A simple model for fluctuating interresponse times is developed and studied. It involves a mechanism that generates regularly spaced excitations, each of which can trigger off a response after a random delay. The excitations are not observable, but their periodicity is reflected in a regular patterning of responses. The probability distribution of the time between responses is derived and its properties are analyzed. Several limiting cases are also examined.

Arguments for using multiple cutting scores are theoretically inapplicable when the selection measures are fallible. The effect of errors of measurement in altering the shape of some optimum selection regions is here investigated mathematically, with numerical illustrations, for the case of two selection variables.

The problem of defining and determining the effective contribution of a component variable to the variance of a composite is briefly reviewed. Another method of dealing with this problem is proposed and illustrated with analysis of a three-component problem. The proposed method is based on systematic application of a series of regressions and is restricted in application to positive manifold systems.

Indexes of skewness and kurtosis for a test-score distribution are expressed in terms of item parameters. Both are shown to depend, in part, on item means, variances, and covariances. The index of skewness depends also on trivariances. A trivariance is a product moment involving first powers of deviation scores for three items. The index of kurtosis depends on quadrivariances, as well as trivariances. A quadrivariance is a product moment involving first powers of deviation scores for four items. Empirical data are presented for responses of groups of subjects to 25 triads and 25 tetrads of items from five tests.

A derivation is presented for correcting correlations for indirect restriction of range when correlations between one selection variable and non-interval measures are involved. Following a systematic investigation of special cases, formulas are given for practical application in validation work. Procedures for the case with several selection variables are briefly discussed.

The negative hypergeometric distribution of raw scores on mental tests is derived from certain assumptions relating to test theory. This result is checked empirically in a number of examples. Further derivations lead to the bivariate distribution of parallel tests which is also verified with actual data. The bivariate distribution of raw score and true score is also derived from a further assumption. This distribution is used to set confidence limits for true scores for persons with a given raw score.

A procedure is derived for estimating the latent parameters of items not initially included in a latent class solution, on the assumption that the relations between those additional items and the original ones are accounted for by the same latent structure. A chi-square test is proposed for evaluating the relatedness of the additional items to the latent structure. The extension is generalized to associate continuous outside variables with the original solution, and this is accompanied by the suggestion that a simple analysis of variance be used to assess the results. Finally, this latent structure extension is compared with the Dwyer extension of factor analysis.

General formulas for obtaining scores for individuals on components (factors derived from correlation matrices with unit communality estimates) are given. They are specialized to give Hotelling's formula for principal component scores. Formulas for scores on components rotated from principal components are developed.

One-absorbing barrier random walks arising from Luce's nonlinear beta model for learning and a linear commuting-operator model (called the alpha model) are considered. Functional equations for various statistics are derived from the branching processes defined by the two models. Solutions to general functional equations, satisfied by statistics of the alpha and beta models, are obtained. The methods presented have application to other learning models.

For the two-absorbing-barrier specialization of Luce's beta learning model, the asymptotic distribution of the response probability has all its density at p = 0 and p = 1. The functional equation for the amount of the density at p = 1 is investigated in this paper.

It is shown how the square root method can be used in the derivation of formulas in multiple prediction and in the computations necessary for their application. A formula derived by Horst indicates the validity a test must possess in order to contribute a specified amount to the multiple correlation of a battery. Then a formula is derived which relates the length of one test in a battery to the multiple correlation of the battery as a whole. The formulas hold for population parameters; in this paper, sampling theory is not discussed. A numerical example is presented; it is shown that one square root analysis provides all the data needed for the formulas.