A review of R-packages for random-intercept probit regression in small clusters

Generalized Linear Mixed Models (GLMMs) are widely used to model clustered categorical outcomes. To tackle the intractable integration over the random effects distributions, several approximation approaches have been developed for likelihood-based inference. As these seldom yield satisfactory results when analyzing binary outcomes from small clusters, estimation within the Structural Equation Modeling (SEM) framework is proposed as an alternative. We compare the performance of R-packages for random-intercept probit regression relying on: the Laplace approximation, adaptive Gaussian quadrature (AGQ), Penalized Quasi-Likelihood (PQL), an MCMC-implementation, and integrated nested Laplace approximation within the GLMM-framework, and a robust diagonally weighted least squares estimation within the SEM-framework. In terms of bias for the fixed and random effect estimators, SEM usually performs best for cluster size two, while AGQ prevails in terms of precision (mainly because of SEM's robust standard errors). As the cluster size increases, however, AGQ becomes the best choice for both bias and precision

A new approach for within-subject mediation analysis in AB/BA crossover designs

Crossover trials are widely used in psychological and medical research to assess the effect of reversible exposures. In such designs, each subject is randomly allocated to a sequence of conditions, enabling the evaluation of treatment differences within each individual. When there are but two possible exposures -each assessed during one of two time periods-, the crossover study is referred to as an AB/BA design. The goal of this presentation is to discuss mediation analysis in such simple crossover studies. We do so by considering within-subject mediation from a counterfactual-based perspective and by deriving expressions for the direct and indirect effects. Employing simulation studies, the performance of several existing methods will be assessed and compared to that of a novel one we propose. We show that the new method yields unbiased and efficient estimators for the direct and indirect effect, under a minimalistic set of `no unmeasured confounding'-assumptions. Finally, we illustrate the different techniques with data from a neurobehavioral study

Estimation methods for generalized linear mixed models with binary outcomes from small clusters

Generalized linear mixed models (GLMMs) have been widely used for the modelling of longitudinal and clustered data in medical research, social and behavioural sciences, as well as across may other disciplines. Statistical inference of the GLMM, however, is hampered due to the incorporation of random effects: the likelihood function of the GLMM involves integrating out these effects from the joint density of the responses and random effects, which is, except for a few cases, analytically intractable. To tackle this intractability of GLMMs, numerous likelihood-based approximation methods have been proposed. One such method, the Laplace approximation, stands out as one of the most popular ones. Alternatively, Taylor expansions aimed to reduce the estimation of a GLMM to that of an approximated linear mixed or penalized quasi-likelihood method could be used (PQL), as well as an adaptive Gaussian quadrature (AGQ) approach. A second class of methodologies next to the above three likelihood-based approximations, is to pursue a Bayesian approach in which MCMC methods are used to make inferences based on the posterior distribution of the parameters, by e.g. relying on Gibbs sampling. Bayesian methods, although they show good frequentist properties when the model is correct, are known to be computationally intensive. To this end, hybrid models using integrated nested Laplace approximations (INLA) were recently proposed to approximate the posterior marginals for latent Gaussian models, as they have shown a steep decline in the computational burden of MCMC algorithms. A third class of methodology finds its origins in the Structural Equation Modelling (SEM) framework, where a limited-information diagonally weighted least squares (DWLS) estimation procedure has been suggested. In this presentation, we focus on the analysis of binary clustered data with small cluster sizes, since this setting is especially known for posing a challenge to the available GLMM methods. Such data structures may for example arise from crossover studies or dyadic studies with binary outcomes. With this in mind, our intent is to explore the performance of the above-proposed methods as they are available in the statistical computing environment R, in this particular setting. More specifically, we will consider the following functions within their respective R-packages: glmer from lme4 (Laplace, AGQ), glmmPQL from MASS (PQL), MCMCglmm from MCMCglmm (MCMC), inla from R-inla (hybrid), and sem from lavaan (DWLS). Since the performances of many of these methods have but been assessed by themselves or within their classes of methodology, an over-arching comparison through simulation studies will be presented here. The above-mentioned approaches will be compared in terms of bias, mean squared error and coverage. These criteria will be reviewed by monitoring different sample sizes (with fixed cluster size), different intra-cluster correlations, dichotomous versus continuous predictors, within-cluster and between-cluster predictors with varying effect sizes, and different event rates

