Regularization and Model Selection in Mixed Models
Presenter: Francis Hui
When: Thursday, July 14, 2016 Time: 4:30 PM - 6:00 PM
Room: Salon A Carson Hall (Level 2)
Session Synopsis:
Joint selection in mixed models using regularized PQL
Mixed models, particularly GLMMs, are a standard technique to use in many areas of applied statistics when analyzing longitudinal data and data from stratified or multilevel designs. When GLMMs were first popularized, penalised quasi-likelihood (PQL) was very much the go-to method of estimation given its ease of implementation and speed. Today however, with increasingly powerful computation, PQL has largely been replaced by more accurate and formal maximum likelihood methods, e.g. adaptive quadrature and MCMC. This talk focuses on how PQL is still an important method in modern statistics, by combining it with penalized likelihood methods for simultaneous selection of fixed and random effects in GLMMs. Since PQL treats the random effects coefficients as ``fixed�, a penalty can be chosen to exploit their inherent group structure. Furthermore, a hybrid information criterion for selecting the tuning parameter is proposed, which combines differing model complexity penalties for the fixed and random effects. Both theoretical and computational aspects of regularized PQL are discussed, including selection consistency, with simulations and an application to forest health monitoring presented, illustrating their strong performance.
Regularization and Model Selection in Mixed Models
Presenter: Howard Bondell
When: Thursday, July 14, 2016 Time: 4:30 PM - 6:00 PM
Room: Salon A Carson Hall (Level 2)
Session Synopsis:
Fully Efficient and Outlier-Robust Estimation in the Linear Model
The triumvirate of outlier resistance, distributional robustness, and efficiency in both small and large samples, constitute the Holy Grail of robust statistics. We show that a two-stage procedure based on an initial robust estimate of scale followed by an application of generalized empirical likelihood comes close to attaining that goal. For the linear model, the resulting estimators are able to attain full asymptotic efficiency at the Normal distribution, while simulations point to the ability to maintain this efficiency down to small sample sizes. Additionally, the estimators are shown to have the maximum attainable finite-sample replacement breakdown point, and thus remain stable in the presence of heavy-tailed distributions and outliers. Although previous proposals with full asymptotic efficiency exist in the literature, their finite sample efficiency can often be low. The method is discussed in detail for linear regression and extended to the mixed model.
Regularization and Model Selection in Mixed Models
Presenter: Tonya Garcia
When: Thursday, July 14, 2016 Time: 4:30 PM - 6:00 PM
Room: Salon A Carson Hall (Level 2)
Session Synopsis:
Modeling the Cholesky Factors of Covariance Matrices of Multivariate Longitudinal Data
Modeling the covariance matrix of multivariate longitudinal data is more challenging as compared to its univariate counterpart due to the presence of correlations among multiple responses. The modified Cholesky block decomposition reduces the task of covariance modeling into parsimonious modeling of its two matrix factors: the regression coefficient matrices and the innovation covariance matrices. These parameters are statistically interpretable, however ensuring positive-definiteness of several (innovation) covariance matrices presents itself as a new challenge. We address this problem using a subclass of Anderson's (1973) linear covariance models and model several covariance matrices using linear combinations of known positive-definite basis matrices with unknown non-negative scalar coefficients. A novelty of this approach is that positive-definiteness is guaranteed by construction; it removes a drawback of Anderson's model and hence makes linear covariance models more realistic and viable in practice. Maximum likelihood estimates are computed using a simple iterative majorization-minimization algorithm. The estimators are shown to be asymptotically normal and consistent. Simulation and a data example illustrate the applicability of the proposed method in providing good models for the covariance structure of a multivariate longitudinal data.
