A shared component model for detecting joint and selective clustering of two diseases

The study of spatial variations in disease rates is a common epidemiological approach used to describe the geographical clustering of diseases and to generate hypotheses about the possible 'causes' which could explain apparent differences in risk. Recent statistical and computational developments have led to the use of realistically complex models to account for overdispersion and spatial correlation. However, these developments have focused almost exclusively on spatial modelling of a single disease. Many diseases share common risk factors (smoking being an obvious example) and, if similar patterns of geographical variation of related diseases can be identified, this may provide more convincing evidence of real clustering in the underlying risk surface. We propose a shared component model for the joint spatial analysis of two diseases. The key idea is to separate the underlying risk surface for each disease into a shared and a disease-specific component. The various components of this formulation are modelled simultaneously by using spatial cluster models implemented via reversible jump Markov chain Monte Carlo methods. We illustrate the methodology through an analysis of oral and oesophageal cancer mortality in the 544 districts of Germany, 1986–1990.     

Modelling count data with overdispersion and spatial effects

In this paper we consider regression models for count data allowing for overdispersion in a Bayesian framework. We account for unobserved heterogeneity in the data in two ways. On the one hand, we consider more flexible models than a common Poisson model allowing for overdispersion in different ways. In particular, the negative binomial and the generalized Poisson (GP) distribution are addressed where overdispersion is modelled by an additional model parameter. Further, zero-inflated models in which overdispersion is assumed to be caused by an excessive number of zeros are discussed. On the other hand, extra spatial variability in the data is taken into account by adding correlated spatial random effects to the models. This approach allows for an underlying spatial dependency structure which is modelled using a conditional autoregressive prior based on Pettitt et al. in Stat Comput 12(4):353–367, (2002). In an application the presented models are used to analyse the number of invasive meningococcal disease cases in Germany in the year 2004. Models are compared according to the deviance information criterion (DIC) suggested by Spiegelhalter et al. in J R Stat Soc B64(4):583–640, (2002) and using proper scoring rules, see for example Gneiting and Raftery in Technical Report no. 463, University of Washington, (2004). We observe a rather high degree of overdispersion in the data which is captured best by the GP model when spatial effects are neglected. While the addition of spatial effects to the models allowing for overdispersion gives no or only little improvement, spatial Poisson models with spatially correlated or uncorrelated random effects are to be preferred over all other models according to the considered criteria.     

Seasonality of hospitalizations due to respiratory diseases: modelling serial correlation all we need is Poisson

The identification of seasonality and trend patterns of the weekly number of hospitalizations may be useful to plan the structure of health care and the vaccination calendar. A generalized additive model with the negative binomial distribution and a generalized additive model with autoregressive terms (GAMAR) and Poisson distribution are fitted including seasonal parameters and nonlinear trend using splines. The GAMAR includes autoregressive terms to take into account the serial correlation, yielding correct standard errors and reducing overdispersion. For the number of hospitalizations of people older than 60 years due to respiratory diseases in São Paulo city, both models present similar estimates but the Poisson-GAMAR presents uncorrelated residuals, no overdispersion and provides smaller confidence intervals for the weekly percentage changes. Forecasts for the next year based on both models are obtained by simulation and the Poisson-GAMAR presented better performance.     

Regression of area mortality rates on explanatory variables: what weighting is appropriate?

"One can often gain insight into the aetiology of a disease by relating mortality rates in different areas to explanatory variables. Multiple regression techniques are usually employed, but unweighted least squares may be inappropriate if the areas vary in population size. Also, a fully weighted regression, with weights inversely proportional to binomial sampling variances, is usually too extreme. This paper proposes an intermediate solution via maximum likelihood which takes account of three sources of variation in death rates: sampling error, explanatory variables and unexplained differences between areas. The method is also adapted for logit (death rates), standardized mortality ratios (SMRs) and log (SMRs). Two [United Kingdom] examples are presented."     

