Beginning students of analysis of variance frequently find that the explanations concerning numbers of degrees of freedom are rather more mystical than explanatory, and that the dependencies and independencies among parameters and linear restraints are not systematically covered or explicitly presented. It is the purpose of this paper to set forth the simple matrix representation of these models in a way which has been found useful for teaching purposes, and may be found of some value in the setting up of mathematical models for experimental designs calling for the analysis of variance or the analysis of covariance.

The system I shall portray is a more or less natural outgrowth of suggestions made by Dr. R. C. Bose, with whom I have discussed some of the features very slightly. I am also indebted to Professor Frederick Mosteller for assistance with the mechanics of some of the models. Any deficiencies which the system may have are not to be laid to the doors of these gentlemen, however, as they have not had any opportunity to go over the material at any length.

A pattern consists of fundaments and relations. Its similarity to other patterns can be expressed either in terms of relations alone—in which case a modified correlation coefficient, rs, suffices—or in terms of fundaments and relations, where new coefficients, rp and r′p are required. The derivation of a coefficient of pattern similarity, rp, is presented and its properties are investigated, both deductively and in terms of actual data on culture patterns.

Standardization scores on the 1937 Revision of the Stanford-Binet are analyzed by the Thurstone centroid factor technique. Correlation matrices at age levels seven, nine, eleven, and thirteen, each based on two hundred subjects, are analyzed. The factors after rotation are interpreted in terms of the items exhibiting high factor saturations. In contrast to the findings of McNemar (1942) no evidence for a general factor in the Stanford-Binet battery is found in the present study. Rather, I.Q. variance can be explained in terms of psychologically meaningful group factors at the various age levels. The results indicate a need for cautious interpretation and use of single score indices of intelligence such as the Stanford-Binet I.Q.