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**ABSTRACT:**Several Monte Carlo methods have been proposed for computing marginal likelihoods in Bayesian analyses. Some of these involve sampling from a sequence of intermediate distributions between the prior and posterior. A difficulty arises if the support in the posterior distribution is a proper subset of that in the prior distribution. This can happen in problems involving latent variables whose support depends upon the data and can make some methods inefficient and others invalid. The correction required for models of this type is derived and its use is illustrated by finding the marginal likelihoods in two examples. One concerns a model for competing risks. The other involves a zero-inflated over-dispersed Poisson model for counts of centipedes, using latent Gaussian variables to capture spatial dependence.Computational Statistics & Data Analysis 03/2014; 71:392–401. DOI:10.1016/j.csda.2013.07.033 - [Show abstract] [Hide abstract]

**ABSTRACT:**Crossover clinical trials can provide substantial benefits by eliminating inter-patient variation from treatment comparisons and by allowing multiple observations of each patient. They are particularly useful when sample sizes are necessarily small. These advantages proved particularly valuable in an assessment of clot prevention in children undergoing haemodialysis. Only small numbers of children are treated at any given time in any single dialysis unit, but each patient is obliged to attend two or three times each week, suggesting the use of a crossover trial with many periods. Standard crossover trials described in the literature (i) typically have fewer than 10 periods and (ii) are based on a model of questionable applicability to this study. This paper describes the derivation of an optimal crossover trial with 30 periods, which was used to compare two anticoagulants using nine patients. There is also a discussion of the analysis of the data obtained in the trial, which had a distribution markedly different from that anticipated when the study was designed. Copyright © 2013 John Wiley & Sons, Ltd.Statistics in Medicine 02/2014; 33(5). DOI:10.1002/sim.5981 - [Show abstract] [Hide abstract]

**ABSTRACT:**A growing realization of the importance of stochasticity in cell and molecular processes has stimulated the need for statistical models that incorporate intrinsic (and extrinsic) variability. In this chapter we consider stochastic kinetic models of reaction networks leading to a Markov jump process representation of a system of interest. Traditionally, the stochastic model is characterized by a chemical master equation. While the intractability of such models can preclude a direct analysis, simulation can be straightforward and may present the only practical approach to gaining insight into a system's dynamics. We review exact simulation procedures before considering some efficient approximate alternatives.Methods in molecular biology (Clifton, N.J.) 05/2013; 1021:169-187. DOI:10.1007/978-1-62703-450-0_9 - [Show abstract] [Hide abstract]

**ABSTRACT:**In common with most forms of designed experiment, crossover trials can be affected by missing data. Attempts to devise designs that can mitigate the possible effects of missing data, such as loss of efficiency, or even inestimability of certain contrasts, have been proposed. However, a potentially serious effect of missing data that has not been addressed in designs hitherto is that the treatment effects may be biassed because of the nature of the missingness process. We investigate this problem in two-treatment, two-period crossover designs. In particular, we consider the robustness of the analysis under a missing at random assumption when, in fact, the data are non-ignorably missing. We show that the conventional AB/BA design still has good properties, although the design with sequences AB, BA, AA, and BB may be preferred if the chance of dropout depends primarily on the difference between the responses in the two periods.Biostatistics 03/2013; 14(4). DOI:10.1093/biostatistics/kxt009 - [Show abstract] [Hide abstract]

**ABSTRACT:**Localized states are found in many pattern forming systems. The aim of this paper is to investigate the occurrence of oscillatory localized states in two-dimensional Boussinesq magnetoconvection. Initially considering an idealized model, in which the vertical structure of the system has been simplified by a projection onto a small number of Fourier modes, we find that these states are restricted to the low ζ regime (where ζ represents the ratio of the magnetic to thermal diffusivities). These states always exhibit bistability with another nontrivial solution branch; in other words, they show no evidence of subcritical behavior. This is due to the weak flux expulsion that is exhibited by these time-dependent solutions. Using the results of this parameter survey, we locate corresponding states in a fully resolved two-dimensional system, although the mode of oscillation is more complex in this case. This is the first time that a localized oscillatory state, of this kind, has been found in a fully resolved magnetoconvection simulation.Physical Review E 02/2013; 87(2-1):023019. DOI:10.1103/PhysRevE.87.023019 - [Show abstract] [Hide abstract]

