The following problem is considered: Given that the frequency distribution of the errors of measurement is known, determine or estimate the distribution of true scores from the distribution of observed scores for a group of examinees. Typically this problem does not have a unique solution. However, if the true-score distribution is “smooth,” then any two smooth solutions to the problem will differ little from each other. Methods for finding smooth solutions are developed a) for a population and b) for a sample of examinees. The results of a number of tryouts on actual test data are summarized.

The objectives of the Psychometric Society are coincident with those of its journal, namely, “the development of psychology as a quantitative rational science.” This general theme has frequently been referred to by the past presidents in their annual addresses. They did so by employing several different approaches: some by proposing directions of future research of the Society; some by making critical-instructive appraisals of past research; some by presenting reports of their own ongoing research; and some by viewing the present scene from an historical or philosophical perspective. Furthermore, the tone of these addresses has run the gamut from the frivolous to the profoundly scholarly.

A brief analysis of the previous addresses disclosed some interesting items. First, there have been eight instances, in our 34 year history, when the address was skipped entirely, beginning with Thurstone’s [1937] published in Science rather than in Psychometrika. Of the addresses that subsequently appeared in Psychometrika, the average was 13 pages (including references) with a range from 5 to 26 pages. (Hopefully, the longer ones were not read in their entirety.) More than half the papers were expository in nature. Although 12 were reports on serious research problems, only four contained heavy mathematical developments. Five addresses could be characterized as facetious or humorous, but nonetheless carried a moral for the Society. In general, suggestions were made in 13 instances for future research or directions of emphasis for the Society, and 9 addressed themselves explicitly to its aims. There was also a sprinkling of criticism, philosophy, and history relating to the Society or to specific psychometric methods. Of course, the foregoing classifications were not mutually exclusive, and merely constitute one man’s opinions.

The usual methods of combining observations to give interpoint distance estimates based on interstimulus differences are shown to lead to a distortion of the stimulus configuration unless all individuals in a group perceive the stimuli in perceptual spaces which are essentially the same. The nature of the expected distortion is shown, and a method of combining individual distance estimates which produces only a linear deformation of the stimulus configuration is given.

This monograph is a part of a more comprehensive treatment of estimation of latent traits, when the entire response pattern is used.

The fundamental structure of the whole theory comes from the latent trait model, which was initiated by Lazarsfeld as the latent structure analysis [Lazarsfeld, 1959], and also by Lord and others as a theory of mental test scores [Lord, 1952]. Similarities and differences in their mathematical structures and tendencies were discussed by Lazarsfeld [Lazarsfeld, 1960] and the recent book by Lord and Novick with contributions by Birnbaum [Lord & Novick, 1968] provides the dichotomous case of the latent trait model in the context of mental measurement.

Abilities are usually assumed to exist in a “positive manifold.” Experimental manipulations of physiological variables, however, suggest that negative relationships exist between certain of the neural processes contributing to simple perceptual-motor vs. perceptual-restructuring tasks. First-order correlative evidence of this phenomenon cannot be obtained because the between-individual differences in general ability level tend to exceed the behavioral effects of the intra-individual opposition between neural processes. Also, since statistical removal of the “g” variance induces bipolarity in the remaining variance, the second-order negative correlations are necessarily regarded as artifactual. A combined correlational-experimental approach is suggested to overcome this difficulty.

In order to study the effectiveness of factor analytic methods, a procedure was developed for computing simulated correlation matrices which are more similar to real data correlation matrices than are those matrices computed from the factor analysis structural model. In the present investigation, three methods of factor extraction were studied as applied to 54 simulated correlation matrices which varied in proportion of variance derived from a major factor domain, number of factors in the major domain, and closeness of the simulation procedure to the factor analysis structural model. While the factor extraction methods differed little from one another in quality of results for matrices more dissimilar to the factor analytic model, major differences in quality of results were associated with fewer factors in the major domain, higher proportion of variance from the major domain, and closeness of the simulation procedure to the factor analysis structural model.

Least squares linear composites of predictors for estimating several criteria are derived, satisfying the restriction that the composites have an arbitrary specified intercorrelation matrix. These composites are compared with the usual unrestricted regression composites. An illustrative example is provided. The derivation depends on a general result, given in an appendix, about best-fitting orthonormal transformations.

Some shortcomings of current methods of estimating the magnitude of perceived difference are considered. A statistical model for perceived difference is derived which avoids these difficulties and employs judgments of ratios of differences as data. Three estimators of squared difference are developed.

