Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.

One of the basic issues in the analysis of three-way arrays by CANDECOMP/PARAFAC (CP) has been the question of uniqueness of the decomposition. Kruskal (1977) has proved that uniqueness is guaranteed when the sum of thek-ranks of the three component matrices involved is at least twice the rank of the solution plus 2. Since then, little has been achieved that might further qualify Kruskal's sufficient condition. Attempts to prove that it is also necessary for uniqueness (except for rank 1 or 2) have failed, but counterexamples to necessity have not been detected. The present paper gives a method for generating the class of all solutions (or at least a subset of that class), given a CP solution that satisfies certain conditions. This offers the possibility to examine uniqueness for a great variety of specific CP solutions. It will be shown that Kruskal's condition is necessary and sufficient when the rank of the solution is three, but that uniqueness may hold even if the condition is not satisfied, when the rank is four or higher.

For any given number of factors, Minimum Rank Factor Analysis yields optimal communalities for an observed covariance matrix in the sense that the unexplained common variance with that number of factors is minimized, subject to the constraint that both the diagonal matrix of unique variances and the observed covariance matrix minus that diagonal matrix are positive semidefinite. As a result, it becomes possible to distinguish the explained common variance from the total common variance. The percentage of explained common variance is similar in meaning to the percentage of explained observed variance in Principal Component Analysis, but typically the former is much closer to 100 than the latter. So far, no statistical theory of MRFA has been developed. The present paper is a first start. It yields closed-form expressions for the asymptotic bias of the explained common variance, or, more precisely, of the unexplained common variance, under the assumption of multivariate normality. Also, the asymptotic variance of this bias is derived, and also the asymptotic covariance matrix of the unique variances that define a MRFA solution. The presented asymptotic statistical inference is based on a recently developed perturbation theory of semidefinite programming. A numerical example is also offered to demonstrate the accuracy of the expressions.

The case of adaptive testing under a multidimensional response model with large numbers of constraints on the content of the test is addressed. The items in the test are selected using a shadow test approach. The 0–1 linear programming model that assembles the shadow tests maximizes posterior expected Kullback-Leibler information in the test. The procedure is illustrated for five different cases of multidimensionality. These cases differ in (a) the numbers of ability dimensions that are intentional or should be considered as “nuisance dimensions” and (b) whether the test should or should not display a simple structure with respect to the intentional ability dimensions.

In this paper we study the interrelationships between two sets of data measured on the same subjects via redundancy analysis. We consider redundancy analysis from an inferential point of view. Under the hypothesis of multinormality, tests of significance are obtained for each successive redundancy component so that only the significant factors are retained for prediction purposes. An example illustrates the method.

Factor analysis is regularly used for analyzing survey data. Missing data, data with outliers and consequently nonnormal data are very common for data obtained through questionnaires. Based on covariance matrix estimates for such nonstandard samples, a unified approach for factor analysis is developed. By generalizing the approach of maximum likelihood under constraints, statistical properties of the estimates for factor loadings and error variances are obtained. A rescaled Bartlett-corrected statistic is proposed for evaluating the number of factors. Equivariance and invariance of parameter estimates and their standard errors for canonical, varimax, and normalized varimax rotations are discussed. Numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures.

Electromagnetic source analysis yields estimates of the sources of the Electro- and/or MagnetoEncephaloGram (EEG/MEG) and thus generates a functional description of the human brain. The standard errors of the source estimates are influenced by the number and position of EEG/MEG sensors, by the number of time samples, and by the number of trials in which EEG/MEG is measured. Therefore, optimal design theory is applied to determine the required number and position of sensors, the required number of samples, and the required number of trials. To that end, the Fedorov exchange algorithm is extended to incorporate multi-response models. A simulation study and an empirical study on visual evoked potentials indicate that the proposed method is fast and reliable, and improves source precision considerably.

A maximum likelihood procedure for combining standardized mean differences based on a noncentratt-distribution is proposed. With a proper data augmentation technique, an EM-algorithm is developed. Information and likelihood ratio statistics are discussed in detail for reliable inference. Simulation results favor the proposed procedure over both the existing normal theory maximum likelihood procedure and the commonly used generalized least squares procedure.

This paper is about the Linear Logistic Test Model (LLTM). We demonstrate that there are infinitely many equivalent ways to specify a model. An implication is that there may well be many ways to change the specification of a given LLTM and achieve the same improvement in model fit. To illustrate this phenomenon, we analyze a real data set using a Lagrange multiplier test for the specification of the model. This Lagrange multiplier test is similar to the modification index used in structural equation modeling.

