Examining collinearities

The cos-max method is a little-known method of identifying collinearities. It is based on the cos-max transformation, which makes minimal adjustment to a set of vectors to create orthogonal components with a one-to-one correspondence between the original vectors and the components. The aim of the transformation is that each vector should be close to the orthogonal component with which it is paired. Vectors involved in a collinearity must be adjusted substantially in order to create orthogonal components, while other vectors will typically be adjusted far less. The cos-max method uses the size of adjustments to identify collinearities. It gives a coherent relationship between collinear sets of variables and variance inflation factors (VIFs) and identifies collinear sets using more information than traditional methods. In this paper we describe these features of the method and examine its performance in examples, comparing it with alternative methods. In each example, the collinearities identified by the cos-max method only contained variables with high VIFs and contained all variables with high VIFs. The collinearities identified by other methods did not have such a close link to VIFs. Also, the collinearities identified by the cos-max method were as simple as or simpler than those given by other methods, with less overlap between collinearities in the variables that they contained.