Non-parametric maximum likelihood estimators for disease mapping

A Non-Parametric Maximum Likelihood approach to the estimation of relative risks in the context of disease mapping is discussed and a NPML approximation to conditional autoregressive models is proposed. NPML estimates have been compared to other proposed solutions (Maximum Likelihood via Monte Carlo Scoring, Hierarchical Bayesian models) using real examples. Overall, the NPML autoregressive estimates (with weighted term) were closer to the Bayesian estimates. The exchangeable NPML model ranked immediately after, even if it implied a greater shrinkage, while the truncated auto-Poisson showed inadequate for disease mapping. The coefficients of the autoregressive term for the different mixtures have clear interpretations: in the breast cancer example, the larger cities in the region showed high rates and very low correlation with the neighbouring areas, while the less populated rural areas with low rates were strongly positively correlated each other. This pattern is expected since breast cancer is strongly correlated with parity and age at first birth, and the female population of the rural areas experienced a decline in fertility much later than those living in the larger cities. The leukemia example highlighted the failure of the Poisson-Gamma model and other general overdispersion tests to detect high risk areas under specific conditions. The NPML approach in Aitkin is very general, simple and flexible. However the user should be warned against the possibility of local maxima and the difficulty in detecting the optimal number of components. Special software (such as CAMAN or DismapWin) had been developed and should be recommended mainly to not experienced users.     

A comparison of conditional autoregressive models used in Bayesian disease mapping

Disease mapping is the area of epidemiology that estimates the spatial pattern in disease risk over an extended geographical region, so that areas with elevated risk levels can be identified. Bayesian hierarchical models are typically used in this context, which represent the risk surface using a combination of available covariate data and a set of spatial random effects. These random effects are included to model any overdispersion or spatial correlation in the disease data, that has not been accounted for by the available covariate information. The random effects are typically modelled by a conditional autoregressive (CAR) prior distribution, and a number of alternative specifications have been proposed. This paper critiques four of the most common models within the CAR class, and assesses their appropriateness via a simulation study. The four models are then applied to a new study mapping cancer incidence in Greater Glasgow, Scotland, between 2001 and 2005.     

贝叶斯时空模型在布鲁氏菌病疫情数据分析中的应用

以2000-2007年内蒙古地区布鲁氏菌病(布病)疫情数据为例,运用空间统计学和传染病流行病学的相关理论,应用贝叶斯理论框架建立时空模型,分析布病在时间和空间上呈现的格局及其演变,以及与之相关联的协变量及其变化,并与传统分析方法进行比较.结果 显示,拟合协变量的贝叶斯时空模型相对较佳(离差信息准则值最小,为2388.000).2000-2007年内蒙古自治区101个旗县的布病疫情呈现较强的空间相关性,时空格局存在较明显的共变现象,每年空间相关性不尽相同,空间相关系数后验中位数位于0.968～0.973之间,总体上随时间变化略呈下降趋势.地区类型和牛羊存栏数量与内蒙古布病流行可能有关,且牛羊存栏数量对布病的影响随年份而变化.与传统描述流行病学分析方法比较,贝叶斯方法对布病发病相对危险度的估计更加稳定.

Abstract：

Based on the number of brucellosis cases reported from the national infectious diseases reporting system in Inner Mongolia from 2000 to 2007, a model was developed. Theories of spatial statistics were used, together with knowledge on infectious disease epidemiology and the frame of Bayesian statistics, before the Bayesian spatio-temporal models were respectively set. The effects of space, time, space-time and the relative covariates were also considered. These models were applied to analyze the brucellosis distribution and time trend in Inner Mongolia during 2000-2007. The results of Bayesian spatio-temporal models was expressed by mapping of the disease and compared to the conventional statistical methods. Results showed that the Bayesian models, under consideration of space-time effect and the relative covariates (deviance information criterion, DIC=2388.000) ,seemed to be the best way to serve the purpose. The county-level spatial correlation of brucellosis epidemics was positive and quite strong in Inner Mongolia. However, the spatial correlation varied with time and the coefficients ranged from 0.968 to 0.973, having a weakening trend during 2000-2007. Types of region and number of stock (cattle and sheep) might be related to the brucellosis epidemics, and the effect on the number of cattle and sheep changed by year. Compared to conventional statistical methods, Bayesian spatio-temporal modeling could precisely estimate the incidence relative risk and was an important tool to analyze the epidemic distribution patterns of infectious diseases and to estimate the incidence relative risk.     

