In the first of two parts in which a general mathematical theory of non-symbolic learning and conditioning is constructed, the sections of the theory dealing with non-symbolic learning and conditioning are presented, and a number of its qualitative implications are compared with available experimental results. In general, the agreement is found to be rather close.

This paper presents a test for determining significance of differences between means of samples which are drawn from positively skewed populations, more specifically, those having a Pearson Type III distribution function. The quantity 2npxg/xp (where p equals the mean squared divided by the variance and n is the number of cases in the sample), which distributes itself as Chi Square for 2np degrees of freedom, may be referred to the tables of Chi Square for testing hypotheses about the value of the true mean. For two independent samples, the larger mean divided by the smaller mean, which distributes itself as F for 2n1p1 and 2n2p2 degrees of freedom, may be referred to the F distribution tables for testing significance of difference between means. The test assumes that the range of possible scores is from zero to infinity. When a lower theoretical score limit (c) exists which is not zero, the quantity (Mean − c) should be used instead of the mean in all calculations.

R. A. Fisher's method of determining a set of weights for the linear combination of the scores in a battery of tests is explained and illustrated. The relationship of this method to combination by multiple regression methods is shown.

This article deals with a factorial analysis of Thurstone's Primary Mental Abilities Tests. The analysis was made in order to determine whether the tests designated to measure a particular ability would be found upon completion of the analysis to be clustered together with significant loadings and be isolated from the tests of the other abilities. The results show that most of the tests specifically designed to measure six of the abilities were isolated with loadings varying in size. The tests for the remaining ability were not isolated. It is also shown that the tests are complex in nature, measuring more than any one single ability.

Tests designed to measure what was conceived to be attention were found by factor analysis to involve a factor which is independent of the factors of rote memory, visual-space, number, or perception. To a large extent, at least, this attention factor is independent of content and of mode of presentation of test material. The tests in which the variance is mainly dependent upon this factor are those involving a high degree of sustained or relatively continuous mental effort.

A case of interaction between two groups of “active” and one group of “passive” individuals, in which the efforts of the influencing active groups decrease with increasing total success in the past, is studied. In that case the numbers of passive individuals, exhibiting respectively the two opposite behaviors, fluctuate periodically, with a positive damping.

The coefficients of rank difference correlation that are barely significant at six different levels of significance are given for N's of 2 to 30. Most of the values were obtained by translation of Olds' tables of probabilities for various values of Σd2. Comparison of these data with those obtained by four other methods indicates that one method yields values more appropriate than those obtained from Olds' data for coefficients significant at the .01 level for N's from 11 to 25. This method also provides a convenient means of obtaining approximate values of coefficients significant at the .01 level for N's above 30. Need for caution in evaluating the significance of coefficients obtained from data involving tie rankings is indicated. The article concludes with recommendations as to choice of methods of determining the significance of rank difference coefficients.

The mathematical derivation of a test for determining the fiducial limits of, and significance of difference between, means when the samples are drawn from exponential populations is presented. The test for differences between means takes the particularly simple form of the F test (the ratio of the larger to the smaller mean) with each mean possessed of 2n degrees of freedom, n being the number of cases in the sample. Random sampling, a range of scores upwards from a lower limit of zero, and independence of means from each other are necessary assumptions for the use of the test. Examples of situations in which the test should be used are given, together with a description of the necessary computational procedures. Comparisons of the results of the application of this test with the erroneous application of the critical ratio on actual data show that rather large discrepancies exist between the two tests. Results obtained by applying tests which assume normality to exponential distributions are subject to much error.

A general formula for the reliability of a weighted composite has been derived by which that reliability can be estimated from a knowledge of the weights whatever their source, reliabilities, dispersions, and intercorrelations of the components. The Spearman-Brown formula has been shown to be a special case of the general statement. The effect of the internal consistency or intercorrelation of the components has been investigated and the conditions defining the set of weights yielding maximum reliability shown to be that the weight of a component is proportional to the sum of its intercorrelations with the remaining components and inversely proportional to its error variance.

An outline for a course in test theory is presented, together with a list of assignments, problems, and a bibliography. The course has been given in the Psychology Department of the University of Chicago. The material is presented in outline form at the present time because of the increased need for training in test theory due to the increase in the use of psychological tests for classification of military personnel, and because much of the material in such a course must be selected from a wide array of articles in the literature. This material is presented in order that an organized body of material for instructional purposes may be readily available to those interested.

In generalization of a previous study, a mathematical approach to the theory of the average size of cities as well as of the distribution of city sizes is outlined.

