Monte Carlo Maximum Likelihood Research Articles

A generalisation of the method of regression calibration and comparison with Bayesian and frequentist model averaging methods

For many cancer sites low-dose risks are not known and must be extrapolated from those observed in groups exposed at much higher levels of dose. Measurement error can substantially alter the dose–response shape and hence the extrapolated risk. Even in studies with direct measurement of low-dose exposures measurement error could be substantial in relation to the size of the dose estimates and thereby distort population risk estimates. Recently, there has been considerable attention paid to methods of dealing with shared errors, which are common in many datasets, and particularly important in occupational and environmental settings. In this paper we test Bayesian model averaging (BMA) and frequentist model averaging (FMA) methods, the first of these similar to the so-called Bayesian two-dimensional Monte Carlo (2DMC) method, and both fairly recently proposed, against a very newly proposed modification of the regression calibration method, the extended regression calibration (ERC) method, which is particularly suited to studies in which there is a substantial amount of shared error, and in which there may also be curvature in the true dose response. The quasi-2DMC with BMA method performs well when a linear model is assumed, but very poorly when a linear-quadratic model is assumed, with coverage probabilities both for the linear and quadratic dose coefficients that are under 5% when the magnitude of shared Berkson error is large (50%). For the linear model the bias is generally under 10%. However, using a linear-quadratic model it produces substantially biased (by a factor of 10) estimates of both the linear and quadratic coefficients, with the linear coefficient overestimated and the quadratic coefficient underestimated. FMA performs as well as quasi-2DMC with BMA when a linear model is assumed, and generally much better with a linear-quadratic model, although the coverage probability for the quadratic coefficient is uniformly too high. However both linear and quadratic coefficients have pronounced upward bias, particularly when Berkson error is large. By comparison ERC yields coverage probabilities that are too low when shared and unshared Berkson errors are both large (50%), although otherwise it performs well, and coverage is generally better than the quasi-2DMC with BMA or FMA methods, particularly for the linear-quadratic model. The bias of the predicted relative risk at a variety of doses is generally smallest for ERC, and largest for the quasi-2DMC with BMA and FMA methods (apart from unadjusted regression), with standard regression calibration and Monte Carlo maximum likelihood exhibiting bias in predicted relative risk generally somewhat intermediate between ERC and the other two methods. In general ERC performs best in the scenarios presented, and should be the method of choice in situations where there may be substantial shared error, or suspected curvature in the dose response.

Identifying childhood malaria hotspots and risk factors in a Nigerian city using geostatistical modelling approach

Malaria ranks high among prevalent and ravaging infectious diseases in sub-Saharan Africa (SSA). The negative impacts, disease burden, and risk are higher among children and pregnant women as part of the most vulnerable groups to malaria in Nigeria. However, the burden of malaria is not even in space and time. This study explores the spatial variability of malaria prevalence among children under five years (U5) in medium-sized rapidly growing city of Akure, Nigeria using model-based geostatistical modeling (MBG) technique to predict U5 malaria burden at a 100 × 100 m grid, while the parameter estimation was done using Monte Carlo maximum likelihood method. The non-spatial logistic regression model shows that U5 malaria prevalence is significantly influenced by the usage of insecticide-treated nets—ITNs, window protection, and water source. Furthermore, the MBG model shows predicted U5 malaria prevalence in Akure is greater than 35% at certain locations while we were able to ascertain places with U5 prevalence > 10% (i.e. hotspots) using exceedance probability modelling which is a vital tool for policy development. The map provides place-based evidence on the spatial variation of U5 malaria in Akure, and direction on where intensified interventions are crucial for the reduction of U5 malaria burden and improvement of urban health in Akure, Nigeria.

Two decades of malaria control in Malawi: Geostatistical Analysis of the changing malaria prevalence from 2000-2022.