Centering lower-level interactions in multilevel models

In hierarchical designs, the effect of a lower level predictor on an outcome may oftentimes be confounded by an (un)measured upper level variable. When such confounding is left unaddressed, the effect of the lower level predictor will be estimated with bias. As to remove any such bias in a linear random intercept model, researchers often separate the lower level effect into a within- and between-component (under a specific set of confounding-assumptions). When the effect of the lower level predictor is additionally moderated by another lower level predictor, an interaction between both predictors needs to be included into the model. To again address any possible unmeasured upper level confounding, this interaction term also requires partitioning into a within- and between- cluster component. This can be achieved by first multiplying both predictors and to consequently centering that product term, or vice versa. We demonstrate that the former centering approach proves much more efficient and robust against misspecification of cross- and upper-level effects, compared to the latter

Mediation analysis in AB/BA crossover studies

Crossover trials are widely used in psychological and medical research to assess the effect of reversible exposures. In such designs, each subject is randomly allocated to a sequence of conditions, enabling the evaluation of treatment differences within each individual. When there are but two possible exposures -each assessed during one of two time periods-, the crossover study is referred to as an AB/BA design. The goal of this presentation is to discuss mediation analysis in such simple crossover studies. We do so by considering within-subject mediation from a counterfactual-based perspective and by deriving expressions for the direct and indirect effects. Employing simulation studies, the performance of several existing methods will be assessed and compared to that of a novel one we propose. We show that the new method yields unbiased and efficient estimators for the direct and indirect effect, under a minimalistic set of `no unmeasured confounding'-assumptions. Finally, we illustrate the different techniques with data from a neurobehavioral study

Preschool predictors of mathematics in first grade children with autism spectrum disorder

AbstractUp till now, research evidence on the mathematical abilities of children with autism spectrum disorder (ASD) has been scarce and provided mixed results. The current study examined the predictive value of five early numerical competencies for four domains of mathematics in first grade. Thirty-three high-functioning children with ASD were followed up from preschool to first grade and compared with 54 typically developing children, as well as with normed samples in first grade. Five early numerical competencies were tested in preschool (5–6 years): verbal subitizing, counting, magnitude comparison, estimation, and arithmetic operations. Four domains of mathematics were used as outcome variables in first grade (6–7 years): procedural calculation, number fact retrieval, word/language problems, and time-related competences. Children with ASD showed similar early numerical competencies at preschool age as typically developing children. Moreover, they scored average on number fact retrieval and time-related competences and higher on procedural calculation and word/language problems compared to the normed population in first grade. When predicting first grade mathematics performance in children with ASD, both verbal subitizing and counting seemed to be important to evaluate at preschool age. Verbal subitizing had a higher predictive value in children with ASD than in typically developing children. Whereas verbal subitizing was predictive for procedural calculation, number fact retrieval, and word/language problems, counting was predictive for procedural calculation and, to a lesser extent, number fact retrieval. Implications and directions for future research are discussed

A quantum-like model for complementarity of preferences and beliefs in dilemma games

We propose a formal model to explain the mutual influence between observed behavior and subjects' elicited beliefs in an experimental sequential prisoner's dilemma. Three channels of interaction can be identified in the data set and we argue that two of these effects have a non-classical nature as shown, for example, by a violation of the sure thing principle. Our model explains the three effects by assuming preferences and beliefs in the game to be complementary. We employ non-orthogonal subspaces of beliefs in line with the literature on positive-operator valued measure. Statistical fit of the model reveals successful predictions

Centering of interactions in lower-level mediation models

When considering multilevel mediation, centring is often applied to lower-level variables. One well-established approach for this separates the lower-level variables into a W(ithin)- and B(etween)-cluster component in linear settings, as to effectively eliminate additive upper level confounding of the mediator M and the outcome Y. When moderated mediation is considered, however, careful thought is needed about the method of centring; partitioning the interaction can be achieved in two ways: multiply the main effects that make up the interaction first, and apply centring within clusters next, or the other way around. Alternatively, M and Y can also be modelled jointly, hereby also allowing for unmeasured additive upper M-Y confounding, but at the same time avoiding any necessity for centring of both the main and the interaction effects. Employing simulations, we study the performance of these three approaches in the presence of interactions under varying data generating mechanisms, and discuss the relative merits of each approach