The derivation of multiple and partial regression statistics from uniqueness-augmented factor loadings, presented in the literature for orthogonal factor solutions, is generalized to oblique solutions. A mathematical rationale for the general case, without restriction to uncorrelated factors, is presented. Use of the general formulation is illustrated with a two-factor, seven-variable example.

A variance-components analysis is presented for paired comparisons in terms of three components: s, the scale value of the stimuli; d, a deviation from the linear model specified by the law of comparative judgment; and b, a binomial error component. Estimates are given for each of the three variances, σs2, σd2, and σb2. Several coefficients, analogous to reliability coefficients, based on these three variances are indicated. The techniques are illustrated in a replicated comparison of handwriting specimens.

Until recently, two happy vices stole a good share of my time. Of those burglars, the first was the reading of science-fiction stories, a source of entertainment that I discovered at the age of eight, but kept manfully in check because I never stumbled upon a continuous supply. Of late, however, real life has mocked this branch of fiction so closely that I now miss in it the elements of fantasy and escape that used to delight. You might say that, being crowded out by the real thing, we armchair spacemen face technological obsolescence.

My second time-stealer has been “Murder for Pleasure,” as Howard Haycraft calls the detective story. Fortunately, I did not discover its existence until my twenties, so I had a little time for study before then. The great value of the detective story, quite apart from its educational aspects– after all, where else can one gather so much data about useful everyday questions such as tasteless poisons, homicidal law, and criminal psychology–is the fine training one gets in applied psychology and in the logic of everyday life.

An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carroll's criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that the normal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure –although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.

The inter-battery method of factor analysis was devised to provide information relevant to the stability of factors over different selections of tests. Two batteries of tests, postulated to depend on the same common factors, but not parallel tests, are given to one sample of individuals. Factors are determined from the correlation of the tests in one battery with the tests in the other battery. These factors are only those that are common to the two batteries. No communality estimates are required. A statistical test is provided for judging the minimum number of factors involved. Rotation of axes is carried out independently for the two batteries. A final step provides the correlation between factors determined by scores on the tests in the two batteries. The correlations between corresponding factors are taken as factor reliability coefficients.

Implications of random error of measurement for the sensitivity of the F test of differences between means are elaborated. By considering the mathematical models appropriate to design situations involving true and fallible measures, it is shown how measurement error decreases the sensitivity of a test of significance. A method of reducing such loss of sensitivity is described and recommended for general practice.

The specification of sample size is an important aspect of the planning of every experiment. When the investigator intends to use the techniques of analysis of variance in the study of treatments effects, he should, in specifying sample size, take into consideration the power of the F tests which will be made. The charts presented in this paper make possible a simple and direct estimate of the sample size required for F tests of specified power.

Guttman's principal components for the weighting system are the item scoring weights that maximize the generalized Kuder-Richardson reliability coefficient. The principal component for any item is effectively the same as the factor loading of the item divided by the item standard deviation, the factor loadings being obtained from an ordinary factor analysis of the item intercorrelation matrix.

It is suggested that the ambiguity of a set of paired comparison judgments may be measured by the quantity √σi2 + σj2 − 2rijσiσi, This quantity is termed the comparatal dispersion. A simultaneous solution for scale values and ratios of comparatal dispersions has been presented and applied to some data on food preferences.

The question is raised as to whether the null hypothesis concerning the number of common factors underlying a given set of correlations should be that this number is small. Psychological and algebraic evidence indicate that a more appropriate null hypothesis is that the number is relatively large, and that smallness should be but an alternative hypothesis. The question is also raised as to why approximation procedures should be aimed primarily at the observed correlation matrix R and not at, say, R−1. What may be best for R may be worst for R−1, and conversely, yet R−1 is directly involved in problems of multiple and partial regressions. It is shown that a widely accepted inequality for the possible rank to which R can be reduced, when modified by communalities, is indeed false.