Malaria remains a public health problem in Malawi and has a serious socio-economic impact on the population. In the past two decades, available malaria control measures have been substantially scaled up, such as insecticide-treated bed nets, artemisinin-based combination therapies, and, more recently, the introduction of the malaria vaccine, the RTS,S/AS01. In this paper, we describe the epidemiology of malaria for the last two decades to understand the past transmission and set the scene for the elimination agenda. A collation of parasite prevalence surveys conducted between the years 2000 and 2022 was done. A spatio-temporal geostatistical model was fitted to predict the yearly malaria risk for children aged 2-10 years (PfPR 2-10) at 1×1 km spatial resolutions. Parameter estimation was done using the Monte Carlo maximum likelihood method. District-level prevalence estimates adjusted for population are calculated for the years 2000 to 2022. A total of 2,595 sampled unique locations from 2000 to 2022 were identified through the data collation exercise. This represents 70,565 individuals that were sampled in the period. In general, the PfPR2_10 declined over the 22 years. The mean modelled national PfPR2_10 in 2000 was 43.93 % (95% CI:17.9 to 73.8%) and declined to 19.2% (95%CI 7.49 to 37.0%) in 2022. The smoothened estimates of PfPR2_10 indicate that malaria prevalence is very heterogeneous with hotspot areas concentrated on the southern shores of Lake Malawi and the country's central region. The last two decades are associated with a decline in malaria prevalence, highly likely associated with the scale-up of control interventions. The country should move towards targeted malaria control approaches informed by surveillance data.

A generalisation of the method of regression calibration

There is direct evidence of risks at moderate and high levels of radiation dose for highly radiogenic cancers such as leukaemia and thyroid cancer. For many cancer sites, however, it is necessary to assess risks via extrapolation from groups exposed at moderate and high levels of dose, about which there are substantial uncertainties. Crucial to the resolution of this area of uncertainty is the modelling of the dose–response relationship and the importance of both systematic and random dosimetric errors for analyses in the various exposed groups. It is well recognised that measurement error can alter substantially the shape of this relationship and hence the derived population risk estimates. Particular attention has been devoted to the issue of shared errors, common in many datasets, and particularly important in occupational settings. We propose a modification of the regression calibration method which is particularly suited to studies in which there is a substantial amount of shared error, and in which there may also be curvature in the true dose response. This method can be used in settings where there is a mixture of Berkson and classical error. In fits to synthetic datasets in which there is substantial upward curvature in the true dose response, and varying (and sometimes substantial) amounts of classical and Berkson error, we show that the coverage probabilities of all methods for the linear coefficient alpha are near the desired level, irrespective of the magnitudes of assumed Berkson and classical error, whether shared or unshared. However, the coverage probabilities for the quadratic coefficient beta are generally too low for the unadjusted and regression calibration methods, particularly for larger magnitudes of the Berkson error, whether this is shared or unshared. In contrast Monte Carlo maximum likelihood yields coverage probabilities for beta that are uniformly too high. The extended regression calibration method yields coverage probabilities that are too low when shared and unshared Berkson errors are both large, although otherwise it performs well, and coverage is generally better than these other three methods. A notable feature is that for all methods apart from extended regression calibration the estimates of the quadratic coefficient beta are substantially upwardly biased.

Parameter estimation procedures for exponential-family random graph models on count-valued networks: A comparative simulation study

The exponential-family random graph models (ERGMs) have emerged as an important framework for modeling social networks for a wide variety of relational types. ERGMs for valued networks are less well-developed than their unvalued counterparts, and pose particular computational challenges. Network data with edge values on the non-negative integers (count-valued networks) is an important such case, with examples ranging from the magnitude of migration and trade flows between places to the frequency of interactions and encounters between individuals. Here, we propose an efficient parallelizable subsampled maximum pseudo-likelihood estimation (MPLE) scheme for count-valued ERGMs, and compare its performance with existing Contrastive Divergence (CD) and Monte Carlo Maximum Likelihood Estimation (MCMLE) approaches via a simulation study based on migration flow networks in two U.S. states. Our results suggest that edge value variance is a key factor in method performance, while network size mainly influences their relative merits in computational time. For small-variance networks, all methods perform well in point estimations while CD greatly overestimates uncertainties, and MPLE underestimates them for dependence terms; all methods have fast estimation for small networks, but CD and subsampled multi-core MPLE provides speed advantages as network size increases. For large-variance networks, both MPLE and MCMLE offer high-quality estimates of coefficients and their uncertainty, but MPLE is significantly faster than MCMLE; MPLE is also a better seeding method for MCMLE than CD, as the latter makes MCMLE more prone to convergence failure. The study suggests that MCMLE and MPLE should be the default approach to estimate ERGMs for small-variance and large-variance valued networks, respectively. We also offer further suggestions regarding choice of computational method for valued ERGMs based on data structure, available computational resources and analytical goals.

