At the heart of attitudinal and strategic explanations of judicial behavior is the assumption that justices have policy preferences. In this paper we employ Markov chain Monte Carlo (MCMC) methods to fit a Bayesian measurement model of ideal points for all justices serving on the U.S. Supreme Court from 1953 to 1999. We are particularly interested in determining to what extent ideal points of justices change throughout their tenure on the Court. This is important because judicial politics scholars oftentimes invoke preference measures that are time invariant. To investigate preference change, we posit a dynamic item response model that allows ideal points to change systematically over time. Additionally, we introduce Bayesian methods for fitting multivariate dynamic linear models (DLMs) to political scientists. Our results suggest that many justices do not have temporally constant ideal points. Moreover, our ideal point estimates outperform existing measures and explain judicial behavior quite well across civil rights, civil liberties, economics, and federalism cases.

We develop a Bayesian procedure for estimation and inference for spatial models of roll call voting. This approach is extremely flexible, applicable to any legislative setting, irrespective of size, the extremism of the legislators ’ voting histories, or the number of roll calls available for analysis. The model is easily extended to let other sources of information inform the analysis of roll call data, such as the number and nature of the underlying dimensions, the presence of party whipping, the determinants of legislator preferences, and the evolution of the legislative agenda; this is especially helpful since generally it is inappropriate to use estimates of extant methods (usually generated under assumptions of sincere voting) to test models embodying alternate assumptions (e.g., log-rolling, party discipline). A Bayesian approach also provides a coherent framework for estimation and inference with roll call data that eludes extant methods; moreover, via Bayesian simulation methods, it is straightforward to generate uncertainty assessments or hypothesis tests concerning any auxiliary quantity of interest or to formally compare models. In a series of examples we show how our method is easily extended to accommodate theoretically interesting models of legislative behavior. Our goal is to provide a statistical framework for combining the measurement of legislative preferences with tests of models of legislative behavior. Modern studies of legislative behavior focus

The term data augmentation refers to methods for constructing iterative optimization or sampling algorithms via the introduction of unobserved data or latent variables. For deterministic algorithms,the method was popularizedin the general statistical community by the seminal article by Dempster, Laird, and Rubin on the EM algorithm for maximizing a likelihood function or, more generally, a posterior density. For stochastic algorithms, the method was popularized in the statistical literature by Tanner and Wong’s Data Augmentation algorithm for posteriorsampling and in the physics literatureby Swendsen and Wang’s algorithm for sampling from the Ising and Potts models and their generalizations; in the physics literature,the method of data augmentationis referred to as the method of auxiliary variables. Data augmentationschemes were used by Tanner and Wong to make simulation feasible and simple, while auxiliary variables were adopted by Swendsen and Wang to improve the speed of iterative simulation. In general,however, constructing data augmentation schemes that result in both simple and fast algorithms is a matter of art in that successful strategiesvary greatlywith the (observed-data) models being considered.After an overview of data augmentation/auxiliary variables and some recent developments in methods for constructing such

Empirical Bayes regression procedures are commonly used in educational and psychological testing as extensions to latent variable models. The National Assessment of Educational Progress (NAEP) is an important national survey using such procedures. NAEP applies empirical Bayes methods to models from item response theory in order to calibrate student responses to questions of varying difficulty. Partially due to the limited computing technology that existed when NAEP was first conceived, NAEP analyses are carried out using a two-stage estimation procedure that ignores uncertainty about some model parameters. Furthermore, the item response theory model NAEP uses ignores the effect of item clustering created by the design of a test form. Using Markov chain Monte Carlo, we simultaneously estimate all parameters of an expanded model that considers item clustering in order to investigate the impact of item clustering and ignored uncertainty about model parameters on NAEP's reported outcome me...