Consideration is given to determination of parameters of a functional relation between two variables by the means of factor analysis techniques. If the function can be separated into a sum of products of functions of the individual parameters and corresponding functions of the independent variable, particular values of the functions of the parameters and of the functions of the independent variables might be found by factor analysis. Otherwise approximate solutions may be determined. These solutions may represent important results from experimental investigations.

Three methods of rotation (the graphical, the Thurstone analytical, and the direct-rotational) were applied to the matrix of centroid loadings for 35 variables, to determine which method is the most efficient from theoretical and practical standpoints. The direct-rotational method provided the most information for determining the rank of the configuration and was most economical with respect to time required to reach a rotational solution. The analytical method required the least number of judgmental decisions and was the most objective. The graphical method was the most laborious but had a slight advantage with regard to the number of near-zero loadings in the rotational solution.

This study compares a quartimax rotation of the centroid factor loadings for Thurstone's Primary Mental Abilities Test Battery with factorings of the same correlation matrix by Thurstone (simple structure), Zimmerman (revised simple structure), Holzinger and Harman (bi-factor analysis), and Eysenck (group factor analysis). The quartimax results agree very closely with the solutions of Holzinger and Harman and of Eysenck, and reasonably well with the two simple structure analyses. The principal difference is the general factor provided by the quartimax solution. Reproduction of the factorial structure is sufficiently good to justify its use at least as the first stage of rotation. More extensive trial of the method will be needed with more varied data before it will be possible to decide whether quartimax factors meet psychological requirements sufficiently well without further rotation.

Two different linear models are presented for the four-dimensional classification system in which correlations exist between certain pairs of observations. Except for the assumption of correlated observations, classical assumptions associated with classification systems are made. The models considered are modifications of those which underlie the split-plot design and the split-split-plot design. In the first model the correlations between observations of the levels of one dimension are all set equal to ρ. In the second model the observations of the levels of one dimension are assumed correlated to degree ρ1, whereas the observations of a second dimension are correlated to degree ρ2. Analyses for the two models and tests of hypotheses for various parameters are indicated.

A theory for discrimination learning which incorporates the concept of an observing response is presented. The theory is developed in detail for experimental procedures in which two stimuli are employed and two responses are available to the subject. Applications of the model to cases involving probabilistic and nonprobabilistic schedules of reinforcement are considered; some predictions are derived and compared with experimental results.

A stochastic process applicable to the learning behavior of an individual subject is discussed. The process describes both the response times and the sequence of choices obtained from a situation involving two alternatives. Parameter estimates and techniques for assessing goodness of fit are considered.

Consistency in paired comparison data is defined. Two types of inconsistency which may arise are defined. Computational formulas for these types of inconsistency are derived, and examples illustrating the use of these formulas are presented.

Nonparametric tests are discussed in relation to parametric tests. A distinction is made between two types of nonparametric tests. One type leads to an exact significance level, the other to an approximate significance level. The failure to distinguish between these two types has led to confusion and error. Examples are cited.

The analytical method for simple structure proposed by Thurstone is applied to four separate cases and found to yield satisfactory results. The simple structure obtained by Thurstone's method is found to match closely that obtained by other methods and corresponds to the true structure of the matrix in those cases where true structure is known. Difficulties about the choice of the correct trial vector led the writer to develop a modification of Thurstone's method, useful where high speed computational facilities are available. Instructions are given for this so-called mass modification, and the procedure is illustrated with a 5-factor, 14-variable example. While the results do not fully correspond to a previous graphical solution, it can be argued that the results obtained by the new method show an improved simple structure. The modified method is applied to three other correlation matrices, yielding in each case a satisfactory simple structure.

A three-component model for comparative judgment which allows for individual differences in preference is proposed. An implication of the model is that errors in the observed proportions due to sampling individuals in paired comparisons experiments are correlated. By neglecting this correlation, Mosteller's test for the method of paired comparisons tends to accept falsely the goodness of fit of the Case V solution. It is shown that bounds may be set for the correlation effect which make a valid test possible in some cases and provide useful standard errors for the estimated affective values.