An approach to the analysis of multivariate time series is presented in which linear structural relationships among multiple stochastic variables are investigated. A number of alternative structural models are considered for the case of two stochastic variables. Each model represents a possible hypothesis concerning the relationship of growth in one variable to growth in the second. Both symmetric and asymmetric models are considered. Extensions of two of the models to three variables are illustrated by means of a numerical example. Implications of the models for the problem of detecting change in multivariate time series are discussed.

Various writers have suggested the use of projective transformations in psychological theories. Among them are Herbst [t970], Atkin [1972] and Levine [in press]. In general these suggestions relate to scaling problems in psychophysics and in social psychology. The purpose of the present note is to extend these applications to test theory. One of the models proposed by Rasch [1960] can be seen as a special case of the projective transformation and this fact has certain implications for the use of the model.

This paper gives a method of estimating the reliability of a test which has been divided into three parts. The parts do not have to satisfy any statistical criteria like parallelism or τ-equivalence. If the parts are homogeneous in content (congeneric), i.e., if their true scores are linearly related and if sample size is large then the method described in this paper will give the precise value of the reliability parameter. If the homogeneity condition is violated then underestimation will typically result. However, the estimate will always be at least as accurate as coefficient α and Guttman's lower bound λ3 when the same data are used. An application to real data is presented by way of illustration. Seven different splits of the same test are analyzed. The new method yields remarkably stable reliability estimates across splits as predicted by the theory. One deviating value can be accounted for by a certain unsuspected peculiarity of the test composition. Both coefficient α and λ3 would not have led to the same discovery.

In a multiple (or multivariate) regression model where the predictors are subject to errors of measurement with a known variance-covariance structure, two-sample hypotheses are formulated for (i) equality of regressions on true scores and (ii) equality of residual variances (or covariance matrices) after regression on true scores. The hypotheses are tested using a large-sample procedure based on maximum likelihood estimators. Formulas for the test statistic are presented; these may be avoided in practice by using a general purpose computer program. The procedure has been applied to a comparison of learning in high schools using achievement test data.

Omitted items cannot properly be treated as wrong when estimating ability and item parameters. A convenient method for utilizing the information provided by omissions is presented. Theoretical and empirical justifications are presented for the estimates obtained by the new method.

Similarity judgments of three-dimensional stimuli were simulated, with the hypothetical subject attending to only some dimensions of stimulus variation (i.e., “subsampling”) on each trial. Recovery of the stimulus configuration by non-metric multidimensional scaling was investigated as a function of subsampling, the amount of random error in the judgments, and the number of stimuli being scaled.

It was found that: (1) dimensions to which the subject often attends were well recovered even when dimensions seldom attended to were not, and (2) measures of recovery based on interpoint distances were inadequate. Several previous Monte Carlo studies were evaluated in light of the results.

On the basis of the idea of common elements to account for similarity, Restle [1959] developed a set-theoretical model on the generation of similarity judgments. Restle differentiates between his “more qualitative discussion” (p. 207), regarding stimuli as sets of elements and using no geometric concepts or assumptions, and metric quantitative developments of similarity, regarding stimuli as points in a geometric space. To the latter we can count the additive difference model of Beals, Krantz, & Tversky [1968] and Tversky & Krantz [1970].

Despite these differences the two models lead to the same conclusions on the characteristics of similarity judgments.

Let n stimuli be linearly ordered by a physical attribute. Consider an individual's “more than” relation based on perceived attribute values. Assuming that the individual's relation agrees with the linear order and is an interval order, the paper examines the extent to which ordered pairs in the linear order but not in the interval order (due to nondetection of “small” differences) can be correctly identified on the basis of the interval order alone. Different methods of constructing strict weak orders from interval orders are compared by the degree of agreement with the underlying linear order.

Within the context of a contingency table, this note describes the relationship between Freeman's measure of association θ and the asymmetric association measures developed by Somers. The θ coefficient is appropriate for a contingency table in which the levels of one factor are ordered and the levels of the other factor are unordered; the indices defined by Somers are usually used when the levels of both factors are ordered and one is assumed to be the independent factor.

The study deals with the problem of determining true dimensionality of data-with-error scaled by Kruskal's multidimensional scaling technique. Artificial data was constructed for 6, 8, 12, 16, and 30 point configurations of 1, 2, or 3 true dimensions by adding varying amounts of error to the true distances. Results show how stress is affected by error, number of points, and number of dimensions, and indicate that stress and the “elbow” criterion are inadequate for purposes of identifying true dimensionality when there is error in the data. The Wagenaar-Padmos procedure for identifying true dimensionality and error level is discussed. A simplified technique, involving a measure called Constraint, is suggested.

Recently, D’Agostino and Cureton {1972] discussed a test of normality against skewed alternatives; the test was based upon g1, the third standardized sample moment. In an earlier paper, D’Agostino [1970] suggested that an experimenter might be interested in testing the underlying assumption of normality associated with an analysis of variance. For such a test, one would be required to decide if k independent random samples were drawn from k normal populations. The purpose of this note is to suggest a simple extension of the test based upon g1, to the case of k independent random samples.

In articles by Homn [1969] and Carroll and Chang [1970] and in a memorandum by Bloxom [1968], a model was proposed for individual differences in multidimensional scaling. The distance in a p-dimensional space between stimuli j and k for person i is assumed to be given by

for N persons and for all pairs of n stimuli, where wim, is the “weight” given to axis m by person i and ajm, is the projection of stimulus j on that axis.

Novick, Lewis and Jackson [1973] propose an interesting Bayesian method for the simultaneous estimation of the probabilities of several binomial distributions. They utilise arc-sine transformations and normal approximations and then refer to the normal theory method of Lindley [1972] which utilises the novel concept of ‘exchangeability’. The posterior estimates for the probabilities shrink the usual proportions towards a central value based on collateral information, and should provide substantial improvements.

The purpose of our comment is to discuss the relative merits of other methods which are available whenever exchangeability of the probabilities is appropriate, The main methods are

(1) The Logistic Approach: Leonard [1972a] uses logistic transformations of the parameters and proposes a simple exact method of estimation which is valid even if some of the cell frequencies are zero. This is because the logistic transformations are to the parameters and not to the observed frequencies. The exact posterior modal estimates replace the usual observed logits in such a way that difficulties involving zeros are removed. The are-sine approximation is of course inaccurate in such cases.

The homogeneous case of the continuous response model is expanded to the multi-dimensional latent space, and the normal ogive model is presented. The operating density characteristic of the continuous item response and the vector of basic functions are developed. It is found out that there is a vector of sufficient statistics for estimating the subject's vector of latent traits, given the item parameter vectors. The relationship between the model and the linear factor analysis is observed. The matrix of item response information functions is introduced. Some additional observations are also made.