This paper explores the robustness of conclusions from a statistical model against variations in model choice (rather than variations in random sampling and random assignment to treatments, which are the usual variations covered by inferential statistics). After the problem formulation in section 1, section 2 presents an example from Box and Tiao in which variation in parent distribution is modeled for a one sample location problem. An adaptive Bayesian procedure permits to use a weighted mixture of parent distributions rather than choosing just one, such as a normal or a uniform distribution.

In section 3 similar considerations are applied to an event history model for the influence of education and gender on age at first marriage, but here the conclusions are hardly influenced by the choice of the duration distribution. In section 4 a brief discussion of model choice in factor analysis and structural equation models is followed by a more elaborate treatment of the choice of integer valued slopes for item response functions in the OPLM model which is an extension of the Rasch model. A modest simulation study suggests that Adaptive Bayesian Modeling with a mixture of sets of slopes works better than fixing one set of postulated slopes.

The paper concludes with some remarks on the roles and sources of prior distributions followed by a short epilogue which argues that simultaneous consideration of a class of models for the same data is sometimes superior to exclusively analyzing the data under one specific model chosen from such a class.

There are various optimization strategies for approximating, through the minimization of a least-squares loss function, a given symmetric proximity matrix by a sum of matrices each subject to some collection of order constraints on its entries. We extend these approaches to include components in the approximating sum that satisfy what are called the strongly-anti-Robinson (SAR) or circular strongly-anti-Robinson (CSAR) restrictions. A matrix that is SAR or CSAR is representable by a particular graph-theoretic structure, where each matrix entry is reproducible from certain minimum path lengths in the graph. One published proximity matrix is used extensively to illustrate the types of approximation that ensue when the SAR or CSAR constraints are imposed.

The subject of factor indeterminacy has a vast history in factor analysis (Guttman, 1955; Lederman, 1938; Wilson, 1928). It has lead to strong differences in opinion (Steiger, 1979). The current paper gives necessary and sufficient conditions for observability of factors in terms of the parameter matrices and a finite number of variables. Five conditions are given which rigorously define indeterminacy. It is shown that (un)observable factors are (in)determinate. Specifically, the indeterminacy proof by Guttman is extended to Heywood cases. The results are illustrated by two examples and implications for indeterminacy are discussed.

Data are ipsative if they are subject to a constant-sum constraint for each individual. In the present study, ordinal ipsative data (OID) are defined as the ordinal rankings across a vector of variables. It is assumed that OID are the manifestations of their underlying nonipsative vector y, which are difficult to observe directly. A two-stage estimation procedure is suggested for the analysis of structural equation models with OID. In the first stage, the partition maximum likelihood (PML) method and the generalized least squares (GLS) method are proposed for estimating the means and the covariance matrix of Acy, where Ac is a known contrast matrix. Based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix, the generalized least squares method is used to estimate the structural parameters in the second stage. A goodness-of-fit statistic is given for testing the hypothesized covariance structure. Simulation results show that the proposed method works properly when a sufficiently large sample is available.

A set of linear conditions on item response functions is derived that guarantees identical observed-score distributions on two test forms. The conditions can be added as constraints to a linear programming model for test assembly that assembles a new test form to have an observed-score distribution optimally equated to the distribution on an old form. For a well-designed item pool and items fitting the IRT model, use of the model results into observed-score pre-equating and prevents the necessity of post hoc equating by a conventional observed-score equating method. An empirical example illustrates the use of the model for an item pool from the Law School Admission Test.

This paper presents a new methodology concerned with the estimation of ultrametric trees calibrated on subjects' pairwise proximity judgments of stimuli, capturing subject heterogeneity using a finite mixture formulation. We assume that a number of unobserved classes of subjects exist, each having a different ultrametric tree structure underlying the pairwise proximity judgments. A new likelihood based estimation methodology is presented for those finite mixtures of ultrametric trees, that accommodates ultrametric as well as other external constraints. Various assumptions on the correlation of the error of the dissimilarities are accommodated. The performance of the method to recover known ultrametric tree structures is investigated on synthetic data. An empirical application to published data from Schiffman, Reynolds, and Young (1981) is provided. The ability to deal with external constraints on the tree-topology is demonstrated, and a comparison with an alternative clustering based method is made.