Spatio-temporal modelling of rates for the construction of disease maps

There have been significant developments in disease mapping in the past few decades. The continual development of statistical methodology in this area is responsible for the growing popularity of disease mapping because of its potential usefulness in regional health planning, disease surveillance and intervention, and allocating health funding. Here we review the area of disease mapping where relative risks pertain to an event such as incidence or mortality over space and time. In particular we briefly discuss the use of generalized additive mixed models, an additive extension of generalized linear mixed models, for spatio-temporal analysis of disease rates. To illustrate the procedures, we present an in-depth analysis of infant mortality data in the province of British Columbia, Canada. The goals of the analysis are to produce more reliable small-area estimates of mortality rates, assess spatial patterns over time, and examine risk trends at both global (provincial) and local (local health area) levels.     

Statistical issues in health impact assessment at the state and local levels

In this work, we discuss the uncertainty in estimating the human health risk due to exposure to air pollution, including personal and population average exposure error, epidemiological designs, and methods of analysis. Different epidemiological models may lead to very different conclusions for the same set of data. Thus, evaluation of the assumptions made and sensitivity analysis are necessary. Short-term health impact indicators may be calculated using concentration–response (C-R) functions. We discuss different methods to combine C-R function estimates from a given locale and time period with the larger body of evidence from other locales and periods and with the literature. A shrunken method is recommended to combine C-R function estimates from multiple locales. This shrunken estimate includes information from the overall and the local estimates, and thus, it characterizes the estimated excess of risk due to heterogeneity between the different locations.     

Interpreting posterior relative risk estimates in disease-mapping studies

There is currently much interest in conducting spatial analyses of health outcomes at the small-area scale. This requires sophisticated statistical techniques, usually involving Bayesian models, to smooth the underlying risk estimates because the data are typically sparse. However, questions have been raised about the performance of these models for recovering the "true" risk surface, about the influence of the prior structure specified, and about the amount of smoothing of the risks that is actually performed. We describe a comprehensive simulation study designed to address these questions. Our results show that Bayesian disease-mapping models are essentially conservative, with high specificity even in situations with very sparse data but low sensitivity if the raised-risk areas have only a moderate (less than 2-fold) excess or are not based on substantial expected counts (> 50 per area). Semiparametric spatial mixture models typically produce less smoothing than their conditional autoregressive counterpart when there is sufficient information in the data (moderate-size expected count and/or high true excess risk). Sensitivity may be improved by exploiting the whole posterior distribution to try to detect true raised-risk areas rather than just reporting and mapping the mean posterior relative risk. For the widely used conditional autoregressive model, we show that a decision rule based on computing the probability that the relative risk is above 1 with a cutoff between 70 and 80% gives a specific rule with reasonable sensitivity for a range of scenarios having moderate expected counts (approximately 20) and excess risks (approximately 1.5- to 2-fold). Larger (3-fold) excess risks are detected almost certainly using this rule, even when based on small expected counts, although the mean of the posterior distribution is typically smoothed to about half the true value.     

