Research in Statistics

University Of California-Davis, Davis CA

Investigators

Abstract

In this project, the principle investigator (PI) proposes research on four important topics aimed to develop methods of statistical inference and model diagnostics in cases of correlated observations. These topics are: (i) partially observed information and inference about non-Gaussian linear mixed models; (ii) iterative weighted least squares procedures for analysis of longitudinal data; (iii) A test of global maximum for dependent observations; and (iv) generalized linear mixed model diagnostics. Linear mixed models are widely used in practice for correlated observations. A typical assumption regarding these models is that the observations are normally distributed. However, the normality assumption is likely to be violated in practice. It is known that normality based methods such as REML still produce consistent estimators even if normality fails. However, the estimation of standard errors of these estimators is complicated, because for non-Gaussian data the asymptotic covariance matrix of the Gaussian estimator involves additional unknown parameters. Project (i) aims to completely solve this long-standing problem of practical interest. Furthermore, project (ii) proposes an iterative weighted least squares procedure for computing efficient estimators of the regression coefficients in linear models for longitudinal data analysis and studies its properties. Project (iii) aims to extend a method developed in the i.i.d. case for checking whether a root to the likelihood equation corresponds to the global maximum of the likelihood function. Project (iv) develops goodness-of-fit tests and methods of informal model checking for generalized linear mixed models. Correlated responses are often encountered in practice. For example, in medical studies repeated measures are often collected from the same individuals over time. It would be reasonable to assume that correlations exist among the observations from the same individual. Linear and generalized linear mixed models are two important classes of statistical models widely used in cases of correlated observations. For linear mixed models methods of inference have been developed, but mostly under the assumption that the observations are normal (or Gaussian). However, the normality assumption is likely to be violated in practice. The PI aims to develop a powerful method for inference about non-Gaussian linear mixed models. For generalized linear mixed models, the PI proposes to develop methods of model checking that fills an important gap in the applications of these models. The methods developed are likely to have impact in other fields of statistics as well as in fields such as biomedical research, genetics, biology, economics, education and social science.