Two decades of malaria control in Malawi: Geostatistical Analysis of the changing malaria prevalence from 2000-2022

Background Malaria remains a public health problem in Malawi and has a serious socio-economic impact on the population. In the past two decades, available malaria control measures have been substantially scaled up, such as insecticide-treated bed nets, artemisinin-based combination therapies, and, more recently, the introduction of the malaria vaccine, the RTS,S/AS01. In this paper, we describe the epidemiology of malaria for the last two decades to understand the past transmission and set the scene for the elimination agenda. Methods A collation of parasite prevalence surveys conducted between the years 2000 and 2022 was done. A spatio-temporal geostatistical model was fitted to predict the yearly malaria risk for children aged 2–10 years (PfPR 2–10) at 1×1 km spatial resolutions. Parameter estimation was done using the Monte Carlo maximum likelihood methods. District level prevalence estimates adjusted for population are calculated for the years 2000 to 2022. Results A total of 2,595 sampled unique locations from 2000 to 2022 were identified through the data collation exercise. This represents 70,565 individuals that were sampled in the period. In general, the PfPR2_10 declined over the 22 years. The mean modeled national PfPR2_10 in 2000 was 43.93 % (95% CI:17.9 to 73.8%) and declined to 19.2% (95%CI 7.49 to 37.0%) in 2022. The smoothened estimates of PfPR2_10 indicate that malaria prevalence is very heterogeneous with hotspot areas concentrated on the southern shores of Lake Malawi and the country's central region. Conclusions The last two decades are associated with a decline in malaria prevalence, highly likely associated with the scale up of control interventions. The country should move towards targeted malaria control approaches informed by surveillance data.

Feasibility of Monte-Carlo maximum likelihood for fitting spatial log-Gaussian Cox processes

Log-Gaussian Cox processes (LGCPs) are a popular and flexible tool for modelling point pattern data. While maximum likelihood estimation of the parameters of such a model is attractive in principle, the likelihood function is not available in closed form. Various Monte Carlo approximations have been proposed, but these have seen very limited use in the literature and are often dismissed as impractical. This article provides a comprehensive study of the computational properties of Monte Carlo maximum likelihood estimation (MCMLE) for LGCPs. We compare various importance sampling algorithms for MCMLE, and also consider their performance against other methods of inference (such as minimum contrast) in numerical studies. We find that the best MCMLE algorithm is a practical proposition for parameter estimation given modern computing power, but the performance of this methodology is rather sensitive to the choice of reference parameters defining the importance sampling distribution.

Highly scalable maximum likelihood and conjugate Bayesian inference for ERGMs on graph sets with equivalent vertices.

The exponential family random graph modeling (ERGM) framework provides a highly flexible approach for the statistical analysis of networks (i.e., graphs). As ERGMs with dyadic dependence involve normalizing factors that are extremely costly to compute, practical strategies for ERGMs inference generally employ a variety of approximations or other workarounds. Markov Chain Monte Carlo maximum likelihood (MCMC MLE) provides a powerful tool to approximate the maximum likelihood estimator (MLE) of ERGM parameters, and is generally feasible for typical models on single networks with as many as a few thousand nodes. MCMC-based algorithms for Bayesian analysis are more expensive, and high-quality answers are challenging to obtain on large graphs. For both strategies, extension to the pooled case—in which we observe multiple networks from a common generative process—adds further computational cost, with both time and memory scaling linearly in the number of graphs. This becomes prohibitive for large networks, or cases in which large numbers of graph observations are available. Here, we exploit some basic properties of the discrete exponential families to develop an approach for ERGM inference in the pooled case that (where applicable) allows an arbitrarily large number of graph observations to be fit at no additional computational cost beyond preprocessing the data itself. Moreover, a variant of our approach can also be used to perform Bayesian inference under conjugate priors, again with no additional computational cost in the estimation phase. The latter can be employed either for single graph observations, or for observations from graph sets. As we show, the conjugate prior is easily specified, and is well-suited to applications such as regularization. Simulation studies show that the pooled method leads to estimates with good frequentist properties, and posterior estimates under the conjugate prior are well-behaved. We demonstrate the usefulness of our approach with applications to pooled analysis of brain functional connectivity networks and to replicated x-ray crystal structures of hen egg-white lysozyme.