As observations and student models become complex, educational assessments that exploit advances in technology and cognitive psychology can outstrip familiar testing models and analytic methods. Within the Portal conceptual framework for assessment design, Bayesian inference networks (BINs) record beliefs about students’ knowledge and skills, in light of what they say and do. Joining evidence model BIN fragments—which contain observable variables and pointers to student model variables—to the student model allows one to update belief about knowledge and skills as observations arrive. Markov Chain Monte Carlo (MCMC) techniques can estimate the required conditional probabilities from empirical data, supplemented by expert judgment or substantive theory. Details for the special cases of item response theory (IRT) and multivariate latent class modeling are given, with a numerical example of the latter. 1

Empirical Bayes regression procedures are often used in educational and psychological testing as extensions to latent variables models. The National Assessment of Educational Progress (NAEP) is an important national survey using such procedures. The NAEP applies empirical Bayes methods to models from item response theory to calibrate student responses to questions of varying difficulty. Due partially to the limited computing technology that existed when the NAEP was first conceived, NAEP analyses are carried out using a two-stage estimation procedure that ignores uncertainty about some model parameters. Furthermore, the item response theory model that the NAEP uses ignores the effect of item clustering created by the design of a test form. Using Markov chain Monte Carlo, we simultaneously estimate all parameters of an expanded model that considers item clustering to investigate the impact of item clustering and ignoring uncertainty about model parameters on an important outcome measure that the NAEP reports. Ignoring these two effects causes substantial underestimation of standard errors and induces a modest bias in location estimates.

At the heart of attitudinal and strategic explanations of judicial behavior is the assumption that justices have well-defined policy preferences. In the literature these preferences have been measured in a handful of ways, including using factor analysis and multidimensional scaling techniques (Schubert 1965, 1974), looking at past votes in a single policy area (Epstein et al.

Abstract. We introduce a new skew-probit link for item response theory (IRT) by considering an accumulated skew-normal distribution. The model extends the symmetric probit-normal IRT model by considering a new item (or skewness) parameter for the item characteristic curve. A special interpretation is given for this parameter, and a latent linear structure is indicated for the model when an augmented likelihood is considered. Bayesian MCMC inference approach is developed and an efficiency study in the estimation of the model parameters is undertaken for a data set from (Tanner 1996, pg. 190) by using the notion of effective sample size (ESS) as defined in Kass et al. (1998) and the sample size per second (ESS/s) as considered in Sahu (2002) The methodology is illustrated using a data set corresponding to a Mathematical Test applied in Peruvian schools for which a sensitivity analysis of the chosen priors is conducted and also a comparison with seven parametric IRT models is conducted. The main conclusion is that the skew-probit item response model seems to provide the best fit.

This article introduces a generalization of Tanner and Wong’s data augmentation algorithm which can be used when the complete data posterior distribution cannot be directly sampled. The algorithm proposes parameter values based on complete data sampling distributions of convenient frequentist esti-mators which ignore some information in the complete data likelihood. The proposals are filtered using a Metropolis-Hastings probability to produce draws from the Bayesian posterior distribution of interest. The result is a method of deriving closed form Metropolis-Hastings proposals which do not need to be tuned. The method is used to sample the parameters of a multinomial logit model from their posterior distribution in a manner similar to Albert and Chib’s (1993) algorithm for probit regression. The algo-rithm converges geometrically ergodically, and its convergence rate can be accelerated through the use of working parameter methods developed for standard data augmentation algorithms. As with standard data augmentation, the method remains useful when multinomial logit models are embedded in more compli-cated settings, such as hierarchical models. The algorithm’s utility in complicated settings is illustrated

A Markov chain Monte Carlo (MCMC) method and a bootstrap method were compared in the estimation of standard errors of item response theory (IRT) true score equating. Three test form relationships were examined: parallel, tauequivalent, and congeneric. Data were simulated based on Reading Comprehension and Vocabulary tests of the Iowa Tests of Basic Skills 1. For parallel and congeneric test forms within valid IRT true score ranges, the pattern and magnitude of standard errors of IRT true score equating estimated by the MCMC method were very close to those estimated by the bootstrap method. For tau-equivalent test forms, the pattern of standard errors estimated by the two methods was also similar. Bias and mean square errors of equating produced by the MCMC method were smaller than those produced by the bootstrap method; however, standard errors were larger. In educational testing, the MCMC method may be used as an additional or alternative procedure to the bootstrap method when evaluating the precision of equating results.