Several procedures have been proposed in the statistical literature for estimating simultaneously the mean of each of k binomial populations. In terms of mental test theory, however, it is not clear that these procedures should be used when an item sampling model applies since the binomial error model is usually viewed as an oversimplification of the “true” situation. In this study we compare empirically several of these estimation techniques. Particular attention is given to situations where observations are generated according to a two-term approximation to the compound binomial distribution.

Empirical Bayes methods are shown to provide a practical alternative to standard least squares methods in fitting high dimensional models to sparse data. An example concerning prediction bias in educational testing is presented as an illustration.

Item response theory models posit latent variables to account for regularities in students' performances on test items. Wilson's “Saltus” model extends the ideas of IRT to development that occurs in stages, where expected changes can be discontinuous, show different patterns for different types of items, or even exhibit reversals in probabilities of success on certain tasks. Examples include Piagetian stages of psychological development and Siegler's rule-based learning. This paper derives marginal maximum likelihood (MML) estimation equations for the structural parameters of the Saltus model and suggests a computing approximation based on the EM algorithm. For individual examinees, empirical Bayes probabilities of learning-stage are given, along with proficiency parameter estimates conditional on stage membership. The MML solution is illustrated with simulated data and an example from the domain of mixed number subtraction.

It has long been part of the item response theory (IRT) folklore that under the usual empirical Bayes unidimensional IRT modeling approach, the posterior distribution of examinee ability given test response is approximately normal for a long test. Under very general and nonrestrictive nonparametric assumptions, we make this claim rigorous for a broad class of latent models.

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters θ= (θ1, …, θn) and item parameters β=(β1, …, βk), where βj may be vector valued. With θ considered a random sample from a prior distribution with parameter ϕ, the estimation of (θ, β) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.

We study a proportional reduction in loss (PRL) measure for the reliability of categorical data and consider the general case in which each of N judges assigns a subject to one of K categories. This measure has been shown to be equivalent to a measure proposed by Perreault and Leigh for a special case when there are two equally competent judges, and the correct category has a uniform prior distribution. We consider a general framework where the correct category is assumed to have an arbitrary prior distribution, and where classification probabilities vary by correct category, judge, and category of classification. In this setting, we consider PRL reliability measures based on two estimators of the correct category—the empirical Bayes estimator and an estimator based on the judges' consensus choice. We also discuss four important special cases of the general model and study several types of lower bounds for PRL reliability.

The asymptotic posterior normality (APN) of the latent variable vector in an item response theory (IRT) model is a crucial argument in IRT modeling approaches. In case of a single latent trait and under general assumptions, Chang and Stout (Psychometrika, 58(1):37–52, 1993) proved the APN for a broad class of latent trait models for binary items. Under the same setup, they also showed the consistency of the latent trait’s maximum likelihood estimator (MLE). Since then, several modeling approaches have been developed that consider multivariate latent traits and assume their APN, a conjecture which has not been proved so far. We fill this theoretical gap by extending the results of Chang and Stout for multivariate latent traits. Further, we discuss the existence and consistency of MLEs, maximum a-posteriori and expected a-posteriori estimators for the latent traits under the same broad class of latent trait models.

A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent variables. The response model generalizes GLMMs to incorporate factor structures in addition to random intercepts and coefficients. As in GLMMs, the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower-level units in the higher-level units and missing data. A wide range of response processes can be modeled including ordered and unordered categorical responses, counts, and responses of mixed types. The structural model is similar to the structural part of a SEM except that it may include latent and observed variables varying at different levels. For example, unit-level latent variables (factors or random coefficients) can be regressed on cluster-level latent variables. Special cases of this framework are explored and data from the British Social Attitudes Survey are used for illustration. Maximum likelihood estimation and empirical Bayes latent score prediction within the GLLAMM framework can be performed using adaptive quadrature in gllamm, a freely available program running in Stata.

