If the lengths of the tests in a battery are altered, their intercorrelations and their validities or correlations with a criterion are also altered. Consequently, the multiple correlation of the battery with the criterion will also be altered. These changes are a function of the reliabilities of the tests. Suppose we have given from a set of experimental data (1) the time allowed for each test in the battery, (2) the reliability of each test, (3) the intercorrelations, and (4) the validities of all the tests. If we specify the over-all testing time we are willing to allow for the test in the future, we can determine the amount by which each test must be altered in order to give the maximum multiple correlation with the criterion. The method is presented, together with numerical examples and the mathematical proof.

A simplified method of computing the inverse of a triangular matrix is presented. It is useful with the multiple-group method of factoring the correlation matrix as well as with other factor-analysis and multiple-correlation problems.

Matrix methods may be applied to the analysis of experimental data concerning group structure when these data indicate relationships which can be depicted by line diagrams such as sociograms. One may introduce two concepts, n-chain and clique, which have simple relationships to the powers of certain matrices. Using them it is possible to determine the group structure by methods which are both faster and more certain than less systematic methods. This paper describes such a matrix method and applies it to the analysis of practical examples. At several points some unsolved problems in this field are indicated.

The Kuder-Richardson formula (20) is rewritten to be identical with the simplest formula, (21), except for the addition of a term involving the standard deviation, σp, of the item p's. If σp can be estimated, a rapid and superior estimate of test reliability is possible in contrast to the simpler formula (21) used when the number of items and mean and standard deviation of test scores are known.

A battery of 20 tests originally analyzed by Alexander (1) was reworked according to the principle of simple structure. His results were sustained in general. Both analyses yielded five factors in the first-order domain. Of these, three factors in the re-analysis (v, X and F) have almost exactly the same loadings as the corresponding factors in the original work, and were interpreted in the same way. The loading pattern of a fourth factor, Z, left uninterpreted in the original study, happened to be more clear in the re-analysis, and an interpretation was attempted. It appears to be a factor of perceptual synthesis, and seems to play an important role in intellectual processes. A fifth factor, not present in Alexander's results, appeared in the new analysis: the reasoning factor, involved in inductive and deductive thinking. All four cognitive factors are related to a general factor that can be thought of as representing abstraction and education of relations and correlates, these processes being, therefore, the essential feature underlying intellectual behavior, at least in that sector surveyed by the tests of the present battery.

A modification of the Wherry-Doolittle test selection method is presented by which tests are included in a multiple correlation (obtained for a given battery of tests) in the sequence in which the rate of return in validity per unit of testing time is greatest, rather than in the order of the size of their contribution to the multiple correlation. It is proposed that the modified method can be utilized profitably when there are economic or practical limits on the time available for test administration.