A new item selection technique is presented which takes into account the intercorrelations of the items as well as their correlations with the criterion. The technique is regarded as superior to comparable techniques in that it is considered to achieve greater economy of time, greater objectivity of procedure, higher validity, and higher reliability. The mathematical theory underlying the method is developed. An approximate solution of the mathematical equations is suggested. An approximation procedure for the complete item selection technique is presented, based on the mathematical solution, but much simpler in procedure. The clerical operations involved in the approximation procedure are outlined and illustrated on a sample worksheet.

A variation of the centroid method is described and illustrated. By the application of new rules for reflecting signs, it may be possible to reduce to insignificance the factor loadings of tests showing insignificant correlation (original or residual) with clusters of tests having relatively high intercorrelations. As a result, a factor common to any one of these clusters may be revealed by the centroid method itself with little or no need for rotation of axes or further calculations.

The evaluation of the level of difficulty of a test item is ordinarily derived from the proportion of a specified population passing or failing the item. With items that have a limited number of alternative responses there must be a correction in this proportion to make allowance for chance success. A table of corrected proportions is given for different numbers of alternatives varying from two to eight.

In continuation of a previous paper, a mechanism of delayed reflexes is considered more in detail. Equations governing such a mechanism are established and approximately solved. The formulae thus obtained describe the phenomenon of “concentration” of a conditioned reflex around a definite time interval after stimulation. Applied along the lines discussed in the previous paper to some simple combinations of stimuli and responses the formulae lead to a description of the elimination of errors by trial. They give a relation between the number of repetitions, necessary to eliminate a wrong act, and other constants, describing the situation.

The implications contained in Richardson's article on item analysis in March 1936 issue of Psychometrika are examined in the light of multiple factor theory. It is shown that item analysis is a necessary, but not a sufficient condition for the construction of a test which shall measure a single trait. The intercorrelations of certain items selected by a method of item analysis are examined, found to contain many zero and some negative correlations. Multiple factor analysis showed that eight traits were measured by the items which had been asserted to measure only one.

The resolution of a set of variables into principal components may be performed upon variables having arbitrary variances, or upon variables obtained from these by reduction of all variances to unity. The two procedures do not yield the same results even if the principal components are reduced to unit variance in both cases, except when the original variances are all equal.

A derivation of equations fundamental to the technique of path coefficients is given. Suggestions are made with respect to the calculations required in the use of the technique. The relations of the technique to those of partial correlation, semi-partial correlation, part correlation, multiple correlation, and factor analysis are discussed. Some consideration is given to the merits and limitations of the technique of path coefficients.