The misuse of count data aggregated over time for disease mapping

The ongoing spread of spatial analysis techniques for small areas has facilitated the publication of mortality and morbidity Atlases based on time periods that group information spanning several years. Although this is a widespread practice, this paper proves that the use of count data aggregated over time for disease mapping may give inappropriate area-specific relative risk. As a result, both decision-making and healthcare policies could be affected by inappropriate model specifications using aggregated information over time.The Poisson distribution properties were used in order to quantify the bias in area-specific relative risk estimation due to count data aggregated over time. A hierarchical Bayesian model with spatio-temporal random structure is proposed as an alternative to smoothing relative risk if the period of study need to span several years. Methods discussed in this paper were applied to a small-area survey on male mortality from all causes in Southern Spain for the period 1985-1999. The results suggest that particular caution should be taken when interpreting risk maps based on clustered annual data that use models with no temporal structure to smooth out the rates.     

A Bayesian generalized age–period–cohort power model for cancer projections

Age–period–cohort (APC) models are the state of art in cancer projections, assessing past and recent trends and extrapolating mortality or incidence data into the future. Nordpred is a well‐established software, assuming a Poisson distribution for the counts and a log‐link or power‐link function with fixed power; however, its predictive performance is poor for sparse data. Bayesian models with log‐link function have been applied, but they can lead to extreme estimates. In this paper, we address criticisms of the aforementioned models by providing Bayesian formulations based on a power‐link and develop a generalized APC power‐link model, which assumes a random rather than fixed power parameter. In addition, a power model with a fixed power parameter of five was formulated in the Bayesian framework. The predictive performance of the new models was evaluated on Swiss lung cancer mortality data using model‐based estimates of observed periods. Results indicated that the generalized APC power‐link model provides best estimates for male and female lung cancer mortality. The gender‐specific models were further applied to project lung cancer mortality in Switzerland during the periods 2009–2013 and 2014–2018. Copyright © 2014 John Wiley & Sons, Ltd.     

Exploring spatio-temporal patterns of mortality using mixed effects models

A linear mixed effects (LME) model previously used for a spatial analysis of mortality data for a single time period is extended to include time trends and spatio-temporal interactions. This model includes functions of age and time period that can account for increasing and decreasing death rates over time and age, and a change-point of rates at a predetermined age. A geographic hierarchy is included that provides both regional and small area age-specific rate estimates, stabilizing rates based on small numbers of deaths by sharing information within a region. The proposed log-linear analysis of rates allows the use of commercially available software for parameter estimation, and provides an estimator of overdispersion directly as the residual variance. Because of concerns about the accuracy of small area rate estimates when there are many instances of no observed deaths, we consider potential sources of error, focusing particularly on the similarity of likelihood inferences using the LME model for rates as compared to an exact Poisson-normal mixed effects model for counts. The proposed LME model is applied to breast cancer deaths which occurred among white women during 1979-1996. For this example, application of diagnostics for multiparameter likelihood comparisons suggests a restriction of age to a minimum of either 25 or 35, depending on whether small area rate estimates are required. Investigation into a convergence problem led to the discovery that the changes in breast cancer geographic patterns over time are related more to urbanization than to region, as previously thought. Published in 2000 by John Wiley & Sons, Ltd.     

Regression B-spline smoothing in Bayesian disease mapping: with an application to patient safety surveillance

In the context of Bayesian disease mapping, recent literature presents generalized linear mixed models that engender spatial smoothing. The methods assume spatially varying random effects as a route to partially pooling data and 'borrowing strength' in small-area estimation. When spatiotemporal disease rates are available for sequential risk mapping of several time periods, the 'smoothing' issue may be explored by considering spatial smoothing, temporal smoothing and spatiotemporal interaction. In this paper, these considerations are motivated and explored through development of a Bayesian semiparametric disease mapping model framework which facilitates temporal smoothing of rates and relative risks via regression B-splines with mixed-effect representation of coefficients. Specifically, we develop spatial priors such as multivariate Gaussian Markov random fields and non-spatial priors such as unstructured multivariate Gaussian distributions and illustrate how time trends in small-area relative risks may be explored by splines which vary in either a spatially structured or unstructured manner. In particular, we show that with suitable prior specifications for the random effects ensemble, small-area relative risk trends may be fit by 'spatially varying' or randomly varying B-splines. A recently developed Bayesian hierarchical model selection criterion, the deviance information criterion, is used to assess the trade-off between goodness-of-fit and smoothness and to select the number of knots. The methodological development aims to provide reliable information about the patterns (both over space and time) of disease risks and to quantify uncertainty. The study offers a disease and health outcome surveillance methodology for flexible and efficient exploration and assessment of emerging risk trends and clustering. The methods are motivated and illustrated through a Bayesian analysis of adverse medical events (also known as iatrogenic injuries) among hospitalized elderly patients in British Columbia, Canada.     

