The multivariate asymptotic distribution of sequential Chi-square test statistics is investigated. It is shown that: (a) when sequential Chi-square statistics are calculated for nested models on the same data, the statistics have an asymptotic intercorrelation which may be expressed in closed form, and which is, in many cases, quite high; and (b) sequential Chi-square difference tests are asymptotically independent. Some Monte Carlo evidence on the applicability of the theory is provided.

In this paper the usual two-set Guttman simplex model is extended to three sets. The axiomatic foundations of this extention are presented. Two cases are discussed. In Case 1 there is a three-set joint order, while in Case 2 there is a two-set joint order consistent across all levels of the third set. Case 2 represents the first clear formulation of a longitudinal developmental scale. The model is discussed in terms of its most straightforward application, longitudinal developmental data, and in terms of other possible applications.

A model for longitudinal latent structure analysis is proposed. We assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time. The purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two-dimensional latent density. The correlation coefficient between the two values of the latent variable can then be estimated. The theory and methods are illustrated by a Danish dataset concerning psychic vulnerability.

Semiparametric models express a set of distributions of event times in terms of (a) a single parameter which varies across distributions and (b) a single function which does not vary across distributions and which has an unspecified form. These models appear to be attractive alternatives to parametric models of response times in psychometrics. However, our use of such models may require incorporating additional functions which do not vary across distributions and may require expressing the models in terms of the joint distribution of response class and response time.

The objective of research in the matching paradigm is to determine whether a one-to-one pairing configuration between objects in two arrays contains more pairings of a particular kind than expected under the null hypothesis. The presence of infeasible pairings, that is, of particular pairings which can not possibly take place, has the effect of altering the null distribution that would otherwise obtain. To enable researchers to determine the appropriate null distribution, a generalization of classical rook methodology is developed which can accommodate infeasible pairings in the matching paradigm.

Four types of metric scales are distinguished: the absolute scale, the ratio scale, the difference scale and the interval scale. A general coefficient of association for two variables of the same metric scale type is developed. Some properties of this general coefficient are discussed. It is shown that the matrix containing these coefficients between any number of variables is Gramian. The general coefficient reduces to specific coefficients of association for each of the four metric scales. Two of these coefficients are well known, the product-moment correlation and Tucker's congruence coefficient. Applications of the new coefficients are discussed.

When the three-parameter logistic model is applied to tests covering a broad range of difficulty, there frequently is an increase in mean item discrimination and a decrease in variance of item difficulties and traits as the tests become more difficult. To examine the hypothesis that this unexpected scale shrinkage effect occurs because the items increase in complexity as they increase in difficulty, an approximate relationship is derived between the unidimensional model used in data analysis and a multidimensional model hypothesized to be generating the item responses. Scale shrinkage is successfully predicted for several sets of simulated data.

A Monte Carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2, 3, 4, or 5 distinct nonoverlapping clusters. To provide a variety of clustering solutions, the data sets were analyzed by four hierarchical clustering methods. External criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level. Thus, the clustering present in the data was quite strong. The simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data. Several procedures worked fairly well, whereas others performed rather poorly. Thus, the latter group of rules would appear to have little validity, particularly for data sets containing distinct clusters. Applied researchers are urged to select one or more of the better criteria. However, users are cautioned that the performance of some of the criteria may be data dependent.

The linear regression model y=β′x+ ε is reanalyzed. Taking the modest position that β′x is an approximation of the “best” predictor of y we derive the asymptotic distribution of b and R2, under mild assumptions.

The method of derivation yields an easy answer to the estimation of β from a data set which contains incomplete observations, where the incompleteness is random.

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.

An algorithmic approach to test design, using information functions, is presented. The approach uses a special branch of linear programming, i.e. binary programming. In addition, results of some benchmark problems are presented. Within the same framework, it is also possible to formulate the problem of individualized testing.

As a method to ascertain the structure of intra-individual variation, P-technique has met difficulties in the handling of a lagged covariance structure. A new statistical technique, coined dynamic factor analysis, is proposed, which accounts for the entire lagged covariance function of an arbitrary second order stationary time series. Moreover, dynamic factor analysis is shown to be applicable to a relatively short stretch of observations and therefore is considered worthwhile for psychological research. At several places the argumentation is clarified through the use of examples.

The study of correlates of change is the investigation of systematic individual differences in growth. Our representation of systematic individual differences in growth is built up in two parts: (a) a model for individual growth and, (b) a model for the dependence of parameters in the individual growth models on individual characteristics. First, explicit representations of correlates of change are constructed for various models of individual growth. Second, for the special case of initial status as a correlate of change, properties of collections of growth curves provide new results on the relation between change and initial status. Third, the shortcomings of previous approaches to the assessment of correlates of change are demonstrated. In particular, correlations of residual change measures with exogenous individual characteristics are shown to be poor indicators of systematic individual differences in growth.

Using data which reflects upon the genetic makeup of patients, Cadoret, Woolson and Winokur (1977) test two theories regarding the genetic makeup of patients suffering from unipolar affective disorder. This hypothesis test is a good example of a problem in order restricted inference. Using some recently developed techniques in order restricted inference we analyze these data and compare the analyses.

A constrained quadratic spline is proposed as an estimator of the hazard function of a random variable. A maximum penalized likelihood procedure is used to fit the estimator to a sample of psychological response times. The results of a small simulation study suggest that, with a sample size of 500, the procedure may provide a reasonably precise estimate of the shape of a hazard function.

A latent class model for rating data is presented which is the analogue of Andrich's binomial Rasch model for Lazarsfeld's latent class analysis (LCA). The response probabilities for the rating categories follow a binomial distribution and depend on class-specific item parameters. The EM-algorithm for parameter estimation as well as goodness of fit tests for the model are described. An example using questionnaire items on interest in physics illustrates the use of the model as an alternative to the latent trait approach of analyzing test data.

While a rotation procedure currently exists to maximize simultaneously Tucker's coefficient of congruence between corresponding factors of two factor matrices under orthogonal rotation of one factor matrix, only approximate solutions are known for the generalized case where two or more matrices are rotated. A generalization and modification of the existing rotation procedure to simultaneously maximize the congruence is described. An example using four data matrices, comparing the generalized congruence maximization procedure with alternative rotation procedures, is presented. The results show a marked improvement of the obtained congruence using the generalized congruence maximization procedure compared to other procedures, without a significant loss of success with respect to the least squares criterion. A computer program written by the author to perform the rotations is briefly discussed.

In the framework of a robustness study on maximum likelihood estimation with LISREL three types of problems are dealt with: nonconvergence, improper solutions, and choice of starting values. The purpose of the paper is to illustrate why and to what extent these problems are of importance for users of LISREL. The ways in which these issues may affect the design and conclusions of robustness research is also discussed.

Loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables. In this paper we will show that there are cases in which these two techniques can be used complementary to each other. More specifically, we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices, each following a specific loglinear model. Therefore, in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models. A generalization of correspondence analysis, recently proposed by Escofier, will also be discussed. With this decomposition, which includes classical correspondence analysis as a special case, it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis. In this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models.

Factor analysis for nonnormally distributed variables is discussed in this paper. The main difference between our approach and more traditional approaches is that not only second order cross-products (like covariances) are utilized, but also higher order cross-products. It turns out that under some conditions the parameters (factor loadings) can be uniquely determined. Two estimation procedures will be discussed. One method gives Best Generalized Least Squares (BGLS) estimates, but is computationally very heavy, in particular for large data sets. The other method is a least squares method which is computationally less heavy. In one example the two methods will be compared by using the bootstrap method. In another example real life data are analyzed.