In many a psychometric problem, particularly in practical problems, the only data conveniently obtainable are the first choices given to certain stimuli among a list of stimuli that are available for selection. By assuming that such choices represent real comparative judgments, we may apply Thurstone's law of comparative judgments, extract experimental proportions from the numbers of first choices, and compute psychological scale values for the stimuli. Two procedures are proposed for estimating such proportions and examples of their applications are given. A procedure for allocating a meaningful zero point on the scale by the use of absolute judgments is explained and demonstrated. Suggestions are added for overcoming certain weaknesses and limitations of the method of choices.

Papers involving factorial methods appearing in current literature often misuse factorial methods. These meaningless results are reached because of neglect of certain conditions basic to factor theory. The conditions are:

1. (1) The number of basic factors must be smaller than the number of tests.

2. (2) The diagonals of the correlation matrix must be regarded as unknown.

3. (3) The axes must be rotated into a simple configuration.

4. (4) Each factor must be overdetermined by appearance in several tests.

5. (5) Tests should have simple factorial composition.

The error function of the theoretical learning curve developed by Thurstone has been fitted to maze data from rats that have been subjected to cerebral cortical insult. There is a high degree of relationship between the number of errors necessary to complete learning as predicted from the curve and the experimental measure of learning, total-errors-minus-errors-first-trial. The procedure provides a method of analysis for individual learning records and it may be used to shorten the training period. The goodness of fit indicates that the underlying logic of the learning curve is plausible.

Differences in magnitude among the communalities of a set of tests may influence the convergence of successive approximations to the limiting hyperplanes of the configuration. A slight alteration in technique is presented, which may be useful in avoiding this effect and perhaps hastening convergence.

A table to facilitate the computation of Fisher's t function for the comparison of two means is presented. The derivations leading to the construction of the table, and the necessary instructions for its use, are given.

Since item values obtained by item analysis procedures are not always stable from one situation to another, it follows that selection of items for validity or difficulty is sometimes useless. An application of Chi Square to testing homogeneity of item values is made, in the case of the UL method, and illustrative data are presented. A method of applying sampling theory to Horst's maximizing function is outlined, as illustrative of author's observation that the results of item analysis by any of various methods may be similarly tested.

The problem of fitting a least-squares linear subspace to a higher dimensional set of points is solved by the use of matrix approximation results, both when the subspace is required to pass through a given point, and when it is not so restricted. In the latter case the subspace is shown to pass through the centroid of the point set. It is noted that the principal axes of the set of points enter into the problem in a natural manner.

The theoretically best estimate of the reliability coefficient is stated in terms of a precise definition of the equivalence of two forms of a test. Various approximations to this theoretical formula are derived, with reference to several degrees of completeness of information about the test and to special assumptions. The familiar Spearman-Brown Formula is shown to be a special case of the general formulation of the problem of reliability. Reliability coefficients computed in various ways are presented for comparative purposes.

Since the factor problem is reduced mathematically to the expression of the obtained correlational matrix in terms of a matrix of lower rank, criteria for the determination of this lower rank are of first importance. These criteria are investigated by means of certain mathematical deductions, and brought into relation with Spearman's and Thurstone's factor theories.

In general, the methods of factor analysis developed during the past five years are based on the reduction of the correlational matrix by successive steps. The first factor loadings are determined and eliminated from the correlational matrix, giving a residual matrix. This process is continued for successive factor loadings until the elements of the last obtained residual matrix may be regarded as due to chance. The method outlined in this paper assumes the maximum number of factors m in the correlational matrix. The m factor vectors are solved for simultaneously. Once the m factor vectors are found, any vectors having only negligible factor loadings may be discarded.

An interpretation of the semi-major axis of the Thurstone-type learning curve fitted by the authors to animal maze data is given. This interpretation affords a quantitative description of “insight”.

The basic conditions and characteristics of measurement are outlined. The concepts entering into the measurement must be operationally defined; there must be a satisfactory representation of the character to be measured, and a quantitative comparison between the observed degree of the character and some reference point or scale. A conception of measurement which is too narrow would unnecessarily restrict investigation in the fields of psychology and education.

The formula for the Pearson correlation coefficient of a dichotomous variable with a multiple-categoried variable is simplified for computational purposes by effecting in the multiple-categoried variable two types of arbitrary distributions: (1) rectangular and (2) proportional to binomial expansion coefficients. The formulas which result are convenient for the selection of test items and are applicable to the objective estimation of the comparative merits of the alternatives in multiple-choice test items. It is shown that the authoritative answer should have a high positive criterion coefficient, while the omissions and several wrong-answer alternatives should each have low (algebraic) negative criterion coefficients.

This paper presents three theorems concerning the relation between results with obtained and corrected correlation coefficients in the Thurstone method of multiple factor analysis. (1) The rank of the correlational matrix, and thus the number of factors involved, is unaffected by correcting the obtained coefficients for attenuation. (2) The communality of a variable when the obtained coefficients have been corrected for attenuation is equal to the communality with obtained coefficients divided by the reliability coefficient of the variable. (3) The relationship is shown between the factorial matrix of a correlational matrix of raw correlation coefficients and the factorial matrix of a correlational matrix of corrected correlation coefficients, and a method of obtaining either of these factorial matrices from the other is indicated.

When several criteria are available, it is ordinarily necessary (1) to select one of them as the criterion, (2) to use several and thus arrive at several different sets of weights, or (3) to combine them into a single measure. A formula is derived for the determination of a unique set of weights. The use of these weights will produce the highest possible average coefficient of correlation between the various criteria and two (or more) weighted independent variables. If desired, the criteria may be assigned any predetermined weights. The weights then derived for the independent variables are such that the weighted average of the correlation coefficients between the various criteria and the independent variable composite will be a maximum. In the use of these formulas, no assumptions are necessary regarding the interrelationships existing among the criteria and it is not necessary to compute the intercorrelations among the criteria. A numerical example is included.

It is shown that in certain cases practice data approximately meet the assumptions involved in Thurstone's method of absolute scaling. An application of the method was accordingly made in the case of four test performances practiced for 39 days by a group of 56 subjects. The manner in which the practice data were scaled is described by using the data on practice in anagrams as an illustration. Scaling had little effect upon the correlations between initial and final score, but produced marked changes in the apparent effect of practice upon individual differences and in the correlations between initial score and gain.

The Bi-factor Method of factor analysis is described and illustrated with a small group of fourteen tests. A detailed illustration is given of how the method may be modified to the case of overlapping group factors. It is advocated that the Bi-factor pattern in unmodified form be used to determine the adequacy of tests for the measurement of unitary traits.

A technique is indicated by which approximations to the factor loadings of a new test may be obtained if factor loadings of a given group of tests and the correlations of the new test with the other tests are known. The technique is applicable to any orthogonal system and is especially adapted to cases in which Σajiajk = 0 when i ≠ k. Application is also made to the simultaneous determination of the factor weights of a group of tests in which no additional common factor is present. The technique is useful in adding tests to a completed factorial solution and in using factorial solutions involving errors to give results which are approximately correct.

Offner's demonstration of the excitation time equivalence, for all currents, of the theories of Hill and Rashevsky is here extended to a more general case which includes both. It is found that any system of this general type may be replaced by an equivalent one of the Rashevsky type, thus effecting considerable simplification in the mathematical detail.

The relations between ability, motivation, and speed in a mental task are represented by a psychometric surface. Ability of a subject is defined as the degree of difficulty for which the probability is 1/2 that he will complete the task in infinite time. Motivation affects the rate at which mental work is done, but not the altitude of performance. Suggestions towards experimental work are made.