Methods of Reordering the Correlation Matrix to Facilitate Visual Inspection and Preliminary Cluster Analysis.

Using the notion of parallel items this paper presents a family of new criteria for cluster analysis. Instead of looking at the item correlation matrix one could look at a matrix of similarity coefficients. These coefficients are standardized raw dot products of columns in the correlation matrix or related numbers. After discussing the properties of this matrix the discussion will move to a classic example of a factor analytic problem in personality assessment, an item pool concerned with Fromm's marketing orientation, and a brief discussion of problems outside of cluster analysis will conclude the paper. One of the most common problems in measurement is the evaluation of a new test or the re-evaluation of an old one. Even if a good deal of care has gone into the specification of the a priori structure of a test, the actual content of a test, when people respond to it, may be different from that desired. For example, suppose we had just constructed a student instructor rating form. We have spent months writing items that tapped different aspects of the instructor's performance. But the item correlation matrix turns out to be flat, i.e. all the items correlate about .70 with all the other items. Then no matter how many dimensions there may be to instructor performance, we have tapped only one, the student's global impression. We would probably regard the test as a failure and begin looking at such things as response categories, whether the items used specific examples or called for global evaluations, etc. On the other hand, suppose that we wish to construct a questionnaire measure of McClelland's (1951) need achievement. We write items to assess each of the dimensions with which achievement motivation should correlate: independence, leadership, etc. The item correlation matrix shows stark simplicity: the items break into ten clusters which are perfectly uncorrelated with each other. The fact that the clusters are uncorrelated is both unanticipated and undesirable. We could respond to this fact in two ways. First we might question the existence of achievement motivation as a unitary trait. Alternately we could sweat blood over the unanswerable question of whether or not our test had the right distribution of items across clusters. Thus, compare a first test that has twenty items in the first cluster and two items in each of the other nine clusters with a second test that has 20 items in the last cluster and two in each of the others. Not only would these tests look quite different, but the corresponding true scores would only correlate .25 with each other. Furthermore the problem is not just a matter of the unequal distribution of items in clusters. A test with four items in each of the ten clusters would be "balanced." But a balanced distribution may be completely inappropriate for meaning achievement motivation. How do we determine the content of a test as people actually respond to it? The traditional answer to this problem is to intercorrelate the items and factor analyze the item correlation matrix. The structure of the test is then determined indirectly by re