I am honored to be able to speak on the 25th anniversary of the founding of the Psychometric Society. It would be appropriate for your president on this occasion to review the 25-year history of our society and of mathematical methods in psychology. Fortunately our committee headed by Charles Wrigley has taken care of this need, and speakers better qualified than I will speak to these issues. I shall turn to a less demanding problem.

In limiting my topic I was faced with certain constraints arising from my training, background, and interests. One constraint arises from a misspent youth during which too many years were spent studying disciplines other than mathematics. A second constraint, which probably stems from my training in experimental psychology, is a tendency to be more interested in data than theory. Finally, there is the Air Force personnel research experience which developed in me an abiding interest in applied prediction problems. If you consider this combination to be an odd one for the President of the Psychometric Society, don't blame me. You elected me!

Formulas are derived for using the available item statistics and score statistics on a test to estimate the moments of the score distribution of a lengthened (or shortened) form of the same test. Other formulas are derived for estimating the bivariate moments of the scatterplot between two parallel test forms using only the data available on either form alone. An empirical study is made showing in each case satisfactory agreement between the theoretical values predicted from the formulas and the values actually observed. These results suggest the utility of the true-score model used in deriving the formulas.

A new paired comparison method, based upon choices between lotteries, is developed for the measurement of utilities of objects with respect to the utility of receiving nothing, i.e., the status quo. The method is used to estimate the utilities of four birthday gifts. These objects had also been studied in an earlier experiment which used choices between single objects and pairs of objects to determine a rational origin. A comparison of the results of the two experiments indicates that both methods scale objects with respect to the same rational origin and unit of measurement.

A generalization of the customary model underlying the Law of Comparative Judgment is established. Methods for estimating location parameters and testing a certain hypothesis are discussed. The problem of the influence of the shape of the curve used to grade responses is raised and a possible approach to its solution is indicated. Two examples are provided.

When multiple significance tests are computed, a certain number of “significant” findings will emerge simply because of chance fluctuations. In the present paper, some factors affecting the number of nominally significant results are elaborated and a general method is suggested which permits unbiased inference as to the significance of a set of findings, as a set. The method advocated employs a high speed computer to generate empirically a sampling distribution tailormade to a particular data matrix. The method is illustrated in the case of dichotomous response to inventory items, where it is found that the statistical model still often used as a basis for estimation is overly conservative. Some problems in the application of the method are discussed.

Intercorrelations among tests nonlinearly related to underlying dimensions require more linear factors than content would demand. For the case of two independent underlying content dimensions, a fictitious example is constructed and made to yield a transformation useful for the nonlinear analysis of certain empirical data. That transformation, when applied to a standard factorization (centroid or principal components if certain symmetries obtain) of the appropriate empirical correlations, yields parameters descriptive of plausible nonplanar regression surfaces for tests on the two underlying dimensions. An empirical example is presented and discussed.

The problem of measuring the association between two characters, one quantitative and the other qualitative, is discussed. The formula for the large sample standard error of the point biserial correlation coefficient under general conditions is derived. The point multiserial correlation coefficient is introduced and some of its properties are examined. Tests of different hypotheses appropriate to these types of problems are formulated.