Impact of uncertainties in exposure assessment on thyroid cancer risk among cleanup workers in Ukraine exposed due to the Chornobyl accident.

A large excess risk of thyroid cancer was observed among Belarusian/Russian/Baltic Chornobyl cleanup workers. A more recent study of Ukraine cleanup workers found more modest excess risks of thyroid cancer. Dose errors in this data are substantial, associated with model uncertainties and questionnaire response. Regression calibration is often used for dose-error adjustment, but may not adequately account for the full error distribution. We aimed to examine the impact of exposure-assessment uncertainties on thyroid cancer among Ukrainian cleanup workers using Monte Carlo maximum likelihood, and compare with results derived using regression calibration. Analyses assessed the sensitivity of results to various components of internal and external dose. Regression calibration yielded an excess odds ratio per Gy (EOR/Gy) of 0.437 (95% CI -0.042, 1.577, p = 0.100), compared with the EOR/Gy using Monte Carlo maximum likelihood of 0.517 (95% CI -0.039, 2.035, p = 0.093). Trend risk estimates for follicular morphology tumors exhibited much more extreme effects of full-likelihood adjustment, the EOR/Gy using regression calibration of 3.224 (95% CI -0.082, 30.615, p = 0.068) becoming ~ 50% larger, 4.708 (95% CI -0.075, 85.143, p = 0.066) when using Monte Carlo maximum likelihood. Results were sensitive to omission of external components of dose. In summary, use of Monte Carlo maximum likelihood adjustment for dose error led to increases in trend risks, particularly for follicular morphology thyroid cancers, where risks increased by ~ 50%, and were borderline significant. The unexpected finding for follicular tumors needs to be replicated in other exposed groups.

A projection‐based Laplace approximation for spatial latent variable models

AbstractLaplace method is a practical tool for obtaining maximum likelihood estimators for a wide class of latent variable models. The main idea is to approximate the integrand using a Gaussian distribution. However, with increasing observations, the Laplace approximation becomes infeasible because the dimension of the correlated latent variables grows, which results in the high‐dimensional optimization problem. One important example is spatial latent variable models, which are widely used in many fields, such as ecology, epidemiology, and sociology. Spatial latent variable models are useful for investigating the relationship between spatial covariates or predicting the unobserved area. Here, we propose a fast Laplace approximation based on the dimension reduction of the latent variables. Our methods are faster and have fewer components to be tuned than simulation‐based methods such as Markov chain Monte Carlo maximum likelihood and Monte Carlo expectation‐maximization. Our approach can be applied to the large non‐Gaussian spatial data sets, commonly used in modern environmental sciences. Especially, we show how we may understand spatial patterns of non‐Gaussian responses for two case studies: confirmed COVID‐19 cases in the United States and thickness of the Antarctic ice sheet. Through simulation studies under different scenarios, we investigate that our method can provide accurate maximum likelihood estimations and predictions quickly. Our study can be widely applicable for practical maximum likelihood inference for high‐dimensional random effect models. We provide a freely available R‐package that can implement the proposed method.

Reduced-Dimensional Monte Carlo Maximum Likelihood for Latent Gaussian Random Field Models

Monte Carlo maximum likelihood (MCML) provides an elegant approach to find maximum likelihood estimators (MLEs) for latent variable models. However, MCML algorithms are computationally expensive when the latent variables are high-dimensional and correlated, as is the case for latent Gaussian random field models. Latent Gaussian random field models are widely used, for example, in building flexible regression models and in the interpolation of spatially dependent data in many research areas such as analyzing count data in disease modeling and presence-absence satellite images of ice sheets. We propose a computationally efficient MCML algorithm by using a projection-based approach to reduce the dimensions of the random effects. We develop an iterative method for finding an effective importance function; this is generally a challenging problem and is crucial for the MCML algorithm to be computationally feasible. We find that our method is applicable to both continuous (latent Gaussian process) and discrete domain (latent Gaussian Markov random field) models. We illustrate the application of our methods to challenging simulated and real data examples for which maximum likelihood estimation would otherwise be very challenging. Furthermore, we study an often overlooked challenge in MCML approaches to latent variable models: practical issues in calculating standard errors of the resulting estimates, and assessing whether resulting confidence intervals provide nominal coverage. Our study therefore provides useful insights into the details of implementing MCML algorithms for high-dimensional latent variable models. Supplementary materials for this article are available online.