Hierarchical Bayesian bivariate disease mapping: analysis of children and adults asthma visits to hospital

In spatial epidemiology, detecting areas with high ratio of disease is important as it may lead to identifying risk factors associated with disease. This in turn may lead to further epidemiological investigations into the nature of disease. Disease mapping studies have been widely performed with considering only one disease in the estimated models. Simultaneous modelling of different diseases can also be a valuable tool both from the epidemiological and also from the statistical point of view. In particular, when we have several measurements recorded at each spatial location, one can consider multivariate models in order to handle the dependence among the multivariate components and the spatial dependence between locations. In this paper, spatial models that use multivariate conditionally autoregressive smoothing across the spatial dimension are considered. We study the patterns of incidence ratios and identify areas with consistently high ratio estimates as areas for further investigation. A hierarchical Bayesian approach using Markov chain Monte Carlo techniques is employed to simultaneously examine spatial trends of asthma visits by children and adults to hospital in the province of Manitoba, Canada, during 2000–2010.     

Empirical Bayes estimates for correlated hierarchical data with overdispersion

An extension of the generalized linear mixed model was constructed to simultaneously accommodate overdispersion and hierarchies present in longitudinal or clustered data. This so‐called combined model includes conjugate random effects at observation level for overdispersion and normal random effects at subject level to handle correlation, respectively. A variety of data types can be handled in this way, using different members of the exponential family. Both maximum likelihood and Bayesian estimation for covariate effects and variance components were proposed. The focus of this paper is the development of an estimation procedure for the two sets of random effects. These are necessary when making predictions for future responses or their associated probabilities. Such (empirical) Bayes estimates will also be helpful in model diagnosis, both when checking the fit of the model as well as when investigating outlying observations. The proposed procedure is applied to three datasets of different outcome types. Copyright © 2014 John Wiley & Sons, Ltd.     

Score test of homogeneity for survival data

If follow-up is made for subjects which are grouped into units, such as familial or spatial units then it may be interesting to test whether the groups are homogeneous (or independent for given explanatory variables). The effect of the groups is modelled as random and we consider a frailty proportional hazards model which allows to adjust for explanatory variables. We derive the score test of homogeneity from the marginal partial likelihood and it turns out to be the sum of a pairwise correlation term of martingale residuals and an overdispersion term. In the particular case where the sizes of the groups are equal to one, this statistic can be used for testing overdispersion. The asymptotic variance of this statistic is derived using counting process arguments. An extension to the case of several strata is given. The resulting test is computationally simple; its use is illustrated using both simulated and real data. In addition a decomposition of the score statistic is proposed as a sum of a pairwise correlation term and an overdispersion term. The pairwise correlation term can be used for constructing a statistic more robust to departure from the proportional hazard model, and the overdispesion term for constructing a test of fit of the proportional hazard model.     

Hierarchical Bayes estimation of mortality rates for disease mapping

Mapping of incidence rates or mortality rates (relative risks) from diseases like cancer and leukemia is of primary importance in an epidemiological study. The usual procedure is to map the standardized mortality ratio (SMR) across different geographical regions. Direct use of SMR may not be worthwhile, particularly for small places, as it does not take into account the high variability for different population sizes over different regions and the spatial patterns of the regions under study. In this paper a hierarchical Bayes approach is presented in smoothing the relative risks and providing the measures of uncertainty associated with these estimates of relative risks.     

Non-stationary partition modeling of geostatistical data for malaria risk mapping

The most common assumption in geostatistical modeling of malaria is stationarity, that is spatial correlation is a function of the separation vector between locations. However, local factors (environmental or human-related activities) may influence geographical dependence in malaria transmission differently at different locations, introducing non-stationarity. Ignoring this characteristic in malaria spatial modeling may lead to inaccurate estimates of the standard errors for both the covariate effects and the predictions. In this paper, a model based on random Voronoi tessellation that takes into account non-stationarity was developed. In particular, the spatial domain was partitioned into sub-regions (tiles), a stationary spatial process was assumed within each tile and between-tile correlation was taken into account. The number and configuration of the sub-regions are treated as random parameters in the model and inference is made using reversible jump Markov chain Monte Carlo simulation. This methodology was applied to analyze malaria survey data from Mali and to produce a country-level smooth map of malaria risk.     

