The item characteristic curve (ICC), defining the relation between ability and the probability of choosing a particular option for a test item, can be estimated by using polynomial regression splines. These provide a more flexible family of functions than is given by the three-parameter logistic family. The estimation of spline ICCs is described by maximizing the marginal likelihood formed by integrating ability over a beta prior distribution. Some simulation results compare this approach with the joint estimation of ability and item parameters.

The paper suggests new methods for comparing the medians corresponding to independent treatment groups. The procedures are based on the Harrell-Davis estimator in conjunction with a slight modification and extension of the bootstrap calibration technique suggested by Loh. Alternatives to the Harrell-Davis estimator are briefly discussed. For the special case of two treatment groups, the proposed procedure always had more power than the Fligner-Rust solution, as well as the procedure examined by Wilcox and Charlin. Included is an illustration, using real data, that comparing medians, rather than means, can yield a substantially different conclusion as to whether two distributions differ in terms of some measure of central location.

Data in the form of zero-one matrices where conditioning on the marginals is relevant arise in diverse fields such as social networks and ecology; directed graphs constitute an important special case. An algorithm is given for the complete enumeration of the family of all zero-one matrices with given marginals and with a prespecified set of cells with structural zero entries. Complete enumeration is computationally feasible only for relatively small matrices. Therefore, a more useable Monte Carlo simulation method for the uniform distribution over this family is given, based on unequal probability sampling and ratio estimation. This method is applied to testing reciprocity of choices in social networks.

Ideal-points are widely used to model choices when preferences are single-peaked. Ideal-point choice models have been typically estimated at the individual-level, or have been based on the assumption that ideal-points are normally distributed over the population of choice makers. We propose two probabilistic ideal-point choice models for the external analysis of preferences that allow for more flexible multimodal distributions of ideal-points, thus acknowledging the existence of subpopulations with distinct preferences. The first model extends the ideal-point probit model for heterogeneous preferences to accommodate a mixture of multivariate normal distributions of ideal-points. The second model assumes that ideal-points are uniformly distributed within finite ranges of the attribute space, leading to a more simplistic formulation and a more flexible distribution. The two models are applied to simulated and actual choice data, and compared to the ideal-point probit model.

An alternating least squares method for iteratively fitting the longitudinal reduced-rank regression model is proposed. The method uses ordinary least squares and majorization substeps to estimate the unknown parameters in the system and measurement equations of the model. In an example with cross-sectional data, it is shown how the results conform closely to results from eigenanalysis. Optimal scaling of nominal and ordinal variables is added in a third substep, and illustrated with two examples involving cross-sectional and longitudinal data.

A number of methods for the analysis of three-way data are described and shown to be variants of principal components analysis (PCA) of the two-way supermatrix in which each two-way slice is “strung out” into a column vector. The methods are shown to form a hierarchy such that each method is a constrained variant of its predecessor. A strategy is suggested to determine which of the methods yields the most useful description of a given three-way data set.

This paper presents a new stochastic multidimensional scaling procedure for the analysis of three-mode, three-way pick any/J data. The method provides either a vector or ideal-point model to represent the structure in such data, as well as “floating” model specifications (e.g., different vectors or ideal points for different choice settings), and various reparameterization options that allow the coordinates of ideal points, vectors, or stimuli to be functions of specified background variables. A maximum likelihood procedure is utilized to estimate a joint space of row and column objects, as well as a set of weights depicting the third mode of the data. An algorithm using a conjugate gradient method with automatic restarts is developed to estimate the parameters of the models. A series of Monte Carlo analyses are carried out to investigate the performance of the algorithm under diverse data and model specification conditions, examine the statistical properties of the associated test statistic, and test the robustness of the procedure to departures from the independence assumptions. Finally, a consumer psychology application assessing the impact of situational influences on consumers' choice behavior is discussed.

A latent trait model is presented for the repeated measurement of ability based on a multidimensional conceptualization of the change process. A simplex structure is postulated to link item performance under a given measurement condition or occasion to initial ability and to one or more modifiabilities that represent individual differences in change. Since item discriminations are constrained to be equal within a measurement condition, the model belongs to the family of multidimensional Rasch models. Maximum likelihood estimators of the item parameters and abilities are derived, and an example provided that shows good recovery of both item and ability parameters. Properties of the model are explored, particularly for several classical issues in measuring change.

A wide variety of paired comparison, triple comparison, and ranking experiments may be viewed as generalized linear models. These include paired comparison models based on both the Bradley-Terry and Thurstone-Mosteller approaches, as well as extensions of these models that allow for ties, order of presentation effects, and the presence of covariates. Moreover, the triple comparison model of Pendergrass and Bradley, as well as models for complete rankings of more than three items, can also be represented as generalized linear models. All such models can be easily fit by maximum likelihood, using the widely available GLIM computer package. Additionally, GLIM enables the computation of likelihood ratio statistics for testing many hypotheses of interest. Examples are presented that cover a variety of cases, along with their implementation on GLIM.

A commonly used method to evaluate the accuracy of a measurement is to provide a confidence interval that contains the parameter of interest with a given high probability. Smallest exact confidence intervals for the ability parameter of the Rasch model are derived and compared to the traditional, asymptotically valid intervals based on the Fisher information. Tables of the exact confidence intervals, termed Clopper-Pearson intervals, can be routinely drawn up by applying a computer program designed by and obtainable from the author. These tables are particularly useful for tests of only moderate lengths where the asymptotic method does not provide valid confidence intervals.