Psychometrika

On viewing Thurstone's psychophysical scale from the point of view of the mathematical theory of one-parameter continuous groups, it is seen that a variety of different psychological or statistical assumptions can all be made to lead to a scale possessing similar properties, though requiring different computational techniques for their determination. The natural extension to multi-dimensional scaling is indicated.

Orthogonal rotation of canonical variates is shown to preserve the major properties of the canonical solution and may increase its interpretability.

A new rotational method, particularly appropriate when a positive manifold can be assumed, is presented in theory and in computational detail. It is applied to a five-dimensional factorial matrix in which a simple configuration is known to exist. The new technique is much more efficient in yielding the simple configuration than are the methods formerly used. Landahl's transformations can be combined with this method to advantage.

In many situations it is desirable or necessary to administer a set of tests to several different groups, and to ask if the results obtained in the different groups may be regarded as being essentially the same in some sense. In the case of two variables (one dependent and one independent) one may, for instance, ask if the errors of estimate and the regression lines may be regarded as being the same for the populations from which the different groups are drawn. For this case, the present article considers tests for three hypotheses regarding the populations from which the different groups are drawn: (a)H A, the hypothesis that all standard errors of estimate are equal; (b)H B, the hypothesis that all regression lines are parallel, (assumingH A); and (c)H C, the hypothesis that the regression lines are identical, (assumingH B). Test criteria for these three hypotheses and their sampling theory for large samples are presented. The results are extended to the case of several independent variables. An illustrative problem is presented for two groups, two independent and one dependent variable.

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.

In order to investigate certain hypotheses concerning the nature of number ability, and, secondarily, the nature of perceptual speed, a battery of thirty-four tests was given to 223 Chicago high school seniors and the data were factored by the centroid method. Seven primary factors were identifiable upon rotation. Several deductions are made relative to the interpretation of the factors and relative to the consistency of the data with the hypotheses which were to be tested.

This discussion paper argues that both the use of Cronbach’s alpha as a reliability estimate and as a measure of internal consistency suffer from major problems. First, alpha always has a value, which cannot be equal to the test score’s reliability given the interitem covariance matrix and the usual assumptions about measurement error. Second, in practice, alpha is used more often as a measure of the test’s internal consistency than as an estimate of reliability. However, it can be shown easily that alpha is unrelated to the internal structure of the test. It is further discussed that statistics based on a single test administration do not convey much information about the accuracy of individuals’ test performance. The paper ends with a list of conclusions about the usefulness of alpha.

The properties of a simple neural mechanism are developed theoretically and interpreted in terms of observable variables. The results are discussed in connection with experimental data in cases for which the latter are available.

A solution for Case III of the Law of Comparative Judgment, modeled after Thurstone's solution for Case IV but eliminating the restrictive assumption of relatively equal discriminal dispersions, is developed.