An approach for multiple factor analysis of dichotomized variables is presented. It is based on the distribution of the first and second order joint probabilities of the binary scored items. The estimator is based on the generalized least squares principle. Standard errors and a test of the fit of the model is given.

Investigation of the effects of a series of treatment conditions upon some social behaviors may require observation of a set of subjects which have been mutually paired, in round-robin fashion. Data arising from such experiments are difficult to analyze, partly because they do not fit neatly into standard designs. A paired compositions scaling model due to Bechtel is adapted to provide a linear model for such round-robin experiments. Both fixed and mixed versions of the model are considered and some results of a Monte Carlo study of the mixed model are reported. The model is applied to illustrative data from the field of physiological psychology.

Through an extension of work by Guttman, common factor theory, image theory, and component theory are derived from distinct minimum subsets of assumptions chosen out of a set of five possible assumptions. It is thence shown that the problem of indeterminacy of factor scores in the common factor model is precisely reflected in the problem of the non-orthogonality of anti-images. Indeed, image scores are determinate for the same reason that the usual estimates of factor scores are determinate, and image scores cannot be used as though they were factor scores for the same reason that factor score estimates cannot be used as though they were factor scores.

The algorithm is applicable to structures such as are obtained from additive conjoint measurement designs, unfolding theory, general Fechnerian scaling, some special types of multidimensional scaling, and ordinal multiple regression. A description is obtained of the space containing all possible numerical representations which can satisfy the structure, the size and shape of which is informative. The Abelson-Tukey maximin r2 solution is provided.

Suppose Pi(i) (i = 1, 2, ..., m, j = 1, 2, ..., n) give the locations of mn points in p-dimensional space. Collectively these may be regarded as m configurations, or scalings, each of n points in p-dimensions. The problem is investigated of translating, rotating, reflecting and scaling the m configurations to minimize the goodness-of-fit criterion Σi=1m Σi=1n Δ2(Pj(i)Gi), where Gi is the centroid of the m points Pi(i) (i = 1, 2, ..., m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special case m = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.

A model is proposed for the description of ordinal test scores based on the definition of true score as expected rank. Derivations from the model are compared with results from clasiscal test theory as developed by Lord and Novick, in particular with respect to parallel tests and composites. An unbiased estimator of population true score from sample data is derived and its variance is shown to decrease with increasing sample size. Population reliability is shown to be analytically related to expected sample reliability, and methods of reliability estimation are discussed.

McNemar’s problem concerns the hypothesis of equal probabilities for the unlike pairs of correlated binary variables.

We consider four different extensions to this problem, each for testing simultaneous equality of proportions of unlike pairs in c independent populations of correlated binary variables, but each under different assumptions and/or additional hypotheses. For each extension both the likelihood ratio test and the goodness-of-fit chi-square test are given. When c = 1, all cases reduce to McNemar’s problem. Forc ≥ 2, however, the tests are quite different, depending on exactly how the hypothesis and alternatives of McNemar are extended.

An example illustrates how widely the results may differ, depending on which extended framework is appropriate.

Estimates of conditional uncertainty, contingent uncertainty, and normed modifications of contingent uncertainty have been proposed for the two-way contingency table. The asymptotic standard errors of the estimates are derived.

A simple method of monotone regression is described based on the principle of minimizing pairwise departures from monotonicity.

A selection of statistical problems commonly encountered in psychological or psychiatric research concerning correlation coefficients are re-evaluated in the light of recently developed simplifications in the forms of the distribution theory of the intraclass correlation coefficient (exact theory), of the product-moment correlation coefficient and of the Spearman rank correlation coefficient (approximate).

A class of related nonmetric (“monotone invariant”) hierarchical grouping methods is presented. The methods are defined in terms of generalized cliques, based on a systematically varying specification of the degree of indirectness of permitted relationships (i.e., degree of “chaining”). This approach to grouping is shown to provide a useful framework for grouping methods based on an a priori specification of the properties of the desired subsets, and includes a natural generalization for “complete linkage” and “single linkage” clustering, such as the methods of Johnson [1967]. The central feature of the class of methods is a simple iterative matrix operation on the original disparities (“inverse-proximities” or “dissimilarities”) matrix, and one of the methods also constitutes a very efficient single linkage clustering procedure.