Existing test statistics for assessing whether incomplete data represent a missing completely at random sample from a single population are based on a normal likelihood rationale and effectively test for homogeneity of means and covariances across missing data patterns. The likelihood approach cannot be implemented adequately if a pattern of missing data contains very few subjects. A generalized least squares rationale is used to develop parallel tests that are expected to be more stable in small samples. Three factors were varied for a simulation: number of variables, percent missing completely at random, and sample size. One thousand data sets were simulated for each condition. The generalized least squares test of homogeneity of means performed close to an ideal Type I error rate for most of the conditions. The generalized least squares test of homogeneity of covariance matrices and a combined test performed quite well also.

This paper develops diagnostic measures to identify those observations in Thurstonian models for ranking data which unduly influence parameter estimates that are obtained by the partition maximum likelihood approach of Chan and Bentler (1998). Diagnostic measures are constructed by employing the local influence approach that uses geometric techniques to assess the effect of small perturbations on a postulated statistical model. Very little additional effort is required to compute the proposed diagnostic measures, because all of the necessary building blocks are readily available after a usual fit of the model.

A Monte Carlo-algorithm realizing a family of nonpaxametric tests for the Rasch model is introduced which are conditional on the item and subject marginals. The algorithm is based on random changes of elements of data matrices without changing the maxginals; most powerful tests against all alternative hypotheses are given for which a monotone characteristic may be computed from the data matrix; alternatives may also be composed. Computation times are long, but exact p-values are approximated with the quality of approximation only depending on calculation time, but not on the number of persons. The power and the flexibility of the procedure is demonstrated by means of an empirical example where, among others, indicators for increased item similarities, the existence of subscales, violations of sufficiency of the raw score as well as learning processes were found. Many of the features described are implemented in the program T-Rasch 1.0 by Ponocny and Ponocny-Seliger (1999).

The problem of choosing the correct number of clusters is as old as cluster analysis itself. A number of authors have suggested various indexes to facilitate this crucial decision. One of the most extensive comparative studies of indexes was conducted by Milligan and Cooper (1985). The present piece of work pursues the same goal under different conditions. In contrast to Milligan and Cooper's work, the emphasis here is on high-dimensional empirical binary data. Binary artificial data sets are constructed to reflect features typically encountered in real-world data situations in the field of marketing research. The simulation includes 162 binary data sets that are clustered by two different algorithms and lead to recommendations on the number of clusters for each index under consideration. Index results are evaluated and their performance is compared and analyzed.

Changes in dichotomous data caused by treatments can be analyzed by means of the so-called linear logistic model with relaxed assumptions (LLRA). The LLRA does not require observable criteria representing a single underlying latent trait, but it postulates the generalizability of the treatment effects over criteria and subjects. To test this latter crucial assumption, the mixture LLRA was proposed that allows directly unobservable types of subjects to have different treatment effects. As the earlier methods for estimating the parameters of the mixture LLRA have specific drawbacks, a further method based on the conditional maximum likelihood principle will be presented here. In contrast to the earlier conditional methods, it uses all of the dichotomous change data while having fewer parameters. Further, its goodness-of-fit tests become more sensitive to a falsely specified number of change-types even though the treatment effects are biased. For typically occurring small to moderate sample sizes, however, parametric bootstrapping of the distributions of the fit statistics is recommended for performing hypotheses tests. Finally, three applications of the new method to empirical data are described: first, about the effect of the so-called Trager psychophysical integration, second, about the effect of autogenic therapy on patients with psychosomatic symptoms, and, third, about the effect of religious education on the attitude towards sects.

The seriation of proximity matrices is an important problem in combinatorial data analysis and can be conducted using a variety of objective criteria. Some of the most popular criteria for evaluating an ordering of objects are based on (anti-) Robinson forms, which reflect the pattern of elements within each row and/or column of the reordered matrix when moving away from the main diagonal. This paper presents a branch-and-bound algorithm that can be used to seriate a symmetric dissimilarity matrix by identifying a reordering of rows and columns of the matrix optimizing an anti-Robinson criterion. Computational results are provided for several proximity matrices from the literature using four different anti-Robinson criteria. The results suggest that with respect to computational efficiency, the branch-and-bound algorithm is generally competitive with dynamic programming. Further, because it requires much less storage than dynamic programming, the branch-and-bound algorithm can provide guaranteed optimal solutions for matrices that are too large for dynamic programming implementations.

A construction method is given for all factors that satisfy the assumptions of the model for factor analysis, including partially determined factors where certain error variances are zero. Various criteria for the seriousness of indeterminacy are related. It is shown that Green's (1976) conjecture holds: For a linear factor predictor the mean squared error of prediction is constant over all possible factors. A simple and general geometric interpretation of factor indeterminacy is given on the basis of the distance between multiple factors. It is illustrated that variable elimination can have a large effect on the seriousness of factor indeterminacy. A simulation study reveals that if the mean square error of factor prediction equals .5, then two thirds of the persons are “correctly” selected by the best linear factor predictor.