Performance of likelihood-based estimation methods for multilevel binary regression models.

By means of a fractional factorial simulation experiment, we. compare the performance of penalised quasi-likelihood (PQL), non-adaptive Gaussian quadrature and adaptive Gaussian quadrature in estimating parameters for multilevel logistic regression models. The comparison is done in terms of bias, mean-squared error (MSE), numerical convergence and computational efficiency. It turns out that in terms of MSE, standard versions of the quadrature methods per-form relatively poorly in comparison with PQL.Bias; Binary regression; Convergence; Efficiency; Factorial; Fractional factorial experiment; Gaussian quadrature; Logistic regression; Methods; Model; Models; Monte Carlo simulation; Multilevel analysis; Parameters; Penalised quasi-likelihood; Performance; Regression; Simulation;

Evaluation of advanced optimisation methods for estimating Mixed Logit models

The performances of different simulation-based estimation techniques for mixed logit modeling are evaluated. A quasi-Monte Carlo method (modified Latin hypercube sampling) is compared with a Monte Carlo algorithm with dynamic accuracy. The classic Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization algorithm line-search approach and trust region methods, which have proved to be extremely powerful in nonlinear programming, are also compared. Numerical tests are performed on two real data sets: stated preference data for parking type collected in the United Kingdom, and revealed preference data for mode choice collected as part of a German travel diary survey. Several criteria are used to evaluate the approximation quality of the log likelihood function and the accuracy of the results and the associated estimation runtime. Results suggest that the trust region approach outperforms the BFGS approach and that Monte Carlo methods remain competitive with quasi-Monte Carlo methods in high-dimensional problems, especially when an adaptive optimization algorithm is used

Performance of likelihood-based estimation methods for multilevel binary regression models.

By means of a fractional factorial simulation experiment, we compare the performance of Penalised Quasi-Likelihood, Non-Adaptive Gaussian Quadrature and Adaptive Gaussian Quadrature in estimating parameters for multi-level logistic regression models. The comparison is done in terms of bias, mean squared error, numerical convergence, and computational efficiency. It turns out that, in terms of Mean Squared Error, standard versions of the Quadrature methods perform relatively poor in comparison with Penalized Quasi-Likelihood.Bias; Binary regression; Convergence; Efficiency; Factorial; Fractional factorial experiment; Gaussian quadrature; Logistic regression; Methods; Model; Models; Monte Carlo simulation; Multilevel analysis; Penalised quasi-likelihood; Performance; Regression; Simulation;

An efficient algorithm for constructing Bayesian optimal choice designs.

Recently, Kessels et al. (2006) developed a way to produce Bayesian G- and V-optimal designs for the multinomial logitmodel. These designs allow for precise response predictions which is the goal of conjoint choice experiments. The authors showed that the G- and V- optimality criteria outperform the D- and A-optimality criteria for prediction. However, their G- and V-optimal design algorithm is computationally intensive, which is a barrier to its use in practice. In this paper, we present an efficient algorithm for calculating Bayesian optimal designs by means of the different criteria. Particularly, the speed of computation for the V-optimality criterion has improved dramatically.The new algorithm makes it possible to use Bayesian D-, A-, G- and V-optimal designs that are tailored to individual respondents in computerized conjoint choice studies.Adaptive algorithm; Bayesian D-,A-,G- and V-optimality; Cholesky decomposition; Conjoint choice design; Coordinate-exchange; Conjoint choice experiments;

Efficient conjoint choice designs in the presence of respondent heterogeneity.

The authors propose a fast and efficient algorithm for constructing D-optimal conjoint choice designs for mixed logit models in the presence of respondent heterogeneity. With this new algorithm, the construction of semi-Bayesian D-optimal mixed logit designs with large numbers of attributes and attribute levels becomes practically feasible. The results from the comparison of eight designs (ranging from the simple locally D-optimal design for the multinomial logit model and the nearly orthogonal design generated by Sawtooth (CBC) to the complex semi-Bayesian mixed logit design) across wide ranges of parameter values show that the semi-Bayesian mixed logit approach outperforms the competing designs not only in terms of estimation efficiency but also in terms of prediction accuracy. In particular, it was found that semi-Bayesian mixed logit designs constructed with large heterogeneity parameters are most robust against the misspecification of the values for the mean of the individual level coefficients for making precise estimations and predictions.Keywords:semi-Bayesianmixedlogitdesign,heterogeneity,predictionaccuracy,multinomiallogitdesign,model-robustdesign,D-optimality,algorithmAlgorithm; D-Optimality; Heterogeneity; Model-robust design; Multinomial logit design; Prediction accuracy; Semi-Bayesian mixed logit design;

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

Variational Bayes with Intractable Likelihood

Variational Bayes (VB) is rapidly becoming a popular tool for Bayesian inference in statistical modeling. However, the existing VB algorithms are restricted to cases where the likelihood is tractable, which precludes the use of VB in many interesting situations such as in state space models and in approximate Bayesian computation (ABC), where application of VB methods was previously impossible. This paper extends the scope of application of VB to cases where the likelihood is intractable, but can be estimated unbiasedly. The proposed VB method therefore makes it possible to carry out Bayesian inference in many statistical applications, including state space models and ABC. The method is generic in the sense that it can be applied to almost all statistical models without requiring too much model-based derivation, which is a drawback of many existing VB algorithms. We also show how the proposed method can be used to obtain highly accurate VB approximations of marginal posterior distributions.Comment: 40 pages, 6 figure