This paper treats the problem of testing an ordered hypothesis based on the ranks of the data. The statistical procedures for the randomized block design with more than one observation per cell are derived. Multiple comparisons and estimation procedures are also included.

Following a general introduction to Tukey’s Jackknife technique and the construction of approximate confidence intervals, ten variants of the procedure are defined for giving interval estimates for coefficient alpha. A data base of known population characteristics was generated and used to compare the robustness of the ten Jackknife alternatives with Feldt’s well-known sampling theory based upon the assumptions of normality. The empirical results indicate that out of the ten variants, defined by five transformations and two methods of data subdivision, only two are justifiable competitors to Feldt’s approach.

The theoretical and practical importance of a double undertaking is discussed: the development of learning and transfer taxonomies with psychometric relevance and the building of psychometric classificatory systems with implications for learning and instruction. Psychometric classifications of human performances are most often based on the covariation of individual differences. The model presented justifies the expectation that the transfer from learning one task to learning another is linearly dependent on the coefficient of intercorrelation between the two tasks when the coefficient is corrected for attenuation. Two studies so far have explicitly confirmed the main deductions from this model. Contrary to the predictions, however, the regression curves yielded negative intercepts. Two empirically testable explanations are offered, one of which would be in full accordance with the model, while the other would call for a further assumption.

A Bayesian Model II approach to the estimation of proportions in m groups (discussed by Novick, Lewis, and Jackson) is extended to obtain posterior marginal distributions for the proportions. It is anticipated that these will be useful in applications (such as Individually Prescribed Instruction) where decisions are to be made separately for each proportion, rather than jointly for the set of proportions. In addition, the approach is extended to allow greater use of prior information than previously and the specification of this prior information is discussed.

A general class of estimation procedures for the factor model is considered. The procedures are shown to yield estimates possessing the same asymptotic sampling properties as those from estimation by maximum likelihood or generalized least squares, both of which are special members of the class. General expressions for the derivatives needed for Newton-Raphson determination of the estimates are derived. Numerical examples are given, and the effect of the choice of estimation procedure is discussed.

Many data analysis problems in psychology may be posed conveniently in terms which place the parameters to be estimated on one side of an equation and an expression in these parameters on the other side. A rule for improving the rate of convergence of the iterative solution of such equations is developed and applied to four problems: the principal axis communality problem, individual differences multidimensional scaling, LP norm multiple regression, and LP norm factor analysis of a data matrix. The rule results in substantially faster solutions or in solutions where none would be possible without the rule.

Kaiser’s iterative algorithm for the varimax rotation fails when (a) there is a substantial cluster of test vectors near the middle of each bounding hyperplane, leading to non-bounding hyperplanes more heavily overdetermined than those at the boundaries of the configuration of test vectors, and/or (b) there are appreciably more than m (m factors) tests whose loadings on one of the factors of the initial F- matrix, usually the first, are near-zero, leading to overdetermination of the hyperplane orthogonal to this initial F- axis before rotation. These difficulties are overcome by weighting the test vectors, giving maximum weights to those likely to be near the primary axes, intermediate weights to those likely to be near hyperplanes but not near primary axes, and near-zero weights to those almost collinear with or almost orthogonal to the first initial F- axis. Applications to the Promax rotation are discussed, and it is shown that these procedures solve Thurstone’s hitherto intractable “invariant” box problem as well as other more common problems based on real data.

A theory of discrimination which assumes that subjects compare psychological values evoked by a stimulus to a subjective referent is proposed. Momentary differences between psychological values for the stimulus and the referent are accumulated over time until one or the other of two response thresholds is first exceeded. The theory is analyzed as a random walk bounded between two absorbing barriers. A general solution to response conditioned expected response times is computed and the important role played by the moment generating function (mgf) for increments to the random walk is examined. From considerations of the mgf it is shown that unlike other random walk models [Stone, 1960; Laming, 1968] the present theory does not imply that response conditioned mean correct and error times must be equal. For two fixed stimuli and a fixed referent it is shown that by controlling values of response thresholds, subjects can produce Receiver Operating Characteristics similar or identical to those predicted by Signal Detection Theory, High Threshold Theory, or Low Threshold Theory.

A Bayesian procedure is given for estimation in unrestricted common factor analysis. A choice of the form of the prior distribution is justified. It is shown empirically that the procedure achieves its objective of avoiding inadmissible estimates of unique variances, and is reasonably insensitive to certain variations in the shape of the prior distribution.