A technique is presented to test the completeness of factor solutions and also to test the significance of common-component loadings. The chisquare test involved is based upon the asymptotic normal properties of the residuals.

The Gleser-DuBois conditions for selecting from a number of test items those which will maximize the correlation between total test score and criterion will degenerate into expressions requiring only item counts on total distributions and the upper halves of distributions. A grouping convention for scores near medians is recommended. The inefficiency of the method is easily compensated for, because, regardless of the size of the sample, only standard test-scoring equipment and brief computations are required. A procedure is outlined, and some applications are discussed.

A statistical problem which frequently arises in educational and psychological experimentation is that of testing the significance of the difference of the mean scores of two groups on some criterion variable, where the differential effects of one or more variables which are correlated with the criterion must be statistically eliminated. The usual analytical technique for this type of problem is the analysis of covariance (9). The Neyman-Johnson technique (7) provides another, and substantially different, approach. A computational procedure is suggested here which utilizes the advantages of both techniques without an undue increase in computational labor. In addition, the Neyman-Johnson technique is generalized to the case of n predictor variables. Its application has heretofore been limited to a maximum of three predictor variables.

This study consists of four factor analyses of the Army Air Forces Aircrew Classification Batteries. The first was an analysis of the 1945 wartime battery, while the other three were analyses of the 1947 postwar battery, consisting of essentially the same variables, but using different samples. Eleven factors were found which had been identified and reported in previous analyses. An additional factor, possibly an artifact, was identified as an age-education doublet. The only factor which differed significantly in the analyses was pilot or flying interest. These factor analyses show that the factorial content of the tests remains quite similar in both wartime and postwar populations.

Tautologies are established for the reliability coefficient ρ2t of the sum of n part scores. It is not assumed that the part scores are experimentally independent of each other nor that the parts are equivalent to each other. The tautologies show the exact role played by experimental dependence and nonequivalence of parts, respectively, in the reliability of the sum. The formal algebra is appropriate to reliability in the sense of repeated trials of the same test, as well as in the sense of a universe of parallel tests, although the empirical meanings are different. Emphasis is on practical formulas that require information from only a single experiment (or test). These can take the form only of lower bounds to ρ2t, four of which are developed.

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 ”.

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.