2004—2019年四川省肺结核时空趋势分析

目的 利用四川省2004—2019年肺结核报告发病数构建时间序列ARIMA模型,分析其时空变化趋势,为肺结核的综合防控提供参考依据。 方法 对2004年1月—2019年12月四川省肺结核发病情况进行分析,构建月度发病数ARIMA时间序列模型和年度发病率空间分布专题地图,分析四川地区肺结核时空流行特征及趋势。 结果 四川省2004—2018年肺结核发病持续下降,2019年略有上升,冬春季发病略高;全省疫情以川西人口稀少区域和川东北为主且相对稳定。ARIMA(2,1,1)(1,1,0)₁₂预测2020年肺结核报告发病数略高于2019年发病数。 结论 ARIMA模型能够较好地对四川省肺结核疫情报告发病数进行拟合和短期预测,肺结核疫情呈现的时空分布特征及其趋势能够为肺结核风险评估、重点地区和重点人群筛选等综合防控提供参考依据。     

Estimates of maternal mortality for 1995

OBJECTIVE: To present estimates of maternal mortality in 188 countries, areas, and territories for 1995 using methodologies that attempt to improve comparability. METHODS: For countries having data directly relevant to the measurement of maternal mortality, a variety of adjustment procedures can be applied depending on the nature of the data used. Estimates for countries lacking relevant data may be made using a statistical model fitted to the information from countries that have data judged to be of good quality. Rather than estimate the Maternal Mortality Ratio (MMRatio) directly, this model estimates the proportion of deaths of women of reproductive age that are due to maternal causes. Estimates of the number of maternal deaths are then obtained by applying this proportion to the best available figure of the total number of deaths among women of reproductive age. FINDINGS: On the basis of this exercise, we have obtained a global estimate of 515,000 maternal deaths in 1995, with a worldwide MMRatio of 397 per 100,000 live births. The differences, by region, were very great, with over half (273,000 maternal deaths) occurring in Africa (MMRatio: > 1000 per 100,000), compared with a total of only 2000 maternal deaths in Europe (MMRatio: 28 per 100,000). Lower and upper uncertainty bounds were also estimated, on the basis of which the global MMRatio was unlikely to be less than 234 or more than 635 per 100,000 live births. These uncertainty bounds and those of national estimates are so wide that comparisons between countries must be made with caution, and no valid conclusions can be drawn about trends over a period of time. CONCLUSION: The MMRatio is thus an imperfect indicator of reproductive health because it is hard to measure precisely. It is preferable to use process indicators for comparing reproductive health between countries or across time periods, and for monitoring and evaluation purposes.     

A spatio-temporal absorbing state model for disease and syndromic surveillance

Reliable surveillance models are an important tool in public health because they aid in mitigating disease outbreaks, identify where and when disease outbreaks occur, and predict future occurrences. Although many statistical models have been devised for surveillance purposes, none are able to simultaneously achieve the important practical goals of good sensitivity and specificity, proper use of covariate information, inclusion of spatio-temporal dynamics, and transparent support to decision-makers. In an effort to achieve these goals, this paper proposes a spatio-temporal conditional autoregressive hidden Markov model with an absorbing state. The model performs well in both a large simulation study and in an application to influenza/pneumonia fatality data.     

