This paper generalizes the p* class of models for social network data to predict individual-level attributes from network ties. The p* model for social networks permits the modeling of social relationships in terms of particular local relational or network configurations. In this paper we present methods for modeling attribute measures in terms of network ties, and so construct p* models for the patterns of social influence within a network. Attribute variables are included in a directed dependence graph and the Hammersley-Clifford theorem is employed to derive probability models whose parameters can be estimated using maximum pseudo-likelihood. The models are compared to existing network effects models. They can be interpreted in terms of public or private social influence phenomena within groups. The models are illustrated by an empirical example involving a training course, with trainees' reactions to aspects of the course found to relate to those of their network partners.

Combinatorial optimization problems in the social and behavioral sciences are frequently associated with a variety of alternative objective criteria. Multiobjective programming is an operations research methodology that enables the quantitative analyst to investigate tradeoffs among relevant objective criteria. In this paper, we describe an interactive procedure for multiobjective asymmetric unidimensional seriation problems. This procedure uses a dynamic-programming algorithm to partially generate the efficient set of sequences for small to medium-sized problems, and a multioperation heuristic to estimate the efficient set for larger problems. The interactive multiobjective procedure is applied to an empirical data set from the psychometric literature. We conclude with a discussion of other potential areas of application in combinatorial data analysis.

Item responses that do not fit an item response theory (IRT) model may cause the latent trait value to be inaccurately estimated. In the past two decades several statistics have been proposed that can be used to identify nonfitting item score patterns. These statistics all yield scalar values. Here, the use of the person response function (PRF) for identifying nonfitting item score patterns was investigated. The PRF is a function and can be used for diagnostic purposes. First, the PRF is defined in a class of IRT models that imply an invariant item ordering. Second, a person-fit method proposed by Trabin & Weiss (1983) is reformulated in a nonparametric IRT context assuming invariant item ordering, and statistical theory proposed by Rosenbaum (1987a) is adapted to test locally whether a PRF is nonincreasing. Third, a simulation study was conducted to compare the use of the PRF with the person-fit statistic ZU3. It is concluded that the PRF can be used as a diagnostic tool in person-fit research.

By assuming a distribution for the subject weights in a diagonal metric (INDSCAL) multidimensional scaling model, the subject weights become random effects. Including random effects in multidimensional scaling models offers several advantages over traditional diagonal metric models such as those fitted by the INDSCAL, ALSCAL, and other multidimensional scaling programs. Unlike traditional models, the number of parameters does not increase with the number of subjects, and, because the distribution of the subject weights is modeled, the construction of linear models of the subject weights and the testing of those models is immediate. Here we define a random effects diagonal metric multidimensional scaling model, give computational algorithms, describe our experiences with these algorithms, and provide an example illustrating the use of the model and algorithms.

The Psychometric Society is “devoted to the development of Psychology as a quantitative rational science”. Engineering is often set in contradistinction with science; art is sometimes considered different from science. Why, then, juxtapose the words in the title:psychometric, engineering, and art? Because an important aspect of quantitative psychology is problem-solving, and engineering solves problems. And an essential aspect of a good solution is beauty—hence, art. In overview and with examples, this presentation describes activities that are quantitative psychology as engineering and art—that is, as design. Extended illustrations involve systems for scoring tests in realistic contexts. Allusions are made to other examples that extend the conception of quantitative psychology as engineering and art across a wider range of psychometric activities.

We relate Thurstonian models for paired comparisons data to Thurstonian models for ranking data, which assign zero probabilities to all intransitive patterns. We also propose an intermediate model for paired comparisons data that assigns nonzero probabilities to all transitive patterns and to some but not all intransitive patterns.

There is a close correspondence between the multidimensional normal ogive model employed in educational testing and Thurstone's model for paired comparisons data under multiple judgment sampling with minimal identification restrictions. Alike the normal ogive model, Thurstonian models have two formulations, a factor analytic and an IRT formulation. We use the factor analytic formulation to estimate this model from the first and second order marginals of the contingency table using estimators proposed by Muthén. We also propose a statistic to assess the fit of these models to the first and second order marginals of the contingency table. This is important, as a model may reproduce well the estimated thresholds and tetrachoric correlations, yet fail to reproduce the marginals of the contingency table if the assumption of multivariate normality is incorrect.

A simulation study is performed to investigate the performance of three alternative limited information estimators which differ in the procedure used in their final stage: unweighted least squares (ULS), diagonally weighted least squares (DWLS), and full weighted least squares (WLS). Both the ULS and DWLS show a good performance with medium size problems and small samples, with a slight better performance of the ULS estimator.

Person fit statistics are considered for dichotomous item response models. The asymptotic null distribution is derived for statistics which are linear in the item responses, and in which the ability parameter is replaced by an estimate. This allows the asymptotically correct standardization of linear person fit statistics with estimated ability parameter. The fact that the ability parameter is estimated usually decreases the asymptotic variance.

