The usual methods of combining observations to give interpoint distance estimates based on interstimulus differences are shown to lead to a distortion of the stimulus configuration unless all individuals in a group perceive the stimuli in perceptual spaces which are essentially the same. The nature of the expected distortion is shown, and a method of combining individual distance estimates which produces only a linear deformation of the stimulus configuration is given.

Some shortcomings of current methods of estimating the magnitude of perceived difference are considered. A statistical model for perceived difference is derived which avoids these difficulties and employs judgments of ratios of differences as data. Three estimators of squared difference are developed.

We describe a general procedure by which any number of parameters of the factor analytic model can be held fixed at any values and the remaining free parameters estimated by the maximum likelihood method. The generality of the approach makes it possible to deal with all kinds of solutions: orthogonal, oblique and various mixtures of these. By choosing the fixed parameters appropriately, factors can be defined to have desired properties and make subsequent rotation unnecessary. The goodness of fit of the maximum likelihood solution under the hypothesis represented by the fixed parameters is tested by a large sample x2 test based on the likelihood ratio technique. A by-product of the procedure is an estimate of the variance-covariance matrix of the estimated parameters. From this, approximate confidence intervals for the parameters can be obtained. Several examples illustrating the usefulness of the procedure are given.

The asymptotic values of response probabilities in the Audley-Jonckheere learning model, as applied to the two-choice learning situation, are derived for (1) the experimenter-controlled events model, (2) the subject-controlled events model, and (3) the experimenter-subject-controlled events model.

Learning-process statistics for absorbing Markov-chain models are developed by using matrix methods exclusively. The paper extends earlier work by Bernbach by deriving the distribution of the total number of errors, u-tuples, autocorrelation of errors, sequential statistics, and the expectation and variance of all statistics presented. The technique is then extended to latency derivations including the latencies of sequential statistics. Suggestions are made for using the sequential-statistic algorithm in a maximum-likelihood estimation procedure. The technique is important because statistics for very large absorbing matrices can be easily computed without going through tedious theoretical calculations to find explicit mathematical expressions.

When magnitude production is used to obtain a psychophysical power function for a group of Ss, the group exponent is shown to be the harmonic mean of individual-S exponents under the conditions usually employed in empirical research. The conditions are (a) the same numbers are presented to each S, (b) responses to each number are pooled over Ss by taking geometric means, and (c) the function is fit by the method of least squares in which regression is of the logarithm of Ss' responses upon the logarithm of numbers presented by E.

This paper presents a procedure for rotating an arbitrary factor matrix to maximum similarity with a specified factor pattern. The sum of squared distances between specified vectors and rotated vectors in oblique Euclidian space is minimized. An example of the application of the procedure is given.