An Efficient Branch and Bound Procedure for Restricted Principal Components Analysis

Abstract

Principal components analysis (PCA) is one of the foremost multivariate methods utilized in social science research for data reduction, latent variable modeling, multicollinearity resolution, etc. However, while its optimal properties make PCA solutions unique, interpreting the results of such analyses can be problematic. A plethora of rotation methods are available for such interpretive uses, but there is no theory as to which rotation method should be applied in any given social science problem. In addition, different rotational procedures typically render different interpretive results. We present restricted principal components analysis (RPCA) as introduced initially by Hausman (1982). RPCA attempts to optimally derive latent components whose coefficients are integer constrained (e.g.: {−1,0,1}, {0,1}, etc). This constraint results in solutions which are sequentially optimal, with no need for rotation. In addition, the RPCA procedure can enhance data reduction efforts since fewer raw variables define each derived component. Unfortunately, the integer programming solution proposed by Hausman can take far to long to solve even medium-sized problems. We augment his algorithm with two efficient modifications for extracting these constrained components. With such modifications, we are able to accommodate substantially larger RPCA problems. A Marketing application to luxury automobile preference analysis is also provided where traditional PCA and RPCA results are more formally compared and contrasted.