A battery of 32 tests was administered to a sample including 144 Air Force Officer Candidates and 139 Air Cadets. The factor analysis, using Thurstone's complete centroid method and Zimmerman's graphic method of orthogonal rotations, revealed 12 interpretable factors. The non-reasoning factors were interpreted as verbal comprehension, numerical facility, perceptual speed, visualization, and spatial orientation. The factors derived from reasoning tests were identified as general reasoning, logical reasoning, education of perceptual relations, education of conceptual relations, education of conceptual patterns, education of correlates, and symbol substitution. The logical-reasoning factor corresponds to what has been called deduction, but eduction of correlates is perhaps closer to an ability actually to make deductions. The area called induction appears to resolve into three eduction-of-relations factors. Reasoning factors do not appear always to transcend the type of test material used.

A mathematical definition of the theoretical relation between the examinee's actual responses to the test items and his “true ability” is selected. A maximum-likelihood solution is obtained for estimating the examinee's “true ability” from his responses to the items. The standard error of the maximum-likelihood estimate is obtained, its relation to the discriminating power of the test is pointed out, and some generalizations are drawn as to the optimum level of item difficulty. The Neyman-Pearson power function is applied to determine which of two psychological tests is the most powerful for the selection of “successful” examinees.

When a numerical transformation of raw data is used only to simplify the arithmetic of curve fitting, the transformation may lead to undesirable and even highly distorted results. This principle is illustrated with an approximation method of fitting parabolic equations to experimental data, as described recently in texts by Johnson and Lewis. Although the approximation method will never yield as good fits as the exact, least-squares method, satisfactory results are in general achieved whenever the transformed scores yield a linear plot as a function of X. The principal difficulty with the method is that some data which fall along a parabola may not yield a linear plot of the transformed scores versus X, and so cannot be fitted satisfactorily by the approximation method.

In his article “Finite Markov Processes in Psychology,”G. A. Miller derived a least-squares “estimate” for a matrix of transitional probabilities. However, the mathematical proof is found to be invalid.

On page 158, Miller defined by the equation

= N + C,

“where the elements of the matrix C are the corrections that must be added to the observed values in N to give the best estimate ”.

By extension of the rotational process, meaningful orthogonally related positions were found for all of the thirteen centroid factors which Thurstone extracted from his original PMA intercorrelations. Most of the original primary ability factors were more sharply delineated and corresponded more closely to the Army Air Force factors that bear similar names (demonstrating greater invariance from analysis to analysis). While such different results obtained by two investigators applying the same methods on the same data may initiate some concern, the results strengthen rather than weaken the idea that more psychological meaningfulness and greater invariance will result if centroid axes are rotated, using the concepts of a simple structure and positive manifold.

The computation of intercorrelation matrices involving large numbers of variables and the subsequent factoring of these matrices present a formidable task. A method for estimating factor loadings without computing the intercorrelation matrix is developed. The estimation procedure is derived from a theoretical model which is shown to be a special case of the multiple-group centroid method of factoring. Empirical checks have indicated that the model, even though it makes some stringent assumptions, can be applied to a variety of variables found in psychological factoring problems. It has been found to be particularly useful in factoring test items.

In his interesting article “Finite Markov Processes in Psychology,” G. A. Miller derived a “least-squares estimate” for a matrix of transitional probabilities (I). However, the mathematical proof seems to be unclear. Since this proof is considered invalid (2), we shall present a somewhat clearer version of the proof of this result. We shall also examine the general problem in some detail.

In the proof we shall assume that the reader is familiar with matrix notation, which enables a considerably shorter presentation. We shall follow the matrix conventions and the terminology adopted by Miller (I).

An index is proposed to measure the extent of agreement of the data of a sociometric test with another test made at an earlier time or on another test criterion. The index is used to define an index of concordance between the two tests. It is shown how the index may be used for either individuals or groups. Tests of the hypothesis that agreement is random are given for all cases and applied to an example.

The right-response scores and wrong-response scores of speeded aptitude tests were factor analyzed to determine whether they differ in factorial content. The information thus obtained was used to derive scoring formulas that yield purer measures of a factor than do scoring formulas derived in other ways.