Inference and Computational Methods for Regression Modelling in Multiple Time Series of Counts with Random Effects

Abstract: 

This talk is concerned with modelling multiple time series of counts in which shared regression effects need to be tested and in which there are random effects and serial dependence. Let Yjt be the observation at time t = 1, . . . , n on the jth time series where j = 1, . . . J. Typically n is much larger than J in the settings we focus on. Health data applications used to motivate this work have values of these ranging from n = 72 to 1500 and J = 5, 10, 50 . For each j assume that, given the state process {Wjt}, the Yjt are independent with exponential family distribution f(yjt|{Wjt}) = exp {[yjtWjt − b(Wjt)] /ajt(_) + c(yjt)}. The state process is assumed to be of the form Wjt = rT t Uj +xT0 ,t_(0) +xT j,t_(j) +_jt in which there are q random effects U _N(0,_U) where _U is a q × q covariance matrix with associated covariates rt = (r1t, . . . , rqt)T , there are fixed effect covariates x0,t common to each individual series and particular covariates xj,t relevant to the j-th series and that the process _jt either follows a transition process _jt = P1 l=1 (j) l (_ (j))ej,t−l with ej,s = (yj,s − μj,s) /s(μj,s) for some normalizing scale function s(·) or is an unobserved stationary Gaussian time series with parameters _j . Methods of inference based on the likelihood will be considered. The primary focus will be on the transition specification of _jt. In this case the likelihood over all series can be readily built up using existing software of the author for fitting transition models to single time series. The talk will describe how this is done and illustrate the methods on an example of assessing the impact, on monthly counts of single vehicle night time fatalities, of lowering the legal blood alcohol level in 17 US states. Estimation for the latent process specification _jt is computationally more difficult requiring as it does approximation of large (n) dimensional integrals. Available computational methods for this specification will be discussed. For either specification of _jt theory of inference is underdeveloped at this stage and what is known will be reviewed. 

About the speaker: William Dunsmuir is Professor at the Department of Statistics, UNSW. He is interested in time series and in longitudinal data analysis. The focus of his research is on models and methods for time series analysis of counts with applications to health outcomes.