Maximum likelihood factor analysis provides an effective method for estimation of factor matrices and a useful test statistic in the likelihood ratio for rejection of overly simple factor models. A reliability coefficient is proposed to indicate quality of representation of interrelations among attributes in a battery by a maximum likelihood factor analysis. Usually, for a large sample of individuals or objects, the likelihood ratio statistic could indicate that an otherwise acceptable factor model does not exactly represent the interrelations among the attributes for a population. The reliability coefficient could indicate a very close representation in this case and be a better indication as to whether to accept or reject the factor solution.

A method of nonmetric multidimensional scaling is described which minimizes pairwise departures from monotonicity. The procedure is relatively simple, both conceptually and computationally. Experience to date suggests that it produces solutions comparable to those of other methods.

In many applications, it is desirable to estimate binomial proportions in m groups where it is anticipated that these proportions are similar but not identical. Following a general approach due to Lindley, a Bayesian Model II aposteriori modal estimate is derived that estimates the inverse sine transform of each proportion by a weighted average of the inverse sine transform of the observed proportion in the individual group and the average of the estimated values. Comparison with a classical method due to Jackson spotlights some desirable features of Model II analyses. The simplicity of the present formulation makes it possible to study the behavior of the Bayesian Model II approach more closely than in more complex formulations. Also, it is possible to estimate the amount of gain afforded by the Model II analyses.

A major justification for the hierarchical clustering methods proposed by Johnson is based upon their invariance with respect to monotone increasing transformations of the original similarity measures. Several alternative procedures are presented in this paper that also share in the same property of invariance. One of these techniques constructs a hierarchy of partitions by sequentially minimizing a monotone invariant goodness-of-fit statistic; the other techniques construct a hierarchy of partitions by successively subdividing the complete set of objects until one partition class is defined for each individual member in the set. A numerical example comparing these alternative procedures with Johnson's two methods is discussed in terms of a simplified computational scheme for obtaining the necessary hierarchies.

The min and the max hierarchical clustering methods discussed by Johnson are extended to include the use of asymmetric similarity values. The first part of the paper presents the basic min and max procedures but in the context of graph theory; this description is then generalized to directed graphs as a way of introducing the less restrictive characterization of the original clustering techniques.

When teaching elementary concepts on information theory a problem keeps reoccurring which, if only for pedagogical reasons, deserves discussion.

A test which allows for errors of measurement is derived for the hypothesis that all the members of a population who possess a certain skill are a sub-set of the members who possess another skill. Formulae are given for one particular case when two questions are used for each skill, and for when three questions are used for each skill. An illustrative example is given for the two-question case.

It is the purpose of this paper to discuss the ‘Lens Mode’ approach to the analysis of subject performance in multiple-cue judgment tasks embedded in probabilistic environments. Various characteristics of the linear lens model are presented in matrix notation and are formulated without reference to standardized variables in order to facilitate analysis and subsequent generalization. Some new results on ‘matching’ behavior are presented. A brief outline of current problems and related work is presented.

We concern ourselves with the hypothesis that two variables have a perfect disattenuated correlation, hence measure the same trait except for errors of measurement. This hypothesis is equivalent to saying, within the adopted model, that true scores of two psychological tests satisfy a perfect linear relation. Statistical tests of this hypothesis are derived when the relation is specified with the exception of the additive constant. Two approaches are presented and various assumptions concerning the error parameters are used. Then the results are reinterpreted in terms of the possible existence of an unspecified perfect linear relation between true scores of two psychological tests. A numerical example is appended by way of illustration.

The interdistances between thirteen places situated in different parts of the world were estimated by 60 subjects. The estimates were analysed by Kruskal's multidimensional technique and, after a cosine transformation, by factor analysis. It was found that both methods yielded the same three-dimensional solution. Also a two-dimensional configuration could describe the data, and it was shown that this configuration contained representations of unidimensional ratings obtained in other studies.

The Rasch model is an item analysis model with logistic item characteristic curves of equal slope, i.e. with constant item discriminating powers. The proposed goodness of fit test is based on a comparison between difficulties estimated from different scoregroups and over-all estimates.

Based on the within scoregroup estimates and the over-all estimates of item difficulties a conditional likelihood ratio is formed. It is shown that—2 times the logarithm of this ratio is x2-distributed when the Rasch model is true.

The power of the proposed goodness of fit test is discussed for alternative models with logistic item characteristic curves, but unequal discriminating items from a scholastic aptitude test.

A counterexample is produced to a conjecture by K. G. Jöreskog concerning sufficiency conditions for uniqueness of a restricted factor matrix. A substitute condition is stated and proved for the most common situation where the restricted elements are specified to be zero.