Principal components, analysis of variance and data structure

The relation between principal components and analysis of variance is examined. It is shown that the model underlying the extended analysis of variance developed by GOLLOB and MANDEL is useful also as a model for principal component analysis. The elucidation of structure of two-factor data using the new analysis of variance model is illustrated by an example taken from thermodynamics. It has been may good fortune to have spent a full year in close association with Professor HAMAKER at the Technological University of Eindhoven. That year was among the most pleasant and most rewarding of my career. I feel honored to be able to join with Professor HAMAKER'S many friends and colleagues in dedicating this issue of Statistica Neerlandica to him. The method of principal components goes back to ideas proposed by PEARSON as early as 1901 (12) and developed systematically by HOTELLING in 1933 [4]. Since then the method has been applied to numerous sets of data, more particularly in the field of psychology, but also in numerous other areas of research, including the physical sciences [e.g. 1,2, 5, 7, 8, 10, 13, 14, IS, 16). In 1968 and 1969 respectively, GOLLOB (3) and MANDEL (8) proposed, independently of each other, an extension of the analysis of variance approach, which GOLLOB called "Fanova", because it combined features of analysis of variance and of factor analysis. This method, too, had been anticipated by some earlier authors [13, 17). It is immediately apparent that this extension of the analysis of variance involves the same matrix calculations as the method of principal components. The question then arises whether a deeper conceptual relationship exists between the two methods. In this paper this question is examined. The result is not only a positive answer to this question but also a clarification of the method of principal components. The author believes, as a result of this work, that interpretations of principal component analyses found in the literature are sometimes incorrect. We will attempt to show that such misinterpretations are due, in no small measure, to a particular terminology that has acquired common usage in inferences drawn from principal component analysis