Creating a reference to a complex emergency situation using time series methods: war in Guinea-Bissau 1998-1999

Impacts of complex emergencies or relief interventions have often been evaluated by absolute mortality compared to international standardized mortality rates. A better evaluation would be to compare with local baseline mortality of the affected populations. A projection of population-based survival data into time of emergency or intervention based on information from before the emergency may create a local baseline reference. We find a log-transformed Gaussian time series model where standard errors of the estimated rates are included in the variance to have the best forecasting capacity. However, if time-at-risk during the forecasted period is known then forecasting might be done using a Poisson time series model with overdispersion. Whatever, the standard error of the estimated rates must be included in the variance of the model either in an additive form in a Gaussian model or in a multiplicative form by overdispersion in a Poisson model. Data on which the forecasting is based must be modelled carefully concerning not only calendar-time trends but also periods with excessive frequency of events (epidemics) and seasonal variations to eliminate residual autocorrelation and to make a proper reference for comparison, reflecting changes over time during the emergency. Hence, when modelled properly it is possible to predict a reference to an emergency-affected population based on local conditions. We predicted childhood mortality during the war in Guinea-Bissau 1998-1999. We found an increased mortality in the first half-year of the war and a mortality corresponding to the expected one in the last half-year of the war.     

A spatially adaptive conditional autoregressive prior for area health data

Different priors have been suggested to reflect spatial dependence in area health outcomes or in spatial regression residuals. However, to assume that residuals demonstrate spatial clustering only is a strong prior belief and alternatives have been suggested. A scheme suggested by Leroux et al. [B. Leroux, X. Lei, N. Breslow, Estimation of disease rates in small areas: A new mixed model for spatial dependence, in: M. Halloran, D. Berry (Eds.), Statistical Models in Epidemiology, the Environment and Clinical Trials, Springer-Verlag, New York, 1999, pp. 135–178] involves a single set of random effects and a spatial correlation parameter with extreme values corresponding to pure spatial and pure unstructured residual variation. This paper considers a spatially adaptive extension of that prior to reflect the fact that the appropriate mix between local and global smoothing may not be constant across the region being studied. Local smoothing will not be indicated when an area is disparate from its neighbours (e.g. in terms of social or environmental risk factors for the health outcome being considered). The prior for varying spatial correlation parameters may be based on a regression structure which includes possible observed sources of disparity between neighbours. A case study considers probabilities of long term illness in 133 small areas in NE London, with disparities based on a measure of socio-economic deprivation.     

Geographically Weighted Negative Binomial Regression—incorporating overdispersion

Global regression assumes that a single model adequately describes all parts of a study region. However, the heterogeneity in the data may be sufficiently strong that relationships between variables can not be spatially constant. In addition, the factors involved are often sufficiently complex that it is difficult to identify them in the form of explanatory variables. As a result Geographically Weighted Regression (GWR) was introduced as a tool for the modeling of non-stationary spatial data. Using kernel functions, the GWR methodology allows the model parameters to vary spatially and produces non-parametric surfaces of their estimates. To model count data with overdispersion, it is more appropriate to use a negative binomial distribution instead of a Poisson distribution. Therefore, we propose the Geographically Weighted Negative Binomial Regression (GWNBR) method for the modeling of data with overdispersion. The results obtained using simulated and real data show the superiority of this method for the modeling of non-stationary count data with overdispersion compared with competing models, such as global regressions, e.g., Poisson and negative binomial and Geographically Weighted Poisson Regression (GWPR). Moreover, we illustrate that these competing models are special cases of the more robust model GWNBR.     

Childhood mortality spatial distribution in Ethiopia

The risk of a child dying before completing five years of age is highest in Sub-Saharan African countries. But Child mortality rates have shown substantial decline in Ethiopia. For this study, the 2000, 2005 and 2011 Ethiopian Demographic Survey (EDHS) was used. Generalized linear mixed model with spatial covariance structure was adapted. The model allowed for spatial correlation, and leads to the more realistic estimate for under-five mortality risk factors. The analysis showed that the risk of under-five mortality shows decline in years. But, some regions showed increase in years. The study highlights the need to implement better education for family planning and child care to improve the under-five mortality situation in some administrative areas.     

