Discussion on "The central role of the identifying assumption in population size estimation" by Serge Aleshin-Guendel, Mauricio Sadinle, and Jon Wakefield.

In this discussion response, we consider some practical implications of the authors' consideration of the no-highest-order interaction (NHOI) model for multiple systems estimation, which permits the authors to derive the explicit (albeit untestable) identifying assumption related to the unobserved (or missing) individuals. In particular, we discuss several aspects, from the standard process of model selection to potential poor predictive performance due to over-fitting and the implications of data reduction. We discuss these aspects in relation to the case study presented by the authors relating to the number of civilian casualties within the Kosovo war, and conduct further preliminary simulations to investigate these issues further. The results suggest that the NHOI models considered, despite having a potentially useful theoretical result in relation to the underlying identifying assumption, may perform poorly in practice.

Bayesian adaptive N-of-1 trials for estimating population and individual treatment effects.

This article proposes a novel adaptive design algorithm that can be used to find optimal treatment allocations in N-of-1 clinical trials. This new methodology uses two Laplace approximations to provide a computationally efficient estimate of population and individual random effects within a repeated measures, adaptive design framework. Given the efficiency of this approach, it is also adopted for treatment selection to target the collection of data for the precise estimation of treatment effects. To evaluate this approach, we consider both a simulated and motivating N-of-1 clinical trial from the literature. For each trial, our methods were compared with the multiarmed bandit approach and a randomized N-of-1 trial design in terms of identifying the best treatment for each patient and the information gained about the model parameters. The results show that our new approach selects designs that are highly efficient in achieving each of these objectives. As such, we propose our Laplace-based algorithm as an efficient approach for designing adaptive N-of-1 trials.

Multivariate emulation of computer simulators: model selection and diagnostics with application to a humanitarian relief model.

We present a common framework for Bayesian emulation methodologies for multivariate output simulators, or computer models, that employ either parametric linear models or non-parametric Gaussian processes. Novel diagnostics suitable for multivariate covariance separable emulators are developed and techniques to improve the adequacy of an emulator are discussed and implemented. A variety of emulators are compared for a humanitarian relief simulator, modelling aid missions to Sicily after a volcanic eruption and earthquake, and a sensitivity analysis is conducted to determine the sensitivity of the simulator output to changes in the input variables. The results from parametric and non-parametric emulators are compared in terms of prediction accuracy, uncertainty quantification and scientific interpretability.

A default prior distribution for contingency tables with dependent factor levels.

A default prior distribution is proposed for the Bayesian analysis of contingency tables. The prior is specified to allow for dependence between levels of the factors. Different dependence structures are considered, including conditional autoregressive and distance correlation structures. To demonstrate the prior distribution, a dataset is considered which involves estimating the number of injecting drug users in the eleven National Health Service board regions of Scotland using an incomplete contingency table where the dependence structure relates to geographical regions.

Incomplete contingency tables with censored cells with application to estimating the number of people who inject drugs in Scotland.

Estimating the size of hidden or difficult to reach populations is often of interest for economic, sociological or public health reasons. In order to estimate such populations, administrative data lists are often collated to form multi-list cross-counts and displayed in the form of an incomplete contingency table. Log-linear models are typically fitted to such data to obtain an estimate of the total population size by estimating the number of individuals not observed by any of the data-sources. This approach has been taken to estimate the current number of people who inject drugs (PWID) in Scotland, with the Hepatitis C virus diagnosis database used as one of the data-sources to identify PWID. However, the Hepatitis C virus diagnosis data-source does not distinguish between current and former PWID, which, if ignored, will lead to overestimation of the total population size of current PWID. We extend the standard model-fitting approach to allow for a data-source, which contains a mixture of target and non-target individuals (i.e. in this case, current and former PWID). We apply the proposed approach to data for PWID in Scotland in 2003, 2006 and 2009 and compare with the results from standard log-linear models.

Injecting drug users in Scotland, 2006: Listing, number, demography, and opiate-related death-rates.

Using Bayesian capture-recapture analysis, we estimated the number of current injecting drug users (IDUs) in Scotland in 2006 from the cross-counts of 5670 IDUs listed on four data-sources: social enquiry reports (901 IDUs listed), hospital records (953), drug treatment agencies (3504), and recent Hepatitis C virus (HCV) diagnoses (827 listed as IDU-risk). Further, we accessed exact numbers of opiate-related drugs-related deaths (DRDs) in 2006 and 2007 to improve estimation of Scotland's DRD rates per 100 current IDUs. Using all four data-sources, and model-averaging of standard hierarchical log-linear models to allow for pairwise interactions between data-sources and/or demographic classifications, Scotland had an estimated 31700 IDUs in 2006 (95% credible interval: 24900-38700); but 25000 IDUs (95% CI: 20700-35000) by excluding recent HCV diagnoses whose IDU-risk can refer to past injecting. Only in the younger age-group (15-34 years) were Scotland's opiate-related DRD rates significantly lower for females than males. Older males' opiate-related DRD rate was 1.9 (1.24-2.40) per 100 current IDUs without or 1.3 (0.94-1.64) with inclusion of recent HCV diagnoses. If, indeed, Scotland had only 25000 current IDUs in 2006, with only 8200 of them aged 35+ years, the opiate-related DRD rate is higher among this older age group than has been appreciated hitherto. There is counter-balancing good news for the public health: the hitherto sharp increase in older current IDUs had stalled by 2006.

A strategy for Bayesian inference for computationally expensive models with application to the estimation of stem cell properties.

Bayesian inference is considered for statistical models that depend on the evaluation of a computationally expensive computer code or simulator. For such situations, the number of evaluations of the likelihood function, and hence of the unnormalized posterior probability density function, is determined by the available computational resource and may be extremely limited. We present a new example of such a simulator that describes the properties of human embryonic stem cells using data from optical trapping experiments. This application is used to motivate a novel strategy for Bayesian inference which exploits a Gaussian process approximation of the simulator and allows computationally efficient Markov chain Monte Carlo inference. The advantages of this strategy over previous methodology are that it is less reliant on the determination of tuning parameters and allows the application of model diagnostic procedures that require no additional evaluations of the simulator. We show the advantages of our method on synthetic examples and demonstrate its application on stem cell experiments.