSAR stu dies. It also considers the analytical aspects of error analysis in least-squares regression, and contrasts principal component regression (PCR) and partial least-squares (PLS) procedures with cross-validation on the issues of error analysis (e.g., the quality of the contribution estimates for each structure descriptor). Further, a methodology for selecting optimal subsets of components in PCR is presented.