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Conventional geostatistical methodology solves the problem of predicting the realized value of a linear functional of a Gaussian spatial stochastic process Sx) based on observations Y i Sx i Z i at sampling locations x i , where the Z i are mutually independent, zero-mean Gaussian random variables. We describe two spatial applications for which Gaussian distributional assumptions are clearly inappropriate. The ®rst concerns the assessment of residual contamination from nuclear weapons testing on a South Paci®c island, in which the sampling method generates spatially indexed Poisson counts conditional on an unobserved spatially varying intensity of radioactivity; we conclude that a conventional geostatistical analysis oversmooths the data and underestimates the spatial extremes of the intensity. The second application provides a description of spatial variation in the risk of campylobacter infections relative to other enteric infections in part of north Lancashire and south Cumbria. For this application, we treat the data as binomial counts at unit postcode locations, conditionally on an unobserved relative risk surface which we estimate. The theoretical framework for our extension of geostatistical methods is that, conditionally on the unobserved process Sx, observations at sample locations x i form a generalized linear model with the corresponding values of Sx i appearing as an offset term in the linear predictor. We use a Bayesian inferential framework, implemented via the Markov chain Monte Carlo method, to solve the prediction problem for non-linear functionals of Sx, making a proper allowance for the uncertainty in the estimation of any model parameters.

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SUMMARY A model is proposed for continuous longitudinal data with non‐ignorable or informative drop‐out (ID). The model combines a multivariate linear model for the underlying response with a logistic regression model for the drop‐out process. The latter incorporates dependence of the probability of drop‐out on unobserved, or missing, observations. Parameters in the model are estimated by using maximum likelihood (ML) and inferences drawn through conventional likelihood procedures. In particular, likelihood ratio tests can be used to assess the informativeness of the drop‐out process through comparison of the full model with reduced models corresponding to random drop‐out (RD) and completely random processes. A simulation study is used to assess the procedure in two settings: the comparison of time trends under a linear regression model with autocorrelated errors and the estimation of period means and treatment differences from a four‐period four‐treatment crossover trial. It is seen in both settings that, when data are generated under an ID process, the ML estimators from the ID model do not suffer from the bias that is present in the ordinary least squares and RD ML estimators. The approach is then applied to three examples. These derive from a milk protein trial involving three groups of cows, milk yield data from a study of mastitis in dairy cattle and data from a multicentre clinical trial on the study of depression. All three examples provide evidence of an underlying ID process, two with some strength. It is seen that the assumption of an ID rather than an RD process has practical implications for the interpretation of the data.

This paper formulates a class of models for the joint behaviour of a sequence of longitudinal measurements and an associated sequence of event times, including single-event survival data. This class includes and extends a number of specific models which have been proposed recently, and, in the absence of association, reduces to separate models for the measurements and events based, respectively, on a normal linear model with correlated errors and a semi-parametric proportional hazards or intensity model with frailty. Special cases of the model class are discussed in detail and an estimation procedure which allows the two components to be linked through a latent stochastic process is described. Methods are illustrated using results from a clinical trial into the treatment of schizophrenia.