Maximum Likelihood Calibration of Stochastic Multipath Radio Channel Models

We propose Monte Carlo maximum likelihood estimation as a novel approach in the context of calibration and selection of stochastic channel models. First, considering a Turin channel model with inhomogeneous arrival rate as a prototypical example, we explain how the general statistical methodology is adapted and refined for the specific requirements and challenges of stochastic multipath channel models. Then, we illustrate the advantages and pitfalls of the method based on simulated data. Finally, we apply our calibration method to wideband signal data from indoor channels.

Study on Driving Cycle Synthesis Method for City Buses considering Random Passenger Load

This paper proposes a novel driving cycle construction method in consideration of velocity, road slope, and passenger load, based on a real-world bus route with a plug-in hybrid electric bus (PHEB). The main purpose is to address the disadvantage that an inaccurate reflection of the real-world driving characteristics for city buses will be caused when ignoring the passenger load in the course of a driving cycle synthesis. Two contributions are supplemented to distinguish from the previous research. Firstly, a novel station-based method is proposed aiming at developing a driving cycle with high accuracy. The kinematic segments are partitioned according to the distance of adjacent bus stops, while a two-dimensional Markov chain Monte Carlo method is employed to synthesize driving cycle between each interval of adjacent bus stops. Secondly, the random passenger load for different bus stops is treated as a discrete Markov chain model, according to the correlation analysis of the measured passenger data which are distinguished for off-peak and peak hours. Meanwhile, Monte Carlo simulation and maximum likelihood estimation are utilized to determine the most likely number of passengers for each bus stop. At last, the fuel consumption of the PHEB is simulated with the best-synthesized driving cycle and contrasted to the mean fuel consumption of the later measured data which is composed of the velocity, road slope, and the passenger load. The results demonstrate that the synthesized driving cycle has a higher accuracy on fuel consumption estimation.

Geostatistical analysis of Malawi's changing malaria transmission from 2010 to 2017.

Background: The prevalence of malaria infection in time and space provides important information on the likely sub-national epidemiology of malaria burdens and how this has changed following intervention. Model-based geostatitics (MBG) allow national malaria control programmes to leverage multiple data sources to provide predictions of malaria prevalance by district over time. These methods are used to explore the possible changes in malaria prevalance in Malawi from 2010 to 2017. Methods: Plasmodium falciparum parasite prevalence ( PfPR) surveys undertaken in Malawi between 2000 and 2017 were assembled. A spatio-temporal geostatistical model was fitted to predict annual malaria risk for children aged 2-10 years ( PfPR 2-10) at 1×1 km spatial resolutions. Parameter estimation was carried out using the Monte Carlo maximum likelihood methods. Population-adjusted prevalence and populations at risk by district were calculated for 2010 and 2017 to inform malaria control program priority setting. Results: 2,237 surveys at 1,834 communities undertaken between 2000 and 2017 were identified, geo-coded and used within the MBG framework to predict district malaria prevalence properties for 2010 and 2017. Nationally, there was a 47.2% reduction in the mean modelled PfPR 2-10 from 29.4% (95% confidence interval (CI) 26.6 to 32.3%) in 2010 to 15.2% (95% CI 13.3 to 18.0%) in 2017. Declining prevalence was not equal across the country, 25 of 27 districts showed a substantial decline ranging from a 3.3% reduction to 79% reduction. By 2017, 16% of Malawi's population still lived in areas that support PfPR 2-10 ≥ 25%. Conclusions: Malawi has made substantial progress in reducing the prevalence of malaria over the last seven years. However, Malawi remains in meso-endemic malaria transmission risk. To sustain the gains made and continue reducing the transmission further, universal control interventions need to be maintained at a national level.