How can an investigator choose a good team of raters to use for measuring a continuous variable when each available rater produces only dichotomous responses? We formulate an underlying model, define an index of goodness for rater teams in terms of average mean square error of the estimate, develop a new estimator and derive the optimal rater terms. The optimal raters have characteristic curves which are linear in form and satisfy the requirements for a Guttman scale.

Recently Markov learning models with two unidentifiable presolution success states, an error state, and an absorbing learned state, have been suggested to handle certain aspects of data better than the three state Markov models of the General All or None model type. In attempting to interpret psychologically, and evaluate statistically the adequacy of various classes of Markov models, a knowledge of the relationship between the classes of models would be helpful. This paper considers some aspects of the relationship between the class of General All or None models and the class of Stationary Absorbing Markov models with N error states, and M presolution success states.

We describe a general procedure by which any number of parameters of the factor analytic model can be held fixed at any values and the remaining free parameters estimated by the maximum likelihood method. The generality of the approach makes it possible to deal with all kinds of solutions: orthogonal, oblique and various mixtures of these. By choosing the fixed parameters appropriately, factors can be defined to have desired properties and make subsequent rotation unnecessary. The goodness of fit of the maximum likelihood solution under the hypothesis represented by the fixed parameters is tested by a large sample x2 test based on the likelihood ratio technique. A by-product of the procedure is an estimate of the variance-covariance matrix of the estimated parameters. From this, approximate confidence intervals for the parameters can be obtained. Several examples illustrating the usefulness of the procedure are given.

Recent advances in computer based psychometric techniques have yielded a collection of powerful tools for analyzing nonmetric data. These tools, although particularly well suited to the behavioral sciences, have several potential pitfalls. Among other things, there is no statistical test for evaluating the significance of the results. This paper provides estimates of the statistical significance of results yielded by Kruskal's nonmetric multidimensional scaling. The estimates, obtained from attempts to scale many randomly generated sets of data, reveal the relative frequency with which apparent structure is erroneously found in unstructured data. For a small number of points (i.e., six or seven) it is very likely that a good fit will be obtained in two or more dimensions when in fact the data are generated by a random process. The estimates presented here can be used as a bench mark against which to evaluate the significance of the results obtained from empirically based nonmetric multidimensional scaling.

This paper presents a way to test the difference between two X2's. The test requires evaluating the difference with respect to the Tm(x) Bessel function. Included is a table of the 5 percent and 1 percent points for the Bessel function with degrees of freedom up to 100.

In a previous paper [Elashoff 1969], we derived optimal rater teams for a particular formulation of the dichotomous rater problem. Here, we describe a computer-based procedure for selecting good rater teams in practice; we apply the procedure to the selection of items for a psychological inventory.

Maximum-likelihood estimators of true score variance and error variance for mental tests are derived for six different models of equivalent measurements. Statistical properties of the estimators are examined. Main emphasis is placed upon essentially τ-equivalent measurements. A statistical criterion for this type of measurement is given. The solution of the comparatively simple maximum-likelihood equations is effected by means of a rapid Newton-Raphson procedure. Two different initial estimators are considered and their relative merits in terms of second moments evaluated. Four numerical examples are appended by way of illustration.

The asymptotic values of response probabilities in the Audley-Jonckheere learning model, as applied to the two-choice learning situation, are derived for (1) the experimenter-controlled events model, (2) the subject-controlled events model, and (3) the experimenter-subject-controlled events model.

Learning-process statistics for absorbing Markov-chain models are developed by using matrix methods exclusively. The paper extends earlier work by Bernbach by deriving the distribution of the total number of errors, u-tuples, autocorrelation of errors, sequential statistics, and the expectation and variance of all statistics presented. The technique is then extended to latency derivations including the latencies of sequential statistics. Suggestions are made for using the sequential-statistic algorithm in a maximum-likelihood estimation procedure. The technique is important because statistics for very large absorbing matrices can be easily computed without going through tedious theoretical calculations to find explicit mathematical expressions.

Control charts facilitating the use of sequential strategies of the updown or staircase type are described. The charts are easy to use and the relevant estimates may be derived very simply from the chart. An example is given in which both the 29.3% and 70.7% points on a response curve are estimated using the Up-Down-Transformed-Response (UDTR) procedure. The charts can be used in psychophysics and are also generally applicable to experiments or control processes involving binary responses.

Let x denote one of the variables in a factor analysis model. In 1935 Roff proved that the communality of x is greater than or equal to the squared multiple correlation coefficient of x with the other variables in the model. In this paper it is shown that the inequality is the first of an infinite sequence of inequalities, each sharper than the one before, and that the condition for equality is the same for all members of the sequence. Also, the limiting inequality is considered and the condition for equality derived. As this condition is always satisfied in a factor analysis model, the inequality is really an identity.