Restricted principal components analysis for marketing research

Principal components analysis (PCA) is one of the foremost multivariate methods utilized in marketing and business 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. The paper aims to introduce restricted PCA (RPCA), which attempts to optimally derive latent components whose coefficients are integer‐constrained (e.g.: {−1,0,1}, {0,1}, etc.).

The paper presents two algorithms for deriving efficient solutions for RPCA: an augmented branch and bound algorithm for sequential extraction, and a combinatorial optimization procedure for simultaneous extraction of these constrained components. The paper then contrasts the traditional PCA‐derived solution with those obtained from both proposed RPCA procedures with respect to a published data set of psychographic variables collected from potential buyers of the Dodge Viper sports car.

This constraint results in solutions which are easily interpretable with no need for rotation. In addition, the proposed procedure can enhance data reduction efforts since fewer raw variables define each derived component.

The paper provides two algorithms for estimating RPCA solutions from empirical data.