Statistical methods for linking health, exposure, and hazards

The Environmental Public Health Tracking Network (EPHTN) proposes to link environmental hazards and exposures to health outcomes. Statistical methods used in case-control and cohort studies to link health outcomes to individual exposure estimates are well developed. However, reliable exposure estimates for many contaminants are not available at the individual level. In these cases, exposure/hazard data are often aggregated over a geographic area, and ecologic models are used to relate health outcome and exposure/hazard. Ecologic models are not without limitations in interpretation. EPHTN data are characteristic of much information currently being collected--they are multivariate, with many predictors and response variables, often aggregated over geographic regions (small and large) and correlated in space and/or time. The methods to model trends in space and time, handle correlation structures in the data, estimate effects, test hypotheses, and predict future outcomes are relatively new and without extensive application in environmental public health. In this article we outline a tiered approach to data analysis for EPHTN and review the use of standard methods for relating exposure/hazards, disease mapping and clustering techniques, Bayesian approaches, Markov chain Monte Carlo methods for estimation of posterior parameters, and geostatistical methods. The advantages and limitations of these methods are discussed.     

Individual-level space-time analyses of emergency department data using generalized additive modeling

ABSTRACT: BACKGROUND: Although daily emergency department (ED) data is a source of information that often includes residence, its potential for space-time analyses at the individual level has not been fully explored. We propose that ED data collected for surveillance purposes can also be used to inform spatial and temporal patterns of disease using generalized additive models (GAMs). This paper describes the methods for adapting GAMs so they can be applied to ED data. METHODS: GAMs are an effective approach for modeling spatial and temporal distributions of point-wise data, producing smoothed surfaces of continuous risk while adjusting for confounders. In addition to disease mapping, the method allows for global and pointwise hypothesis testing and selection of statistically optimum degree of smoothing using standard statistical software. We applied a two-dimensional GAM for location to ED data of overlapping calendar time using a locally-weighted regression smoother. To illustrate our methods, we investigated the association between participants' address and the risk of gastrointestinal illness in Cape Cod, Massachusetts over time. RESULTS: The GAM space-time analyses simultaneously smooth in units of distance and time by using the optimum degree of smoothing to create data frames of overlapping time periods and then spatially analyzing each data frame. When resulting maps are viewed in series, each data frame contributes a movie frame, allowing us to visualize changes in magnitude, geographic size, and location of elevated risk smoothed over space and time. In our example data, we observed an underlying geographic pattern of gastrointestinal illness with risks consistently higher in the eastern part of our study area over time and intermittent variations of increased risk during brief periods. CONCLUSIONS: Spatial-temporal analysis of emergency department data with GAMs can be used to map underlying disease risk at the individual-level and view changes in geographic patterns of disease over time while accounting for multiple confounders. Despite the advantages of GAMs, analyses should be considered exploratory in nature. It is possible that even with a conservative cutoff for statistical significance, results of hypothesis testing may be due to chance. This paper illustrates that GAMs can be adapted to measure geographic trends in public health over time using ED data.     

Modelling to estimate future trends in cancer prevalence

Recent estimates suggest that there are 2 million people in the UK living with or beyond a diagnosis of cancer and there is increasing attention being given to assessing the health and social care needs of this growing population. A simple analytical model has been constructed to estimate future trends in cancer prevalence, using existing prevalence estimates and trends in cancer incidence and survival. Separate estimates are generated for the contribution to future prevalence due to the current prevalent population and that due to future diagnoses. For the current prevalent population, we adopt a conditional survival model incorporating time since diagnosis in addition to age, tumour type and other factors. We discuss the analytical framework that has been constructed and its intended use in providing information that is useful to those planning health and social care services.     

