Factor analysis has been proposed and used as a method of statistical analysis of several measurements made on one individual repeatedly over time. This paper discusses some difficulties in applying factor analysis to multiple time series and attempts to indicate to what extent such methods can accomplish the goals of time series analysis. Some other methods are suggested.

This paper is a presentation of the statistical sampling theory of stepped-up reliability coefficients when a test has been divided into any number of equivalent parts. Maximum-likelihood estimators of the reliability are obtained and shown to be biased. Their sampling distributions are derived and form the basis of the definition of new unbiased estimators with known sampling distributions. These unbiased estimators have a smaller sampling variance than the maximum-likelihood estimators and are, because of this and some other favorable properties, recommended for general use. On the basis of the variances of the unbiased estimators the gain in accuracy in estimating reliability connected with further division of a test can be expressed explicitly. The limits of these variances and thus the limits of accuracy of estimation are derived. Finally, statistical small sample tests of the reliability coefficient are outlined. This paper also covers the sampling distribution of Cronbach's coefficient alpha.

A quantitative system is presented to permit the determination of separate multidimensional perceptual spaces for individuals having different viewpoints about stimulus interrelationships. The structure of individual differences in the perception of stimulus relationships is also determined to provide a framework for ascertaining the varieties of consistent individual viewpoints and their relationships with other variables.

This paper is a presentation of an essential part of the sampling theory of the error variance and the standard error of measurement. An experimental assumption is that several equivalent tests with equal variances are available. These may be either final forms of the same test or obtained by dividing one test into several parts. The simple model of independent and normally distributed errors of measurement with zero mean is employed. No assumption is made about the form of the distributions of true and observed scores. This implies unrestricted freedom in defining the population. First, maximum-likelihood estimators of the error variance and the standard error of measurement are obtained, their sampling distributions given, and their properties investigated. Then unbiased estimators are defined and their distributions derived. The accuracy of estimation is given special consideration from various points of view. Next, rigorous statistical tests are developed to test hypotheses about error variances on the basis of one and two samples. Also the construction of confidence intervals is treated. Finally, Bartlett's test of homogeneity of variances is used to provide a multi-sample test of equality of error variances.

A subject in a two-choice situation characteristically makes several observing responses before performing the final choice. This behavior can be described by means of a random walk model. The present paper explores some possibilities as to how this model can be extended to include choice time. The assumption is made that the duration of each step in the random walk is a random variable which is exponentially distributed. With this assumption, one can predict the probability distributions of the choice times as well as the moments of these distributions.

Criterion measures are frequently obtained by averaging ratings, but the number and kind of ratings available may differ from individual to individual. This raises issues as to the appropriateness of any single regression equation, about the relation of variance about regression to number and kind of criterion observations, and about the preferred estimate of regression parameters. It is shown that if criterion ratings all have the same true score the regression equation for predicting the average is independent of the number and kind of criterion scores averaged.

Two cases are distinguished, one where criterion measures are assumed to have the same true score, and the other where criterion measures have the same magnitude of error of measurement as well. It is further shown that the variance about regression is a function of the number and kind of criterion ratings averaged, generally decreasing as the number of measures averaged increases. Maximum likelihood estimates for the regression parameters are derived for the two cases, assuming a joint normal distribution for predictors and criterion average within each subpopulation of persons for whom the same type of criterion average is available.

Forty-two choice models, each representing stimuli by one-dimensional probability distributions, are obtained by relaxing the assumptions of Thurstone's Case V Law of Comparative Judgment. The models which imply or fail to imply each of nine testable probabilistic conditions are determined. Stochastic transitivity is vulnerable in most of these models. The results suggest discarding weak stochastic transitivity, and in its place using the conjunction of weak stochastic transitivity and the triangular condition. However, unless it is possible to predict which stimuli will produce violations of the conditions, none of the conditions can be rejected on the basis of too frequent intransitive triads of choices.

A generalization of direct ratio scaling methods to multidimensional ratio scaling is described. This method requires an observer to report the proportion of a standard percept that is contained in a given percept and vice versa. The method was developed to meet requirements for experimentation in such areas as color vision, gustation, and olfaction.

Proof is given for a theorem stated but not proved by Eckart and Young in 1936, which has assumed considerable importance in the theory of lower-rank approximations to matrices, particularly in factor analysis.

A general formulation of the power law is presented which has two special features: (1) negative exponents are admissible; and (2) the log law is a special limiting case. Estimation procedures, which provide joint estimates of the exponent and the absolute threshold, are derived for the direct ratio scaling methods. A solution is provided for the averaging problem for ratio production and bisection scaling, two methods generating observations on the physical scale, and Monte Carlo methods are used to evaluate the resulting estimators.

A method is illustrated for estimating correlation coefficients and mean criterion scores in a full-range population from bivariate distributions available in a selected sample when the criterion is a dichotomy. The proposed method requires only the assumptions needed for use of formulas for correcting correlation coefficients for restriction of range when both variables are continuous and is suitable for use when restriction is due to either direct or indirect selection.

The defects of the least-squares or multiple-regression equation approach to estimating orthogonal factors are discussed and transformations of the beta weights are derived which remove these defects with minimum loss in correlations between estimators and true factor scores.

An axiomatization of mental test theory is presented. The presentation deals explicitly with the following concepts: true score, parallel forms, reliability, validity, and combined test score.

Three models for preference behavior are developed within the framework of Coombs' theory of data, an absolute difference model, a ratio model, and a two-stage model. Each of these models describes a mechanism by which unilateral preferences may be determined on a unidimensional J scale. Differential implications of the models for response latencies are derived, and some early data employing an application of the unfolding technique are presented in support of the two-stage model.

Moses has proposed a nonparametric test of the null hypothesis that two samples, say a treatment and a control group, have been drawn from the same population; his test is designed to be sensitive to the alternative hypothesis that the treatment group is extreme in either or both directions as compared with the controls. This paper presents a test of similar character, except that rather than being based on a range of the control group ranks, it depends upon the variance of those ranks. This has the advantage of making ties less troublesome. Tables of significance values are presented for some small sample sizes. A normal approximation is developed and seen to be useful.

The purpose of this paper is to develop mathematical formulations of test reliability when the estimation of reliability requires successive observations and these observations are not independent of the process (or act) of measurement. The specific application of concern arises in performance testing where performance varies as a function of practice and where practice accrues as a result of measurement. Repeated measures are experimentally nonindependent in such cases. Formulations for the estimation of reliability are developed for each of two cases. The first of these assumes a linear model for acquisition; the second assumes a nonlinear (quadratic) model. Illustrative data are presented whereby insights may be gained as to the inadequacy of the models.

A set of definitions leading to an axiom set which makes possible distinction between two or more forms of lexicographic choice is offered. It is shown that, by treating value as quantized and only taking a finite number of levels for each attribute of a multi-attribute choice situation, lexicographic evaluation might be subsumed under more familiar forms of choice model. Special cases and implications are discussed.

The linear discriminant function and the generalized distance function, two special cases of discriminant technique, require multivariate normality and homogeneous variance-covariance matrices, and hence utilize only mean differences among groups. The more general methods can also utilize differences in variances and/or covariances. Tables are given showing the discriminatory value of differences in means, variances, and intercorrelations, taken singly. Equations which utilize all such differences are given for the normal multivariate distribution.