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 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.

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.

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.

In a recent article Bentler and Woodward (1983) discussed computational and statistical issues related to the greatest lower bound ρ+ to reliability. Although my work (Shapiro, 1982) was cited frequently some results presented were misunderstood. A sample estimate + of ρ + was considered and it was claimed (Bentler & Woodward) that: “Since + is not a closed form expression ... an exact analytic expression for h has not been found” (p. 247). (h is a vector of partial derivatives of ρ+ as a function of the covariance matrix.) Therefore Bentler and Woodward proposed to use numerical derivatives in order to evaluate the asymptotic variance avar ( +) of +.

Certain ambiguities in a recent paper on the computation and statistics of the greatest lower bound are clarified.

Lord and Wingersky have developed a method for computing the asymptotic variance-covariance matrix of maximum likelihood estimates for item and person parameters under some restrictions on the estimates which are needed in order to fix the latent scale. The method is tedious, but can be simplified for the Rasch model when one is only interested in the item parameters. This is demonstrated here under a suitable restriction on the item parameter estimates.