Geostatistical analysis of Malawi’s changing malaria transmission from 2010 to 2017

Background: The prevalence of malaria infection in time and space provides important information on the likely sub-national epidemiology of malaria burdens and how this has changed following intervention. Model-based geostatitics (MBG) allow national malaria control programmes to leverage multiple data sources to provide predictions of malaria prevalance by district over time. These methods are used to explore the possible changes in malaria prevalance in Malawi from 2010 to 2017. Methods: Plasmodium falciparum parasite prevalence ( PfPR) surveys undertaken in Malawi between 2000 and 2017 were assembled. A spatio-temporal geostatistical model was fitted to predict annual malaria risk for children aged 2–10 years ( PfPR 2–10) at 1×1 km spatial resolutions. Parameter estimation was carried out using the Monte Carlo maximum likelihood methods. Population-adjusted prevalence and populations at risk by district were calculated for 2010 and 2017 to inform malaria control program priority setting. Results: 2,237 surveys at 1,834 communities undertaken between 2000 and 2017 were identified, geo-coded and used within the MBG framework to predict district malaria prevalence properties for 2010 and 2017. Nationally, there was a 47.2% reduction in the mean modelled PfPR 2-10 from 29.4% (95% confidence interval (CI) 26.6 to 32.3%) in 2010 to 15.2% (95% CI 13.3 to 18.0%) in 2017. Declining prevalence was not equal across the country, 25 of 27 districts showed a substantial decline ranging from a 3.3% reduction to 79% reduction. By 2017, 16% of Malawi’s population still lived in areas that support PfPR 2-10 ≥ 25%. Conclusions: Malawi has made substantial progress in reducing the prevalence of malaria over the last seven years. However, Malawi remains in meso-endemic malaria transmission risk. To sustain the gains made and continue reducing the transmission further, universal control interventions need to be maintained at a national level.

A systematic review of outcome in intramedullary ependymoma and astrocytoma

The objective was to determine the impact of surgical resection and adjuvant therapies on survival in intramedullary ependymoma and astrocytoma. Secondary goals were to determine predictors of survival in surgical patients including histological grading, age and gender. Searching of Medline, Embase and Clinicaltrials.gov databases were performed. Multivariate analyses were performed for overall survival (OS) and progression-free survival (PFS) through Monte Carlo methods and Maximum Likelihood Estimation. 57 articles detail results for 3022 patients. Meta-analysis revealed the following factors to have a statistically significant effect on OS. Patients undergoing gross-total resection (GTR) are 5.37 times more likely to survive than patients with lesser volumes of tumor resected (HR for OS 1.68, p < 0.01). High-grade tumors were associated with a 14 times risk of death over low-grade tumors (HR for OS 2.64, p < 0.01). Radiation increased the risk of mortality in low-grade tumors (HR for OS 5.20, p < 0.01), but decreased mortality in high-grade lesions (HR for OS 2.46, p < 0.01). Adult patients were more likely to die from disease compared with pediatric patients by a factor of 1.6 (HR for OS 0.47, p < 0.01). In PFS, radiotherapy was associated with a reduced time to recurrence (HR for PFS 1.90, p < 0.01). There was a male predominance of 58%. Gender did not influence survival. 79% of patients demonstrated stable or improved functional neurological outcomes six months post-operatively. Our data indicates GTR improves OS in all tumor grades. Radiation improves OS only in the presence of high-grade histology. Advancing age and high-grade histology are negative prognostic indicators.

Noise contrastive estimation: Asymptotic properties, formal comparison with MC-MLE

A statistical model is said to be un-normalised when its likelihood function involves an intractable normalising constant. Two popular methods for parameter inference for these models are MC-MLE (Monte Carlo maximum likelihood estimation), and NCE (noise contrastive estimation); both methods rely on simulating artificial data-points to approximate the normalising constant. While the asymptotics of MC-MLE have been established under general hypotheses (Geyer, 1994), this is not so for NCE. We establish consistency and asymptotic normality of NCE estimators under mild assumptions. We compare NCE and MC-MLE under several asymptotic regimes. In particular, we show that, when $m\rightarrow \infty $ while $n$ is fixed ($m$ and $n$ being respectively the number of artificial data-points, and actual data-points), the two estimators are asymptotically equivalent. Conversely, we prove that, when the artificial data-points are IID, and when $n\rightarrow \infty $ while $m/n$ converges to a positive constant, the asymptotic variance of a NCE estimator is always smaller than the asymptotic variance of the corresponding MC-MLE estimator. We illustrate the variance reduction brought by NCE through a numerical study.