The effect of rate denominator source on US fatal occupational injury rate estimates

BACKGROUND: The Current Population Survey (CPS) is often used as a source of denominator information for analyses of US fatal occupational injury rates. However, given the relatively small sample size of the CPS, analyses that examine the cross-classification of occupation or industry with demographic or geographic characteristics will often produce highly imprecise rate estimates. The Decennial Census of Population provides an alternative source for rate denominator information. We investigate the comparability of fatal injury rates derived using these two sources of rate denominator information. METHODS: Information on fatal occupational injuries that occurred between January 1, 1983 and December 31, 1994 was obtained from the National Traumatic Occupational Fatality surveillance system. Annual estimates of employment by occupation, industry, age, and sex were derived from the CPS, and by linear interpolation and extrapolation from the 1980 and 1990 Census of Population. Fatal injury rates derived using these denominator data were compared. RESULTS: Fatal injury rates calculated using Census-based denominator data were within 10% of rates calculated using CPS data for all major occupation groups except farming/forestry/fishing, for which the fatal injury rate calculated using Census-based denominator data was 24.69/100,000 worker-years and the rate calculated using CPS data was 19.97/100,000 worker-years. The choice of denominator data source had minimal influence on estimates of trends over calendar time in the fatal injury rates for most major occupation and industry groups. CONCLUSIONS: The Census offers a reasonable source for deriving fatal injury rate denominator data in situations where the CPS does not provide sufficiently precise data, although the Census may underestimate the population-at-risk in some industries as a consequence of seasonal variation in employment.     

基于时空核估计方法识别2018年新疆维吾尔自治区手足口病发病的高风险时空区域

  目的  描述2018年1月—2018年12月新疆维吾尔自治区(简称新疆)手足口病(hand, foot and mouth disease, HFMD)发病风险的时空变化,识别新疆HFMD的高风险时空区域,为今后HFMD的防治提供参考依据。  方法  收集2018年1月—2018年12月新疆100个市县(市辖区)HFMD的发病数据,描述数据三间分布的特征;使用本文所提出的时空核估计(spatio-temporal kernel estimation, STKE)方法探测高风险时空区域,并与传统时空扫描分析方法的结果进行对比。  结果  2018年1月—2018年12月新疆100个市县(市辖区)HFMD年发病率为45.01/10万,病例男女性别比例为1.49∶1;0~5岁儿童为主要的发病人群,占全部病例的80.16%;病例主要来源于幼托儿童和散居儿童,分别占总病例数的45.76%和43.94%。基于STKE方法识别出的高风险时空区域主要包含了两个高峰时段(5—7月、9—11月)及一个变化的高风险空间区域(新疆东北部),时空扫描结果验证了该方法的可行性。  结论  新疆HFMD的发生存在时间、空间及人群差异,建议加大对HFMD高峰时段及高发地区的防控力度,增强家庭、幼托机构等对0~5岁婴幼儿的预防措施,以降低HFMD发病风险。     

Bayesian age–period–cohort models with versatile interactions and long‐term predictions: mortality and population in Finland 1878–2050

Age–period–cohort (APC) models are widely used for studying time trends of disease incidence or mortality. Model identifiability has become less of a problem with Bayesian APC models. These models are usually based on random walk (RW1, RW2) smoothing priors. For long and complex time series and for long predicted periods, these models as such may not be adequate. We present two extensions for the APC models. First, we introduce flexible interactions between the age, period and cohort effects based on a two‐dimensional conditional autoregressive smoothing prior on the age/period plane. Our second extension uses autoregressive integrated (ARI) models to provide reasonable long‐term predictions. To illustrate the utility of our model framework, we provide stochastic predictions for the Finnish male and female population, in 2010–2050. For that, we first study and forecast all‐cause male and female mortality in Finland, 1878–2050, showing that using an interaction term is needed for fitting and interpreting the observed data. We then provide population predictions using a cohort component model, which also requires predictions for fertility and migration. As our main conclusion, ARI models have better properties for predictions than the simple RW models do, but mixing these prediction models with RW1 or RW2 smoothing priors for observed periods leads to a model that is not fully consistent. Further research with our model framework will concentrate on using a more consistent model for smoothing and prediction, such as autoregressive integrated moving average models with state‐space methods or Gaussian process priors. Copyright © 2013 John Wiley & Sons, Ltd.