Conjoint analysis is widely used for estimating the effects of a large number of treatments on multidimensional decision-making. However, it is this substantive advantage that leads to a statistically undesirable property, multiple hypothesis testing. Existing applications of conjoint analysis except for a few do not correct for the number of hypotheses to be tested, and empirical guidance on the choice of multiple testing correction methods has not been provided. This paper first shows that even when none of the treatments has any effect, the standard analysis pipeline produces at least one statistically significant estimate of average marginal component effects in more than 90% of experimental trials. Then, we conduct a simulation study to compare three well-known methods for multiple testing correction, the Bonferroni correction, the Benjamini–Hochberg procedure, and the adaptive shrinkage (Ash). All three methods are more accurate in recovering the truth than the conventional analysis without correction. Moreover, the Ash method outperforms in avoiding false negatives, while reducing false positives similarly to the other methods. Finally, we show how conclusions drawn from empirical analysis may differ with and without correction by reanalyzing applications on public attitudes toward immigration and partner countries of trade agreements.

Simulations of future climate contain variability arising from a number of sources, including internal stochasticity and external forcings. However, to the best of our abilities climate models and the true observed climate depend on the same underlying physical processes. In this paper, we simultaneously study the outputs of multiple climate simulation models and observed data, and we seek to leverage their mean structure as well as interdependencies that may reflect the climate’s response to shared forcings. Bayesian modeling provides a fruitful ground for the nuanced combination of multiple climate simulations. We introduce one such approach whereby a Gaussian process is used to represent a mean function common to all simulated and observed climates. Dependent random effects encode possible information contained within and between the plurality of climate model outputs and observed climate data. We propose an empirical Bayes approach to analyze such models in a computationally efficient way. This methodology is amenable to the CMIP6 model ensemble, and we demonstrate its efficacy at forecasting global average near-surface air temperature. Results suggest that this model and the extensions it engenders may provide value to climate prediction and uncertainty quantification.

Estimation of farm prevalence is common in veterinary research. Typically, not all animals within the farm are sampled, and imperfect tests are used. Often, assumptions about herd sizes and sampling proportions are made, which may be invalid in smallholder settings. We propose an alternative method for estimating farm prevalence in the context of Brucella seroprevalence estimation in an endemic region of Kazakhstan. We collected 210 milk samples from Otar district, with a population of about 1000 cattle and 16 000 small ruminants, and tested them using an indirect ELISA. Individual-level prevalence and 95% confidence intervals were estimated using Taylor series linearization. A model was developed to estimate the smallholding prevalence, taking into account variable sampling proportions and uncertainty in the test accuracy. We estimate that 73% of households that we sampled had at least one Brucella-seropositive animal (95% credible interval 68–82). We estimate that 58% (95% confidence interval 40–76) of lactating small ruminants and 14% (95% confidence interval 1–28) of lactating cows were seropositive. Our results suggest that brucellosis is highly endemic in the area and conflict with those of the official brucellosis-testing programme, which found that in 2013 0% of cows and 1·7% of small ruminants were seropositive.

We derive some decision rules to select best predictive regression models in a credibility context, that is, in a “random effects’ linear regression model with replicates. In contrast to usual model selection techniques on a collective level, our proposal allows to detect individual structures, even if they disappear in the collective.

We give exact, non-asymptotic results for the expected squared error loss for a predictor based on credibility estimation in different models. This involves correct accounting of random model parameters and the study of expected loss for shrinkage estimation. We support the theoretical properties of the new model selectors by a small simulation experiment.

This paper considers the histogram of unit cell size built up from m independent observations on a Poisson (μ) distribution. The following question is addressed: what is the limiting probability of the event that there are no unoccupied cells lying to the left of occupied cells of the histogram? It is shown that the probability of there being no such isolated empty cells (or isolated finite groups of empty cells) tends to unity as the number m of observations tends to infinity, but that the corresponding almost sure convergence fails. Moreover this probability does not tend to unity when the Poisson distribution is replaced by the negative binomial distribution arising when μ is randomized by a gamma distribution. The relevance to empirical Bayes statistical methods is discussed.