Formulas are derived for unbiased sample estimators of any raw or central moment of the frequency distribution of true scores. A general method is developed for obtaining from each examinee's observed score a least squares estimate of his true score.

It currently seems to me that the heart of mental test theory is the concept of true score.

The trouble starts as soon as it is realized that the actual test score obtained by a particular individual might just as well have been some numerical value other than the one actually observed. The individual examinee might have guessed differently, might have been less nervous, or might have slept better the night before. The testing conditions might have been different–lighting, facilities for use of paper and pencil, presence or absence of distraction. Finally, any one of many different but equally appropriate psychological tests might have been constructed and administered.

Mental test theory must deal with all these kinds of disturbing influences. However, we are not really interested in each of the different test scores that an examinee might obtain under all sorts of conditions. We are interested in something approximated by these scores. This something may be called the true score on the test.

This paper discusses the item selection problem when the item responses follow a linear multiple factor model. Because of this restrictive assumption, not too unrealistic in situations such as mental testing, it is possible to select optimal sets of items without going through all possible combinations. A method proposed by Elfving to accomplish this is analyzed and then demonstrated through the use of two illustrations. The common and often used procedure of observing the magnitude of the correlation coefficient as an index in item selection is shown to have some merit in the single-factor case.

This paper is concerned with methods for analyzing quantitative, non-categorical profile data, e.g., a battery of tests given to individuals in one or more groups. It is assumed that the variables have a multinormal distribution with an arbitrary variance-covariance matrix. Approximate procedures based on classical analysis of variance are presented, including an adjustment to the degrees of freedom resulting in conservative F tests. These can be applied to the case where the variance-covariance matrices differ from group to group. In addition, exact generalized multivariate analysis methods are discussed. Examples are given illustrating both techniques.

A theoretical approach to the understanding of human behavior in uncertain outcome situations is suggested, an approach which draws upon utility theory, decision-making theory, and statistical association theory. Experimental evidence supporting this approach as opposed to alternative approaches is summarized. Three different formalizations are presented, and a variety of experimental tests is suggested.

Domain sampling principles permit formulation of a general method of multidimensional analysis. Cluster and factor analysis methods are special cases stemming from decisions made at different stages of the general method, especially in defining an independent dimension. Key cluster analyses define a dimension as a selection of s variables drawn from the full n set. Centroid, principal axes, and maximum likelihood analyses define it by the n variables (raw or residual, weighted or unweighted); bifactor and second-order analysis, by both types of selection; square root analysis, by one variable. Key cluster methods can be designed to test hypotheses.

Beginning with sets of arbitrary elements, concepts of distance and betweenness of sets are defined. Since betweenness as defined is not transitive, an investigation is made of the conditions which ensure desirable regularity. It is found that a straight line or linear array of sets is a generalization of nested sets (Guttman scales). Close relationships among the notions of distance, betweenness, and linear arrays are demonstrated. Parallel and perpendicular arrays, dimensions, and multidimensional spaces are characterized.

Under the assumption that responses to different dichotomous items are statistically independent, exact distributions for coefficients of reproducibility, CR, are derived. These distributions are useful in testing whether an observed coefficient of reproducibility differs significantly from chance expectations and whether further scaling manipulations are warranted. The effects of purification on sampling distributions of errors are calculated to demonstrate a relationship between scaling operations and expected chance values of the CR.

To explain changes in the sociometric configuration of a group through time, a problem arises of the extent to which such changes may be viewed as the aggregation of part-processes occurring at the level of two-person choice structures. A possible model is a Markov chain in which three possible states are mutual choice, one-way choice, and indifference, one realization for each pair of choosing individuals in the group. Choice data for an eighth-grade classroom are fitted to this model and are used to answer questions of constancy of transition probabilities, order of the chain, and sex differences.

The logistic function is proposed as an alternative to the integrated normal function when estimating parameters of test items. The logistic curve is described; an iterative method for finding maximum likelihood estimates of its parameters is given, and an example of its use is presented.

The usual least squares estimate of the transitional probability matrix of a finite Markov process is given for the case in which, for each point in time, only the proportions of the sample in each state are known. The purpose of this paper is to give another estimate of this matrix and to investigate the properties of this estimate. It is shown that this estimate is consistent and asymptotically more efficient than the previously considered estimate in a sense defined in this paper.

Simple statistical methods are developed to test whether coefficients of reproducibility, of homogeneity, or of consistency differ significantly from what can be expected if responses to different items are statistically independent. Simple methods are also developed for estimating the variance of coefficients of reproducibility when it is not assumed that responses to different items are independent. These estimates are used to test whether a coefficient differs significantly from any prescribed value, and also to obtain confidence intervals for these coefficients. The rationale for the measurement of reproducibility is also discussed.

The factor analysis model and Lazarsfeld's latent structure scheme for analyzing dichotomous attributes are derived to show how the latter model avoids three knotty problems in factor analysis: communality estimation, rotation, and curvilinearity. Then the latent structure model is generalized into latent profile analysis for the study of interrelations among quantitative measures. Four latent profile examples are presented and discussed in terms of their limitations and the problems of latent metric and dimensionality thereby raised. The possibility of treating higher order empirical relations in a manner paralleling their various uses in the latent structure model is indicated.

A quantitative approach to the psychology of pattern recognition requires knowledge of the number of possible variants of any particular pattern. The general solution for the number [p/m2] of pattern variants that p counters can form on a square network of m2 positions is obtained by elementary group theory. The exact solution is given in terms of the different types (symmetric, asymmetric, etc.) of patterns possible and an approximate formula for the total number of patterns is also developed.

A method is presented for securing a hierarchical factor solution which achieves simple structure at each hierarchical level without rotation or even preliminary arbitrary orthogonal or oblique solutions. The method is based upon the assumption that if overlap is removed from clusters the remaining specifics will achieve simple structure automatically. The problem presented earlier by Schmid and Leiman, using oblique simple structural rotation as a basis, is reworked by this new approach.

The Estes model for spontaneous recovery and regression is modified by relaxing the assumption that within-experiment stimulus fluctuation can be treated as negligible in the general case. The augmented model allows description of stimulus fluctuation both between and within experimental periods. Application of the model to mock experimental situations illustrates some of its properties.

A generalization of the Bush-Mosteller learning model is proposed in connection with two-alternative learning situations with continuous reinforcement. The problem is to test the hypothesis that reward and non-reward are equally effective in promoting learning. Statistically this reduces to testing a hypothesis about the value of a single parameter θ, while a set of other parameters remains unspecified. The test presented has the property of being asymptotically locally most powerful among all tests of the same size and asymptotically similar. The application of the test is illustrated.

A battery of 18 tests of intermediate algebra and 20 reference tests was administered to two successive second-year algebra classes. Each battery was separately factor analyzed by Thurstone methods, and the two analyses were synthesized by the Tucker method. The five congruent factors obtained were identified as: Verbal Comprehension, Deductive Reasoning, Algebraic Manipulative Skill, Number Ability, and Adaptability to a New Task.

A class of strategies is defined, each member of which possesses a certain plausibility. If a subject follows any strategy in this class in a two-choice learning experiment of the type dealt with by the Estes model, the subject's long-run behavior will be the same as that predicted by the Estes model.

This paper presents a model of individual choice behavior for application to experimental situations in which a subject is required to compare utility intervals (differences in subjective value). This model is contrasted with a weaker model, which is also derived. Both models generate ordered metric scales, but differ in predictive power. An experiment on the utility of grades, which provides a test and comparison of the models, is presented.