Three techniques are commonly employed to capitalize on a concomitant variate and improve the precision of treatment comparisons: (1) stratification of the experimental samples and use of a factorial design, (2) analysis of covariance, and (3) analysis of variance of difference scores. The purpose of this paper is to compare the effectiveness of these alternatives in improving experimental precision, to identify the most precise design and the conditions under which its advantage holds, and to derive, in the case of the factorial approach, recommendations as to the optimal numbers of levels.

A method of deriving from the item score matrix all the usual statistics describing the performance on a test of a group of examinees is given. Since this matrix usually is not actually written out, but is implicit in a set of punched cards, a method of working from a more compact matrix F is described. A numerical example is presented. Applications and advantages of the method are cited, as compared with that of recording only the examinees' test scores and the item difficulties.

A significance test is proposed for determining whether a correlation coefficient is less than unity by an amount greater than that attributable to errors of measurement.

A disposition index, DI, which provides information about each possible placement to be considered in a personnel classification situation is discussed. The index is readily computed by machine methods and can be used by counselors required to make assignments. The use of the disposition index provides an adequate approximation to optimal solutions obtained by other methods.

Case IV of the Kuder-Richardson series, their formula (21), is derived as a generalized split-half Spearman-Brown coefficient. The basic assumption employed is shown to be sufficient to justify the various assumptions used in derivations by other authors. Some of the implications of this assumption are discussed.

A formal set of axioms is presented for the method of successive intervals, and directly testable consequences of the scaling assumptions are derived. Then by a systematic modification of basic axioms the scaling model is generalized to non-normal stimulus distributions of both specified and unspecified form.

This note presents the average Spearman rank correlation between m independent rankings and an untied criterion ranking, corrected for ties in any or all of the independent rankings.

A method of studying the problem of correction for guessing and other problems associated with behavior in the test situation is described and an illustrative example presented. As far as the writers are aware this method of approach is novel but, at the same time, it covers many of the practical and theoretical points raised by other writers as reviewed in the introduction.

The expected value of mean square concept is used to determine the effects of the presence of interactions in the single Latin square design on F tests. The results indicate that as the number of random effects included in the experiment increase, more F tests are unbiased, and that some of these are valid F tests. However, when F test bias does occur it is almost always of a negative nature so that the conclusions stated are conservative ones. Positive F test bias may occur when the triple interaction is extant and when zero or one random variate is included in the experiment.

A method of measuring function fluctuation is suggested and an appropriate test of significance is indicated. The proposed method is compared with bi-factor analysis and with some other suggested methods of measuring function fluctuation.

A coefficient of association τ′ is described for a contingency table containing data classified into two sets of ordered categories. Within each of the two sets the number of categories or the number of cases in each category need not be the same. τ′=+1 for perfect positive association and has an expectation of 0 for chance association. In many cases τ′ also has a -1 as a lower limit. The limitations of Kendall&s τ′a and τ′b and Stuart&s τ′c are discussed, as is the identity of these coefficients to τ′ under certain conditions. Computational procedure for τ′ is given.

Derivations are presented relating the length of a test to its weight in a composite. Tests of varying length are constructed so that their weights will be of predetermined magnitudes, and the results compared with expectations. Weighting schemes involving standard deviations of raw scores and of true scores are compared. An important secondary derivation is presented from which it is possible to estimate test reliability knowing only the relative length of a test, its shortened form, and the standard deviation of each.