A distinction is made between statistical inference and psychometric inference in factor analysis. After reviewing Rao's canonical factor analysis (CFA), a fundamental statistical method of factoring, a new method of factor analysis based upon the psychometric concept of generalizability is described. This new procedure (alpha factor analysis, AFA) determines factors which have maximum generalizability in the Kuder-Richardson, or alpha, sense. The two methods, CFA and AFA, each have the important property of giving the same factors regardless of the units of measurement of the observable variables. In determining factors, the principal distinction between the two methods is that CFA operates in the metric of the unique parts of the observable variables while AFA operates in the metric of the common (“communality”) parts.

On the other hand, the two methods are substantially different as to how they establish the number of factors. CFA answers this crucial question with a statistical test of significance while AFA retains only those alpha factors with positive generalizability. This difference is discussed at some length. A brief outline of a computer program for AFA is described and an example of the application of AFA is given.

The researcher in behavioral or biological science has available a large arsenal of methods for studying the nature and statistical significance of differences between various groups of individuals. He may use any of a wide variety of univariate methods to test hypotheses about differences in location, in dispersion, in correlation between pairs of attributes, etc. In a multivariate situation methods are available to test hypotheses about mean vectors, variance-covariance matrices, correlation matrices, factor structures, etc.

There are two points of view involved in studying the nature of differences between groups. One point of view is descriptive, and has traditionally been treated under the topic of estimation in statistics. The other usually involves a question as to whether it is reasonable to assume that some hypothesized model about the nature of differences between populations is in fact true, and has usually been treated under the ~pic of tests of hypotheses. The descriptive and inferential points of view must of necessity often overlap.

Although probit analysis has been applied in the field of psychometrics, its use has been restricted by the requirement of linear regression of probits on the independent variate. This paper shows that polynomial probit analysis, a natural extension of the idea of probit analysis, can sometimes overcome the difficulty of a nonlinear plot and does not involve excessive computational labor.

One of the major concerns of reliability theory has been the estimation of the reliability of a composite measure from the degree of agreement among its component parts. In the classical theory, formulas were developed under the assumption that the parts are strictly equivalent. It was later shown that the same formulas follow from various sets of weaker assumptions which require the composites to be strictly equivalent and require the parts to have a certain homogeneity of statistical properties, but not necessarily to be equivalent. An alternative model which has received increasing attention in recent years regards a given measure as a random sample from a universe of measures whose homogeneity or equivalence is not specified a priori, and a composite test as a random sample of items from a universe of not-necessarily-equivalent items. This too permits an internal-consistency estimate of reliability. Both the equivalent-composites model and the randomsampling model appear to be unduly restrictive and unrealistic; we propose here to develop the implications of a third model in which a test is considered to have been formed by stratified sampling of items.

While the traditional multiple correlation coefficient appears to be inherently an asymmetrical statistic, it is actually a special case of a more general measure of linear relationship between two sets of variables. Another symmetric generalization of linear correlation is to the total relatedness within a set of variables. Both of these developments rest upon the generalized variance of a multivariate distribution, which is seen to be the fundamental concept of linear correlational theory.

This paper seeks to meet the need for a general treatment of the problem of error in classification. Within an m-attribute classificatory system, an object's typical subclass is that subclass to which it is most often allocated under repeated experimentally independent applications of the classificatory criteria. In these terms, an error of classification is an atypical subclass allocation. This leads to definition of probabilities O of occasional subclass membership, probabilities T of typical subclass membership, and probabilities E of error or, more generally, occasional subclass membership conditional upon typical subclass membership. In the relationship f: (O, T, E) the relative incidence of independent O, T, and E values is such that generally one can specify O values given T and E, but one cannot generally specify T and E values given O. Under the restrictions of homogeneity of E values for all members of a given typical subclass, mutual stochastic independence of errors of classification, and suitable conditions of replication, one can find particular systems O = f (T, E) which are solvable for T and E given O. A minimum of three replications of occasional classification is necessary for a solution of systems for marginal attributes, and a minimum of two replications is needed with any cross-classification. Although for such systems one can always specify T and E values given O values, the solution is unique for dichotomous systems only.