For the point distribution model of Lazarsfeld's latent structure analysis, the general matrix equation is stated which relates the manifest data in the form of joint occurrence matrices to the latent parameters. The relationship of the item responses and these joint occurrence matrices is also indicated in matrix form. A general solution for the latent parameters is then presented, which is based on the notion of factoring two joint occurrence matrices. The solution is valid under certain conditions which will usually be fulfilled. The solution assumes that estimates are available for the elements in the joint occurrence matrices with recurring subscripts, analogous to item communality or reliability. Some alternative methods of obtaining these estimates are discussed. Finally a fictitious 3-class, 8-item example is presented in detail.

Non-spurious methods are needed for estimating the coefficient of equivalence for speeded tests from single-trial data. Spuriousness in a split-half estimate depends on three conditions; the split-half method may be used if any of these is demonstrated to be absent. A lower-bounds formula, rc, is developed. An empirical trial of this coefficient and other bounds proposed by Gulliksen demonstrates that, for moderately speeded tests, the coefficient of equivalence can be determined approximately from single-trial data. It is proposed that the degree to which tests are speeded be investigated explicitly, and an index τ is advanced to define this concept.

Having given a fixed amount of total testing time it is important to know how long each test in the battery should be so that the correlation of the battery with the criterion will be a maximum. The precise solution for the test lengths will depend on a particular set of conditions which may be specified. The writer has previously presented solutions for two sets of conditions. This article presents the solution for a third set of conditions. These are: (1) The total number of items or testing time is fixed. (2) The score is the total number of items correctly answered. (3) The test lengths are determined in such a way that the correlation of total score with the criterion is a maximum. The solutions for the two previous sets of conditions, together with the current set, are summarized. A set of experimental data is submitted to each solution and the three sets of results are compared.

If customary methods of solution are used on the method of paired comparisons for Thurstone's Case V (assuming equal standard deviations of sensations for each stimulus), when in fact one or more of the standard deviations is aberrant, all stimuli will be properly spaced except the one with the aberrant standard deviation. A formula is given to show the amount of error due to the aberrant stimulus.

A test of goodness of fit is developed for Thurstone's method of paired comparisons, Case V.

The test involves the computation of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi ^2= n\Sigma(\theta '' - \theta ')^2 /821,$$\end{document}, where n is the number of observations per pair, and θ″ and θ′ are the angles obtained by applying the inverse sine transformation to the fitted and the observed proportions respectively. The number of degrees of freedom is (k−1) (k−2)/2.

Rate of addition was studied as a function of difficulty as measured by problem length. The hypothesis was tested that the rate of addition would decline as a function of the logarithm of the number of addition operations per problem. The test material required the rapid addition of single columns of digits ranging from two to twenty-five digits in length. Rate of uncorrected addition declined as a power function of problem length and the rate of correct addition declined as an exponential function of length. Results indicated that subjects who varied in age and mental status could be differentiated according to the parameters defining the curves of addition rate as a function of length.

A mechanical model is described for illustrating changes in the configuration of points when the reference axes are rotated obliquely and the values of the orthogonal projections of the points on the reference axes are maintained constant. This type of transformation is important in factor analysis.

A description is given of a diagram (available separately) for computing biserial or point biserial correlation coefficients. The diagram is maximally useful where large numbers of coefficients are to be calculated in test item analysis. The diagram is entered with the mean criterion score of the group passing the item and the proportion of correct answers to the item.