**ABSTRACT:**Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations.Physical Review Letters 11/2012; 109(20):205304. DOI:10.1103/PhysRevLett.109.205304 - [Show abstract] [Hide abstract]

**ABSTRACT:**Missing data arise in crossover trials, as they do in any form of clinical trial. Several papers have addressed the problems that missing data create, although almost all of these assume that the probability that a planned observation is missing does not depend on the value that would have been observed; that is, the data are missing at random (MAR). In many applications, this assumption is likely to be untenable; in which case, the data are missing not at random (MNAR). We investigate the effect on estimates of the treatment effect that assume data are MAR when data are actually MNAR. We also propose using the assumption of no carryover treatment effect, which is usually required for this design, to permit the estimation of a treatment effect when data are MNAR. The results are applied to a trial comparing two treatments for neuropathic pain and show that the estimate of treatment effect is sensitive to the assumption of MAR.Statistics in Medicine 07/2012; 31(16):1675-87. DOI:10.1002/sim.4497 - [Show abstract] [Hide abstract]

**ABSTRACT:**We discuss inference for longitudinal clinical trials subject to possibly informative dropout. A selection of available methods is reviewed for the simple case of trials with two timepoints. Using data from two such clinical trials, each with two treatments, we demonstrate that different analysis methods can at times lead to quite different conclusions from the same data. We investigate properties of complete-case estimators for the type of trials considered, with emphasis on interpretation and meaning of parameters. We contrast longitudinal and crossover designs and argue that for crossover studies there are often good reasons to prefer a complete case analysis. More generally, we suggest that there is merit in an approach in which no untestable assumptions are made. Such an approach would combine a dropout analysis, an analysis of complete-case data only, and a careful statement of justified conclusions.Statistical Methods in Medical Research 04/2012; 23(1). DOI:10.1177/0962280212445838 - [Show abstract] [Hide abstract]

**ABSTRACT:**A theory is presented which provides a model for the appearance of critical layers within the flow below a water wave. The wave propagates over constant depth, with constant (non-zero) vorticity. The mechanism described here involves adjusting the surface-pressure boundary condition; two models are discussed. In the first, the pressure at the surface is controlled (mimicking the movement of a low-pressure region associated with a storm) so that the speed and development of the pressure region ensure the appearance of a critical layer. In the second, the pressure boundary condition is allowed to accommodate the reduction of pressure with altitude, although the effects have to be greatly enhanced for this mechanism to produce a critical layer. These two problems are analysed using formal parameter asymptotics. In the second problem, this leads to a Korteweg-de Vries equation for the surface wave, and then the evolution of appropriate solutions of this equation gives rise to the appearance of a critical layer near the bottom; the corresponding problem at the surface can be formulated but not completely resolved. The appearance of a stagnation point and then a critical layer, either at the surface or the bottom, are discussed; the nature of the flow, and the corresponding streamlines are obtained and some typical flow fields are depicted.Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 04/2012; 370(1964):1638-60. DOI:10.1098/rsta.2011.0456 - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider a wavefront model for the spread of Neolithic culture across Europe, and use Bayesian inference techniques to provide estimates for the parameters within this model, as constrained by radiocarbon data from Southern and Western Europe. Our wavefront model allows for both an isotropic background spread (incorporating the effects of local geography), and a localized anisotropic spread associated with major waterways. We introduce an innovative numerical scheme to track the wavefront, and use Gaussian process emulators to further increase the efficiency of our model, thereby making Markov chain Monte Carlo methods practical. We allow for uncertainty in the fit of our model, and discuss the inferred distribution of the parameter specifying this uncertainty, along with the distributions of the parameters of our wavefront model. We subsequently use predictive distributions, taking account of parameter uncertainty, to identify radiocarbon sites which do not agree well with our model. These sites may warrant further archaeological study, or motivate refinements to the model.Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 03/2012; 86(1). DOI:10.1103/PhysRevE.86.016105

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