The Impact of Systematic Trend and Uncertainty on Mortality and Disability in a Multistate Latent Factor Model for Transition Rates

Multiple state functional disability models do not generally include systematic trend and uncertainty. We develop and estimate a multistate latent factor intensity model with transition and recovery rates depending on a stochastic frailty factor to capture trend and uncertainty. We estimate the model parameters using U.S. Health and Retirement Study data between 1998 and 2012 with Monte Carlo maximum likelihood estimation method. The model shows significant reductions in disability and mortality rates during this period and allows us to quantify uncertainty in transition rates arising from the stochastic frailty factor. Recovery rates are very sensitive to the stochastic frailty. There is an increase in expected future lifetimes as well as an increase in future healthy life expectancy. The proportion of lifetime spent in disability on average remains stable with no strong support in the data for either morbidity compression or expansion. The model has widespread application in costing of government-funded aged care and pricing and risk management of long-term-care insurance products.

A comparison of estimation methods for multilevel models of spatially structured data

Two recent contributions (Dong et al., 2015; Osland et al., 2016) point to the relevance of multilevel models for spatially structured data. In Osland et al. (2016) these models are used to examine the importance of district-level covariates for house prices in Stavanger, Norway, in Dong et al. (2015) similarly for land parcel prices in Beijing; we use these data sets in our comparison. In Osland et al. (2016), a district-level spatial random effect was fitted using an intrinsic CAR model estimated using WinBUGS. Dong et al. (2015) used R code provided in supplementary materials to their article, and subsequently improved in an R package (Dong et al., 2016a); computation there used custom MCMC C++ code to fit a SAR district-level spatial random effect. This article compares approaches to estimating models of this kind, using the R packages R2WinBUGS, HSAR, INLA, R2BayesX, hglm and the new package mclcar for Monte Carlo maximum likelihood estimation (Sha, 2016b). We show that multilevel models of spatially structured data may be estimated readily using a variety of approaches, not only the intrinsic CAR model more typically found in the existing literature. We also point to a range of issues for further research in situations in which data acquired at different levels of spatial resolution are combined.

Bayesian Monte Carlo and maximum likelihood approach for uncertainty estimation and risk management: Application to lake oxygen recovery model

Model uncertainty estimation and risk assessment is essential to environmental management and informed decision making on pollution mitigation strategies. In this study, we apply a probabilistic methodology, which combines Bayesian Monte Carlo simulation and Maximum Likelihood estimation (BMCML) to calibrate a lake oxygen recovery model. We first derive an analytical solution of the differential equation governing lake-averaged oxygen dynamics as a function of time-variable wind speed. Statistical inferences on model parameters and predictive uncertainty are then drawn by Bayesian conditioning of the analytical solution on observed daily wind speed and oxygen concentration data obtained from an earlier study during two recovery periods on a eutrophic lake in upper state New York. The model is calibrated using oxygen recovery data for one year and statistical inferences were validated using recovery data for another year. Compared with essentially two-step, regression and optimization approach, the BMCML results are more comprehensive and performed relatively better in predicting the observed temporal dissolved oxygen levels (DO) in the lake. BMCML also produced comparable calibration and validation results with those obtained using popular Markov Chain Monte Carlo technique (MCMC) and is computationally simpler and easier to implement than the MCMC. Next, using the calibrated model, we derive an optimal relationship between liquid film-transfer coefficient for oxygen and wind speed and associated 95% confidence band, which are shown to be consistent with reported measured values at five different lakes. Finally, we illustrate the robustness of the BMCML to solve risk-based water quality management problems, showing that neglecting cross-correlations between parameters could lead to improper required BOD load reduction to achieve the compliance criteria of 5 mg/L.