Nonparametric estimation of mean-squared prediction error in nested-error regression models

Nested-error regression models are widely used for analyzing clustered data. For example, they are often applied to two-stage sample surveys, and in biology and econometrics. Prediction is usually the main goal of such analyses, and mean-squared prediction error is the main way in which prediction performance is measured. In this paper we suggest a new approach to estimating mean-squared prediction error. We introduce a matched-moment, double-bootstrap algorithm, enabling the notorious underestimation of the naive mean-squared error estimator to be substantially reduced. Our approach does not require specific assumptions about the distributions of errors. Additionally, it is simple and easy to apply. This is achieved through using Monte Carlo simulation to implicitly develop formulae which, in a more conventional approach, would be derived laboriously by mathematical arguments.Comment: Published at http://dx.doi.org/10.1214/009053606000000579 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian Covariance Structure Modeling of Multi-Way Nested Data

A Bayesian multivariate model with a structured covariance matrix for multi-way nested data is proposed. This flexible modeling framework allows for positive and for negative associations among clustered observations, and generalizes the well-known dependence structure implied by random effects. A conjugate shifted-inverse gamma prior is proposed for the covariance parameters which ensures that the covariance matrix remains positive definite under posterior analysis. A numerically efficient Gibbs sampling procedure is defined for balanced nested designs, and is validated using two simulation studies. For a top-layer unbalanced nested design, the procedure requires an additional data augmentation step. The proposed data augmentation procedure facilitates sampling latent variables from (truncated) univariate normal distributions, and avoids numerical computation of the inverse of the structured covariance matrix. The Bayesian multivariate (linear transformation) model is applied to two-way nested interval-censored event times to analyze differences in adverse events between three groups of patients, who were randomly allocated to treatment with different stents (BIO-RESORT). The parameters of the structured covariance matrix represent unobserved heterogeneity in treatment effects and are examined to detect differential treatment effects.Comment: 30 pages, 5 figures, 4 table

Partially observed information and inference about non-Gaussian mixed linear models

In mixed linear models with nonnormal data, the Gaussian Fisher information matrix is called a quasi-information matrix (QUIM). The QUIM plays an important role in evaluating the asymptotic covariance matrix of the estimators of the model parameters, including the variance components. Traditionally, there are two ways to estimate the information matrix: the estimated information matrix and the observed one. Because the analytic form of the QUIM involves parameters other than the variance components, for example, the third and fourth moments of the random effects, the estimated QUIM is not available. On the other hand, because of the dependence and nonnormality of the data, the observed QUIM is inconsistent. We propose an estimator of the QUIM that consists partially of an observed form and partially of an estimated one. We show that this estimator is consistent and computationally very easy to operate. The method is used to derive large sample tests of statistical hypotheses that involve the variance components in a non-Gaussian mixed linear model. Finite sample performance of the test is studied by simulations and compared with the delete-group jackknife method that applies to a special case of non-Gaussian mixed linear models.Comment: Published at http://dx.doi.org/10.1214/009053605000000543 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org