Mixed models take the dependency between observations based on the same cluster into account by introducing 1 or more random effects. Common item response theory (IRT) models introduce latent person variables to model the dependence between responses of the same participant. Assuming a distribution for the latent variables, these IRT models are formally equivalent with nonlinear mixed models. It is shown how a variety of IRT models can be formulated as particular instances of nonlinear mixed models. The unifying framework offers the advantage that relations between different IRT models become explicit and that it is rather straightforward to see how existing IRT models can be adapted and extended. The approach is illustrated with a self-report study on anger.Mixed models (see, e.g., Diggle, Heagerty, Liang, & Zeger, 2002;Goldstein, 1995;Longford, 1993;Verbeke & Molenberghs, 2000) are a collection of statistical tools that are well suited for analyzing clustered data, such as, for example, data from students nested within schools or repeated measurement data (measurements nested within participants). Modeling clustered data under a model that assumes independent observations is inappropriate because the observations on the subunits (students, measurements) of the same unit (school, participant) tend to be more homogeneous than the observations on subunits of different units. The heterogeneity between units can be taken into account by assuming that (some of) the parameters of the model follow some random distribution over the population of units. Hence, (some of) the parameters of the model are random variables, and the model is a random effects model, or mixed model.The random effects represent the unit effects. The parameters that are not assumed to be random realizations from a distribution, and hence are not considered to be unit specific, are called fixed effects.Mixed models and related methods were first developed in the context of analysis of variance and regression analysis, leading to the linear mixed model. Other commonly used terms are multilevel models (Goldstein, 1995), hierarchical models (Raudenbush & Bryk, 2002), and random coefficient models (Longford, 1993). More recently, nonlinear mixed models were also developed. A nonlinear mixed model is a model with random coefficients in which the fixed and/or random effects enter the model nonlinearly (Davidian & Giltinan, 1995). As we explain below, a subset of nonlinear mixed models is the class of generalized linear mixed models (McCulloch & Searle, 2001).The main purpose of this article is to explain how item response theory (IRT) models can be conceptualized as nonlinear mixed models and to provide a framework for this conceptualization. There are four important assets of this approach. First, this conceptualization relates IRT to the broad statistical literature on mixed models. Second, applying the same framework to different IRT models can help in the understanding of their differences and commonalities. Third, using this framework, one can read...
This study shows no benefits of case management for older adults with dementia symptoms and their primary informal caregivers. One possible explanation is that case management, which has been recommended among diagnosed dementia patients, may not be beneficial if offered too early. However, on the other hand, it is possible that: (1) case management will be effective in this group if more fully implemented and adapted or aimed at informal caregivers who experience more severe distress and problems; (2) case management is beneficial but that it is not seen in the timeframe studied; (3) case management might have undetected small benefits. This has to be established. Trial registration ISCRTN83135728.
We present a review of statistical inference in generalized linear mixed models (GLMMs). GLMMs are an extension of generalized linear models and are suitable for the analysis of non-normal data with a clustered structure. A GLMM contains parameters common to all clusters (fixed regression effects and variance components) and cluster-specific parameters. The latter parameters are assumed to be randomly drawn from a population distribution. The parameters of this population distribution (the variance components) have to be estimated together with the fixed effects. We focus on the case in which the cluster-specific parameters are normally distributed. The cluster-specific effects are integrated out of the likelihood so that the fixed effects and variance components can be estimated. Unfortunately, the integral over the cluster-specific effects is intractable for most GLMMs with a normal mixing distribution. Within a classical statistical framework, we distinguish between two broad classes of methods to handle this intractable integral: methods that rely on a numerical approximation to the integral and methods that use an analytical approximation to the integrand. Finally, we present an overview of available methods for testing hypotheses about the parameters of GLMMs.
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 model. In turn, both models are constrained bi‐factor models. Therefore, the efficient full maximum likelihood estimation method that has been established for the bi‐factor model can be modified to estimate the parameters of the two other models. An application on a testlet‐based international English assessment indicated that the bi‐factor model was the preferred model for this particular data set.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.