Estimating the effects of randomized experiments and, by extension, their mediating mechanisms, is often complicated by treatment noncompliance. Two estimation methods for causal mediation in the presence of noncompliance have recently been proposed, the instrumental variable method (IV-mediate) and maximum likelihood method (ML-mediate). However,…

This article summarizes a conceptual framework and simple mathematical methods of estimating the probability that one event was a necessary cause of another, as interpreted by lawmakers. We show that the fusion of observational and experimental data can yield informative bounds that, under certain circumstances, meet legal criteria of causation.…

Hierarchical or multilevel linear models are widely used for longitudinal or cross-sectional data on students nested in classes and schools, and are particularly important for estimating treatment effects in cluster-randomized trials, multi-site trials, and meta-analyses. The models can allow for variation in treatment effects, as well as…

Asymptotic expansions of the maximum likelihood estimator (MLE) and weighted likelihood estimator (WLE) of an examinee's ability are derived while item parameter estimators are treated as covariates measured with error. The asymptotic formulae present the amount of bias of the ability estimators due to the uncertainty of item parameter estimators.…

In item response theory (IRT) modeling, the item parameter error covariance matrix plays a critical role in statistical inference procedures. When item parameters are estimated using the EM algorithm, the parameter error covariance matrix is not an automatic by-product of item calibration. Cai proposed the use of Supplemented EM algorithm for…

The problem of factor score indeterminacy implies that the factor and the error scores cannot be completely disentangled in the factor model. It is therefore proposed to compute Harman's factor score predictor that contains an additive combination of factor and error variance. This additive combination is discussed in the framework of classical…

Researchers in psychology are increasingly using model selection strategies to decide among competing models, rather than evaluating the fit of a given model in isolation. However, such interest in model selection outpaces an awareness that one or a few cases can have disproportionate impact on the model ranking. Though case influence on the fit…

Testlet effects can be taken into account by incorporating specific dimensions in addition to the general dimension into the item response theory model. Three such multidimensional models are described: the bi-factor model, the testlet model, and a second-order model. It is shown how the second-order model is formally equivalent to the testlet…

Full-information item bifactor analysis is an important statistical method in psychological and educational measurement. Current methods are limited to single-group analysis and inflexible in the types of item response models supported. We propose a flexible multiple-group item bifactor analysis framework that supports a variety of…

We propose a new limited-information goodness of fit test statistic C[subscript 2] for ordinal IRT models. The construction of the new statistic lies formally between the M[subscript 2] statistic of Maydeu-Olivares and Joe (2006), which utilizes first and second order marginal probabilities, and the M*[subscript 2] statistic of Cai and Hansen…

Maximum likelihood is the most common estimation method in structural equation modeling. Standard errors for maximum likelihood estimates are obtained from the associated information matrix, which can be estimated from the sample using either expected or observed information. It is known that, with complete data, estimates based on observed or…

A marginal maximum a posteriori (MMAP) procedure was implemented to estimate item parameters in the generalized graded unfolding model (GGUM). Estimates from the MMAP method were compared with those derived from marginal maximum likelihood (MML) and Markov chain Monte Carlo (MCMC) procedures in a recovery simulation that varied sample size,…

Markov chain Monte Carlo (MCMC) methods enable a fully Bayesian approach to parameter estimation of item response models. In this simulation study, the authors compared the recovery of graded response model parameters using marginal maximum likelihood (MML) and Gibbs sampling (MCMC) under various latent trait distributions, test lengths, and…

A latent variable modeling approach for examining population similarities and differences in observed variable relationship and mean indexes in incomplete data sets is discussed. The method is based on the full information maximum likelihood procedure of model fitting and parameter estimation. The procedure can be employed to test group identities…

Davidian-curve item response theory (DC-IRT) is introduced, evaluated with simulations, and illustrated using data from the Schedule for Nonadaptive and Adaptive Personality Entitlement scale. DC-IRT is a method for fitting unidimensional IRT models with maximum marginal likelihood estimation, in which the latent density is estimated,…