Six motor tests and six nonverbal tests were administered four times to the same subjects. Subjective reports of the subjects are discussed, changes in mean scores and in variability and score correlations from trial to trial are surveyed, and factor analyses of results on the first and fourth trials are presented and compared. Implications of the findings with respect to correction for attenuation are pointed out.

Given a battery of n tests that has already been resolved into r orthogonal common factors and n unique factors, procedures are outlined for computing the following types of linear multiple regressions directly from the factor loadings: (i) the regression of any one test on the n−1 remaining tests; (ii) all the n different regressions of order n−1 for the n tests, computed simultaneously; (iii) the regression of any common factor on the n tests; (iv) the regressions of all the common factors on the n tests computed simultaneously; (v) the regression of any unique factor on the n tests; (vi) the regressions of all the unique factors on the n tests, computed simultaneously. Multiple and partial correlations are then determined by ordinary formulas from the regression coefficients. A worksheet with explicit instructions is provided, with a completely worked out example. Computing these regressions directly from the factor loadings is a labor-saving device, the efficiency of which increases as the number of tests increases. The amount of labor depends essentially on the number of common factors. This is in contrast to computations based on the original test intercorrelations, where the amount of labor increases more than proportionately as the number of tests increases. The procedures evaluate formulas developed in a previous paper (2). They are based essentially on a shortened way of computing the inverse of the test intercorrelation matrix by use of the factor loadings.

It is shown that approaches other than the internal consistency method of estimating test reliability are either less satisfactory or lead to the same general results. The commonly attendant assumption of a single factor throughout the test items is challenged, however. The consideration of a test made up of K sub-tests each composed of a different orthogonal factor disclosed that the assumption of a single factor produced an erroneous estimate of reliability with a ratio of (n−K)/(n−1) to the correct estimate. Special difficulties arising from this error in application of current techniques to short tests or to test batteries are discussed. Application of this same multi-factor concept to item-analysis discloses similar difficulties in that field. The item-test coefficient approaches √1/K as an upper limit rather than 1.00 and approaches √1/n as a lower limit rather than .00. This latter finding accounts for an over-estimation error in the Kuder-Richardson formula (8). A new method of isolating sub-tests based upon the item-test coefficient is proposed and tentatively outlined. Either this new method or a complete factor analysis is regarded as the only proper approach to the problem of test reliability, and the item-sub-est coefficient is similarly recommended as the proper approach for item analysis.

By application of the technique presented earlier, the factors found by the Guilfords in two studies of personality factors are reanalyzed. The rotation technique for the isolation of meaningful factors relates the factors of the two studies to each other through three common items, and to other factors and traits appearing in the literature of personality measurement. For each of four of the factors resulting from the analysis of the first battery there is close agreement with its counterpart from the analysis of the second battery with respect to loadings of the three common items. These factors are interpreted as perseverance, surgency, flexibility, and tension. Other factors appearing in only one or the other of the batteries are tentatively identified.

The concept of simple structure is criticized for lack of objectivity and for failure to produce invariance of (a) factor loadings under change of battery, and (b) individual scores on primary traits under change of tests. It is also criticized on the grounds that simple structure yields at best the factors which were put in and that, by suitable manipulation of tests, any set of factors may be represented in a battery by tests which will yield a simple structure. A procedure for rotation is developed which locates the first rotated axis to pass through a cluster of tests which, by hypothesis, contain a common factor and to project all tests into a hyperplane orthogonal to this factor. The second factor is then located to pass through the projections of a second cluster of tests defining the second factor, and so on, until all hypothecated factors have been located. Any residual interrelations may then be rotated graphically to the most plausible arrangement.

Directions are given for constructing a very simple nomograph for computing medians, which is entered with information from the cumulative frequency distribution. It gives a linear interpolation within the class interval containing the median.

A matching method proposed by Dr. C. E. Stuart is presented in some detail and the essentials for a test of significance are derived. This method differs from the older matching methods in that partial credit is allowed for a near miss. A slight variation of the method permits the matching of one item with M sets of n traits.

This article presents a Coefficient of Imbalance applicable to any type of communication that may be classified into favorable content, unfavorable content, neutral content, and non-relevant content. The combined influence of the average presentation of relevant content and the average presentation of total content is reduced to two components, the coefficients of favorable imbalance and of unfavorable imbalance. A precise definition of imbalance is developed and measured against ten criteria.

Tables are given for σr√N for the tetrachoric correlation coefficient for the following values of the correlation in the population: .00, ± .10, ± .20, ..., ± .80, ± .90, ± .95.