Cholesky Residuals for Assessing Normal Errors in a Linear Model with Correlated Outcomes: Technical Report

Despite the widespread popularity of linear models for correlated outcomes (e.g. linear mixed models and time series models), distribution diagnostic methodology remains relatively underdeveloped in this context. In this paper we present an easy-to-implement approach that lends itself to graphical displays of model fit. Our approach involves multiplying the estimated margional residual vector by the Cholesky decomposition of the inverse of the estimated margional variance matrix. The resulting "rotated" residuals are used to construct an empirical cumulative distribution function and pointwise standard errors. The theoretical framework, including conditions and asymptotic properties, involves technical details that are motivated by Lange and Ryan (1989), Pierce (1982), and Randles (1982). Our method appears to work well in a variety of circumstances, including models having independent units of sampling (clustered data) and models for which all observations are correlated (e.g., a single time series). Our methods can produce satisfactory results even for models that do not satisfy all of the technical conditions stated in our theory.

Underestimation of Standard Errors in Multi-Site Time Series Studies

Multi-site time series studies of air pollution and mortality and morbidity have figured prominently in the literature as comprehensive approaches for estimating acute effects of air pollution on health. Hierarchical models are generally used to combine site-specific information and estimate pooled air pollution effects taking into account both within-site statistical uncertainty, and across-site heterogeneity. Within a site, characteristics of time series data of air pollution and health (small pollution effects, missing data, highly correlated predictors, non linear confounding etc.) make modelling all sources of uncertainty challenging. One potential consequence is underestimation of the statistical variance of the site-specific effects to be combined. In this paper we investigate the impact of variance underestimation on the pooled relative rate estimate. We focus on two-stage normal-normal hierarchical models and on under- estimation of the statistical variance at the first stage. By mathematical considerations and simulation studies, we found that variance underestimation does not affect the pooled estimate substantially. However, some sensitivity of the pooled estimate to variance underestimation is observed when the number of sites is small and underestimation is severe. These simulation results are applicable to any two-stage normal-normal hierarchical model for combining information of site-specific results, and they can be easily extended to more general hierarchical formulations. We also examined the impact of variance underestimation on the national average relative rate estimate from the National Morbidity Mortality Air Pollution Study and we found that variance underestimation as much as 40% has little effect on the national average.

Fixed-Width Output Analysis for Markov Chain Monte Carlo

Markov chain Monte Carlo is a method of producing a correlated sample in order to estimate features of a complicated target distribution via simple ergodic averages. A fundamental question in MCMC applications is when should the sampling stop? That is, when are the ergodic averages good estimates of the desired quantities? We consider a method that stops the MCMC sampling the first time the width of a confidence interval based on the ergodic averages is less than a user-specified value. Hence calculating Monte Carlo standard errors is a critical step in assessing the output of the simulation. In particular, we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We describe sufficient conditions for the strong consistency and asymptotic normality of both methods and investigate their finite sample properties in a variety of examples.