This paper presents a synthesis of Bock's (1972) nominal categories model and Luce's (1959) choice model for mixed-effects analyses of rank-ordered data. It is shown that the proposed ranking model is both parsimonious and flexible in accounting for preference heterogeneity as well as fixed and random effects of covariates. Relationships to other approaches, including Takane's (1987) ideal point discriminant model and Croon's (1989) latent-class version of Luce's ranking model, are also discussed. The application focuses on a ranking study of behavioral traits that parents find desirable in children.

Methodology is described for fitting a fuzzy consensus partition to a set of partitions of the same set of objects. Three models defining median partitions are described: two of them are obtained from a least-squares fit of a set of membership functions, and the third (proposed by Pittau and Vichi) is acquired from a least-squares fit of a set of joint membership functions. The models are illustrated by application to both a set of hard partitions and a set of fuzzy partitions and comparisons are made between them and an alternative approach to obtaining a consensus fuzzy partition proposed by Sato and Sato; a discussion is given of some interesting differences in the results.

The paper takes up the problem of performing all pairwise comparisons among J independent groups based on 20% trimmed means. Currently, a method that stands out is the percentile-t bootstrap method where the bootstrap is used to estimate the quantiles of a Studentized maximum modulus distribution when all pairs of population trimmed means are equal. However, a concern is that in simulations, the actual probability of one or more Type I errors can drop well below the nominal level when sample sizes are small. A practical issue is whether a method can be found that corrects this problem while maintaining the positive features of the percentile-t bootstrap. Three new methods are considered here, one of which achieves the desired goal. Another method, which takes advantage of theoretical results by Singh (1998), performs almost as well but is not recommended when the smallest sample size drops below 15. In some situations, however, it gives substantially shorter confidence intervals.

Corrections of correlations for range restriction (i.e., selection) and unreliability are common in psychometric work. The current rule of thumb for determining the order in which to apply these corrections looks to the nature of the reliability estimate (i.e., restricted or unrestricted). While intuitive, this rule of thumb is untenable when the correction includes the variable upon which selection is made, as is generally the case. Using classical test theory, we show that it is the nature of the range restriction, not the nature of the available reliability coefficient, that determines the sequence for applying corrections for range restriction and unreliability.

Three classes of polytomous IRT models are distinguished. These classes are the adjacent category models, the cumulative probability models, and the continuation ratio models. So far, the latter class has received relatively little attention. The class of continuation ratio models includes logistic models, such as the sequential model (Tutz, 1990), and nonlogistic models, such as the acceleration model (Samejima, 1995) and the nonparametric sequential model (Hemker, 1996). Four measurement properties are discussed. These are monotone likelihood ratio of the total score, stochastic ordering of the latent trait by the total score, stochastic ordering of the total score by the latent trait, and invariant item ordering. These properties have been investigated previously for the adjacent category models and the cumulative probability models, and for the continuation ratio models this is done here. It is shown that stochastic ordering of the total score by the latent trait is implied by all continuation ratio models, while monotone likelihood ratio of the total score and stochastic ordering on the latent trait by the total score are not implied by any of the continuation ratio models. Only the sequential rating scale model implies the property of invariant item ordering. Also, we present a Venn-diagram showing the relationships between all known polytomous IRT models from all three classes.

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model say M0 implies on a less restricted one M1. If T0 and T1 denote the goodness-of-fit test statistics associated to M0 andM1, respectively, then typically the difference Td=T0−T1 is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the models M0 andM1. As in the case of the goodness-of-fit test, it is of interest to scale the statistic Td in order to improve its chi-square approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of models M0 and M1. A Monte Carlo study is provided to illustrate the performance of the competing statistics.

An important assumption in IRT model-based adaptive testing is that matching difficulty levels of test items with an examinee's ability makes a test more efficient. According to Lord, “An examinee is measured most effectively when the test items are neither too difficult nor too easy for him”. This assumption is examined and challenged through a class of one-parameter IRT models including those of Rasch and the normal ogive. It is found that for a specific model, the validity of the fundamental assumption is closely related to the so-called van Zwet tail ordering of symmetric distributions. In this connection, the cosine distribution serves as the borderline between those satisfying the assumption and those violating the assumption. Graphic and numerical illustrations are presented to demonstrate the theoretic results.

One of the most vexing problems in cluster analysis is the selection and/or weighting of variables in order to include those that truly define cluster structure, while eliminating those that might mask such structure. This paper presents a variable-selection heuristic for nonhierarchical (K-means) cluster analysis based on the adjusted Rand index for measuring cluster recovery. The heuristic was subjected to Monte Carlo testing across more than 2200 datasets with known cluster structure. The results indicate the heuristic is extremely effective at eliminating masking variables. A cluster analysis of real-world financial services data revealed that using the variable-selection heuristic prior to the K-means algorithm resulted in greater cluster stability.

The linear logistic test model (LLTM) specifies the item parameters as a weighted sum of basic parameters. The LLTM is a special case of a more general nonlinear logistic test model (NLTM) where the weights are partially unknown. This paper is about the identifiability of the NLTM. Sufficient and necessary conditions for global identifiability are presented for a NLTM where the weights are linear functions, while conditions for local identifiability are shown to require a model with less restrictions. It is also discussed how these conditions are checked using an algorithm due to Bekker, Merckens, and Wansbeek (1994). Several illustrations are given.