This paper traces the course of the consequences of viewing test responses as simply providing dichotomous data concerning ordinal relations. It begins by proposing that the score matrix is best considered to be items-plus-persons by items-plus-persons, and recording the wrongs as well as the rights. This shows how an underlying order is defined, and was used to provide the basis for a tailored testing procedure. It also was used to define a number of measures of test consistency. Test items provide person dominance relations, and the relations provided by one item can be in one of three relations with a second one: redundant, contradictory, or unique. Summary statistics concerning the number of relations of each kind are easy to get and provide useful information about the test, information which is related to but different from the usual statistics. These concepts can be extended to form the basis of a test theory which is based on ordinal statistics and frequency counts and which invokes the concept of true scores only in a limited sense.

This paper is concerned with the development of a measure of the precision of a multidimensional euclidean structure. The measure is a precision index for each point in the structure, assuming that all the other points are precisely located. The measure is defined and two numerical methods are presented for its calculation. A small Monte Carlo study of the measure's behavior is performed and findings discussed.

A maximum likelihood method of estimating the parameters of the multiple factor model when data are missing from the sample is presented. A Monte Carlo study compares the method with 5 heuristic methods of dealing with the problem. The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.

The method of scoring is used to obtain maximum likelihood estimates of the parameters in the White and Clark learning hierarchy validation model. From the estimate of the proportion of the population possessing only the superordinate skill in a pair of hierarchical skills, and its variance, the hypothesis of inclusion is tested. An illustrative example of the procedure is given.

A modified beta binomial model is presented for use in analyzing ramdom guessing multiple choice tests and certain forms of taste tests. Detection probabilities for each item are distributed beta across the population subjects. Properties for the observable distribution of correct responses are derived. Two concepts of true score estimates are presented. One, analogous to Duncan’s empirical Bayes posterior mean score, is appropriate for assessing the subject’s performance on that particular test. The second is more suitable for predicting outcomes on similar tests.

The polychoric correlation is discussed as a generalization of the tetrachoric correlation coefficient to more than two classes. Two estimation methods are discussed: Maximum likelihood estimation, and what may be called “two-step maximum likelihood” estimation. For the latter method, the thresholds are estimated in the first step. For both methods, asymptotic covariance matrices for estimates are derived, and the methods are illustrated and compared with artificial and real data.

Coefficient κ is generally defined in terms of procedures of computation rather than in terms of a population. Here a population definition is proposed. On this basis, the interpretation of κ as a measure of diagnostic reliability in characterizing an individual, and the effect of reliability, as measured by κ, on estimation bias, precision, and test power are examined. Factors influencing the magnitude of κ are identified. Strategies to improve reliability are proposed, including that of combining multiple unreliable diagnoses.

A method is developed for estimating the response time distribution of an unobserved component in a two-component serial model, assuming the components are stochastically independent. The estimate of the component’s density function is constrained only to be unimodal and non-negative. Numerical examples suggest that the method can yield reasonably accurate estimates with sample sizes of 300 and, in some cases, with sample sizes as small as 100.