Estimation in mixed Poisson regression models

This thesis considers estimation of the parameters associated with models for count data displaying over-dispersion relative to the Poisson distribution where the over-dispersion is modelled using mixing. It is divided into seven chapters with Chapters Two to Five specific to the over-dispersed Poisson problem whilst Chapter Seven, which uses results from Chapter Six, is more general. The motivation for some of this work was the modelling of repeat counts of the number of fibres contained on microscopic slides as obtained by asbestos fibre counters and the subsequent estimation of mean fibre concentrations and counter variability. Chapter One introduces the above mentioned asbestos fibre problem and follows this with an overview of the thesis. In Chapter Two a model for repeated measures count data over-dispersed relative to the Poisson distribution appropriate to the asbestos problem is given. To accommodate the over-dispersion a Poisson random variable is compounded with a positive random variable with mean equal one and variance linked linearly, via a log function, to a set of covariates. Maximum likelihood estimators of the parameters are obtained for the case where the compounding distribution is gamma and extended quasi-likelihood parameter estimators are obtained when the compounding distribution is unspecified. These two sets of parameter estimators are then shown to be comparable in certain circumstances. In Chapter Three a special case of the general model in Chapter Two with a gamma compounding distribution is considered. Here repeat counts for a “subject” are taken as independent Poisson random variables with constant mean. The means are then modelled as independent observations from a gamma distribution. Two sets of moment estimators for the parameters of the model are obtained and generalized variances of the limiting distribution of the moment estimators are compared with the corresponding quantity for the maximum likelihood estimators. Also in this chapter we derive asymptotic results that explain some of the erratic behaviour of the moment estimators. Chapter Four considers the estimation of the shape parameter of the negative binomial distribution (NBD), this distribution being a special case of the model in Chapter Three. Here the results are given for a simulation study comparing four estimators for the shape parameter of the NBD distribution. Two criteria are used to compare the estimates obtained in the simulations, one being the traditional moment based criterion whilst the other is based on a new measure termed the “percentile measure”. This measure, based on the difference between the percentiles of the true and estimated distribution function, is argued to be more appropriate in many cases. In Chapter Five we continue studying the NBD and obtain some quantile related results. First we obtain bounds for the median in terms of the mean that are improvements on the bounds obtained by Payton, Young and Young (1989). Second we obtain percentile related bounds for the mean and use this to obtain a robust estimator for the mean of the NBD when the shape parameter is known. The remaining two chapters are devoted to robust estimation in (generalized) linear mixed models. In Chapter Six a modification to the Fellner (1991) procedure for robustly estimating variance components in normal linear mixed models is proposed and studied. Also given is a robust moment based method. These robust methods are then applied in Chapter Seven to the generalized linear mixed model to obtain robust parameter estimators and the behaviour of these new estimators is studied via a simulation study. From this simulation study in Chapter Seven it is concluded that the extension to generalized linear mixed models of the modification to Fellner’s method has merit. There should though be scope for improvement in the method and this could be a subject for further research. In particular, a possible mechanism for achieving an improvement would be to have more robust starting values for the variance components in the iterative procedure proposed. One solution would be to develop quantile based variance components estimates in the generalized linear mixed model and to use these as starting values. Another project for further research would be to obtain expressions for the variances of the fixed effects estimates for the linear mixed model obtained using the Fellner (1991) method. This would necessarily be an asymptotic result and of interest in its own right. However, once this was available the modified Fellner method of Chapter Six and its extension to generalized linear mixed models given in Chapter Seven could be improved. This is the case as the modification to Fellner’s method given in Chapter Six currently uses for these values the variances of the BLUP estimates of the fixed effects. Finally, an alternative robust estimation procedure based on the results of Windham (1995) could be another subject for further research. That paper, which assumes the data are independent and identically distributed (iid), proposes an estimation procedure that weights datum according to the value of the estimated likelihood at that datum point. The procedure seems to have merit based on the examples considered in Windham’s paper, which include some skewed distributions (e.g. gamma). Further research could consider extending the results to the non it'd case, in particular data from generalized linear mixed model problems. It may be possible using such a procedure in generalized linear mixed model cases to reduce some of the bias that is inherent in procedures based on Winsorization

Sheep Updates 2006 - part 2

This session covers six papers from different authors: GENETICS 1. Novel selection traits - what are the possible side effects?, Darryl Smith, Kathryn Kemper, South Australian Research and Development Institute, David Rutley, University of Adelaide. 2. Genetic Changes in the Australian Merino since 1900, Sheep Genetics Australia Technical Committee, R.R. Woolaston Pullenvale, Queensland, D.J. Brown, Animal Genetics and Breeding Unit*, University of New England, K.D. Atkins, A.E. Casey, NSW Department of Primary Industries, A.J. Ball, Meat and Livestock Australia, University of New England 3. Influence of Sire Growth Estimated Breeding Value (EBV0 on Progeny Growth, David Hopkins, David Stanley, Leonie Martin, NSW Department Primary Industries, Centre for Sheep Meat Development, Arthur Gilmour, Remy van de Ven, NSW Department Primary Industries, Orange Agricultural Institute FINISHING 4. Predicting Input Sensitivity on Lamb Feedlot Profitability by Using Feedlot Calculator, David Stanley, NSW Department Primary Industries, Centre for Sheep Meat Development, Geoff Duddy, NSW Department Primary Industries, Yanco Agricultural Institute, Steve Semple, NSW Department Primary Industries, Orange Agricultural Institute, David Hopkins, NSW Department Primary Industries, Centre for Sheep Meat Development 5. Annual ryegrass toxicity (ARGT) in WA - 2006, David Kessell, Meat & Livestock Australia ARGT Project, Northam, WA 6. Poor ewe nutrition during pregnancy increases fatness of their progeny, Andrew Thompson, Department of Primary Industries, Victori

HST Observations of Gravitationally Lensed QSOs

Thanks to its sharp view, HST has significantly improved our knowledge of tens of gravitationally lensed quasars in four different respects: (1) confirming their lensed nature; (2) detecting the lensing galaxy responsible for the image splitting; (3) improving the astrometric accuracy on the positions of the unresolved QSO images and of the lens; (4) resolving extended lensed structures from the QSO hosts into faint NIR or optical rings or arcs. These observations have helped to break some degeneracies on the lens potential, to probe the galaxy evolution and to reconstruct the true shape of the QSO host with an increased angular resolution