Link and Heath [1975] have analysed a random walk model for two-choice reaction time on the assumption that the two probability density functions (p.d.f.s) of the step-size, each p.d.f. corresponding to one stimulus, are mirror reflections of each other; and they have demonstrated the critical role played by the symmetry of the moment generating function (m.g.f.) of the step size in the determination of whether or not error and correct reaction times are equal. It is shown here that, given reflection symmetry, m.g.f. symmetry is necessary and sufficient for the random walk model to be equivalent to a sequential probability ratio test.

Using an approach nearly identical to one adopted by Guttman, it is established that within the framework of classical test theory the squared multiple correlation for predicting an element of a composite measure from the n − 1 remaining elements is a lower-bound to the reliability of the element. The relationship existing between the reliabilities of the elements of a composite measure and their squared-multiple correlations with remaining elements is used to derive Guttman’s sixth lower bound (λ6) to the reliability of a composite measure. It is shown that Harris factors of a correlation matrix R are associated with a set of (observable) uncorrelated latent variables having maximum coefficients λ6.

All attempts to study the stability of factors depend on having some useful statistic that measures the degree of similarity between factors. This study attempts to provide some normative data about the distribution of one measure of similarity, the congruence coefficient, through a Monte Carlo technique. The matching of “chance” factor patterns was done by the method of Tucker. Statistical tests of the results, based on similarities of the method to canonical and multiple correlation, seemed satisfactory. The tabled results can be used as guides to the significance of congruence coefficients for some cases. The consistencies of the data indicate that a functional resolution may be possible, but none was found.

This brief report provides a comparison of the Z-variance and Box-Scheffé tests for homogeneity of variance. Both procedures are relatively simple to perform and both may be readily utilized in complex, multifactor designs. The Z-variance test is not robust against non-normality; the Box-Scheffé test is robust against non-normality but is not nearly as powerful as the Z-variance test.

Empirical Bayes point estimates of latent trait scores, derived under the assumptions of one of several test theory models, display a certain degree of instability unless the sample size is sufficiently large. A measure of this instability over repeated sampling is the distribution of the overall expected squared error loss which converges, both in probability and in the mean, to the minimum (Bayes) overall expected loss as sample size increases. An asymptotic distribution theory is developed, and the resulting large sample approximation is compared with results obtained from simulated data. Attention is also given to the effects of using a smoothing procedure.

A modified technique of separable programming was used to maximize the squared correlation ratio of weighted responses to partially ordered categories. The technique employs a polygonal approximation to each single-variable function by choosing mesh points around the initial approximation supplied by Nishisato's method. The major characteristics of this approach are: (i) it does not require any grid refinement; (ii) the entire process of computation quickly converges to the acceptable level of accuracy, and (iii) the method employs specific sets of mesh points for specific variables, whereby it reduces the number of variables for the separable programming technique. Numerical examples were provided to illustrate the procedure.

For the six available sets of empirical data, the discrimination (slope) parameter of the logistic item characteristic curve was found to have a significant positive correlation over items with the difficulty (location) parameter. This unpleasant situation can be eliminated by a suitably chosen transformation of the ability scale. An example of the transformation is given and some empirical results of using the transformed scale are shown.

It is argued that (1) the halo-model of Micko is probably too restrictive to fit empirical data, (2) the model misrepresents unrelated percepts as bipolar structures, (3) all variations will probably be interpreted as qualitative, (4) the model requires all dimensions to be bipolar, (5) interpretations of orthogonality of factors and of factor loadings become problematic, (6) the halo may be regarded as a sort of “surface variation,” (7) the common part of two percepts is inadequately defined, and (8) the opposite part of two percepts is neglected.

The small-world problem revolves around the tracing of a line of acquaintances linking any two persons chosen at random. This paper discusses the statistical analysis of data arising out of two experimental studies of the small-world problem. The estimation of parameters in two previously proposed models is discussed, and the goodness-of-fit of these models is considered.

A branch of probability theory that has been studied extensively in recent years, the theory of conditional expectation, provides just the concepts needed for mathematical derivation of the main results of the classical test theory with minimal assumptions and greatest economy in the proofs. The collection of all random variables with finite variance defined on a given probability space is a Hilbert space; the function that assigns to each random variable its conditional expectation is a linear operator; and the properties of the conditional expectation needed to derive the usual test-theory formulas are general properties of linear operators in Hilbert space. Accordingly, each of the test-theory formulas has a simple geometric interpretation that holds in all Hilbert spaces.

A method for dealing with the problem of missing observations in multivariate data is developed and evaluated. The method uses a transformation of the principal components of the data to estimate missing entries. The properties of this method and four alternative methods are investigated by means of a Monte Carlo study of 42 computer-generated data matrices. The methods are compared with respect to their ability to predict correlation matrices as well as missing entries.

The results indicate that whenever there exists modest intercorrelations among the variables (i.e., average off diagonal correlation above .2) the proposed method is at least as good as the best alternative (a regression method) while being considerably faster and simpler computationally. Models for determining the best alternative based upon easily calculated characteristics of the matrix are given. The generality of these models is demonstrated using the previously published results of Timm.

In some situations where reliability must be estimated it is impossible to divide the measuring instrument into more than two separately scoreable parts. When this is the case, the parts may be homogeneous in content but clearly unequal in length. The resultant scores will not be essentially τ-equivalent, and hence total test reliability cannot be satisfactorily estimated via Cronbach's coefficient alpha. Limitation on the number of parts rules out Kristof's three-part approach. A technique is developed for estimating reliability in such situations. The approach is shown to function very well when applied to five achievement tests.

McDonald [1974] obtained Maximum Likelihood (ML) estimates of the free parameters, and an asymptotic likelihood-ratio test, for the hypothesis that one or more elements of a covariance matrix are zero, and/or that groups of two or more of its elements are equal. Estimation was by Newton's method, starting from a closed-form Least Squares (LS) solution that is typically close to the ML solution point. The hypothesis can also be tested using the general model for the analysis of covariance structures given by Jöreskog [1970], but the combination of a closed-form LS starting point and the classical Newton method given by McDonald yields estimates in about a quarter of the computer time needed by JOreskog's program, ACØVS.

The relationship between variables in applied and experimental research is often investigated by the use of extreme (i.e., upper and lower) groups. Earlier analytical work has demonstrated that the extreme groups procedure is more powerful than the standard correlational approach for some values of the correlation and extreme group size. The present article provides methods for using the covariance information that is usually discarded in the classical extreme groups approach. Essentially, then, the new procedure combines the extreme groups approach and the correlational approach. Consequently, it includes the advantages of each and is shown to be more powerful than either approach used alone.