Spatial measurement error in infectious disease models

Individual-level models (ILMs) for infectious disease can be used to model disease spread between individuals while taking into account important covariates. One important covariate in determining the risk of infection transfer can be spatial location. At the same time, measurement error is a concern in many areas of statistical analysis, and infectious disease modelling is no exception. In this paper, we are concerned with the issue of measurement error in the recorded location of individuals when using a simple spatial ILM to model the spread of disease within a population. An ILM that incorporates spatial location random effects is introduced within a hierarchical Bayesian framework. This model is tested upon both simulated data and data from the UK 2001 foot-and-mouth disease epidemic. The ability of the model to successfully identify both the spatial infection kernel and the basic reproduction number (R ₀) of the disease is tested.     

Latent variable model for mixed correlated power series and ordinal longitudinal responses with non ignorable missing values

We propose a joint model based on a latent variable for analyzing mixed power series and ordinal longitudinal data with and without missing values. A bivariate probit regression model is used for the missing mechanisms. Random effects are used to take into account the correlation between longitudinal responses. A full likelihood-based approach is used to yield maximum-likelihood estimates of the model parameters. Our model is applied to a medical data set, obtained from an observational study on women where the correlated responses are the ordinal response of osteoporosis of the spine and the power series response of the number of joint damages. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms and overdispersion of the model on likelihood displacement.     

A parametric model for heterogeneity in paired Poisson counts

We present a model for data in the form of matched pairs of counts. Our work is motivated by a problem in fission-track analysis, where the determination of a crystal's age is based on the ratio of counts of spontaneous and induced tracks. It is often reasonable to assume that the counts follow a Poisson distribution, but typically they are overdispersed and there exists a positive correlation between the numbers of spontaneous and induced tracks in the same crystal. We propose a model that allows for both overdispersion and correlation by assuming that the mean densities follow a bivariate Wishart distribution. Our model is quite general, having the usual negative-binomial and Poisson models as special cases. We propose a maximum-likelihood estimation method based on a stochastic implementation of the EM algorithm, and we derive the asymptotic standard errors of the parameter estimates. We illustrate the method with a data set of fission-track counts in matched areas of zircon crystals.     

Analysis of hypoglycemic events using negative binomial models

Negative binomial regression is a standard model to analyze hypoglycemic events in diabetes clinical trials. Adjusting for baseline covariates could potentially increase the estimation efficiency of negative binomial regression. However, adjusting for covariates raises concerns about model misspecification, in which the negative binomial regression is not robust because of its requirement for strong model assumptions. In some literature, it was suggested to correct the standard error of the maximum likelihood estimator through introducing overdispersion, which can be estimated by the Deviance or Pearson Chi‐square. We proposed to conduct the negative binomial regression using Sandwich estimation to calculate the covariance matrix of the parameter estimates together with Pearson overdispersion correction (denoted by NBSP). In this research, we compared several commonly used negative binomial model options with our proposed NBSP. Simulations and real data analyses showed that NBSP is the most robust to model misspecification, and the estimation efficiency will be improved by adjusting for baseline hypoglycemia. Copyright © 2013 John Wiley & Sons, Ltd.     

企业财务风险预测模型实证比较分析

企业财务风险一直是风险管理理论和实务界关心的热点话题。运用判别分析和计量经济方法对重庆市某商业银行的461个样本企业2002-2005年的违约特征进行实证检验和预测。结果发现最重要的决定变量是资产负责率、酸性试验比率、资产净利率等7个财务比率以及企业所处的产业部门,考虑了异方差性的probit模型有更好的预测能力。     

Poisson regression models with errors-in-variables: implication and treatment

Overdispersion has been a common phenomenon in count data and usually treated with the negative binomial model. This paper shows that measurement errors in covariates in general also lead to overdispersion on the observed data if the true data generating process is indeed the Poisson regression. This kind of overdispersion cannot be treated using the negative binomial model, as otherwise, biases will occur. To provide consistent estimates, we propose a new type of corrected score estimator assuming that the distribution of the latent variables is known. The consistency and asymptotic normality of the proposed estimator are established. Simulation results show that this estimator has good finite sample performance. We also illustrate that the Akaike information criterion and Bayesian information criterion work well for selecting the correct model if the true model is the errors-in-variables Poisson regression.     

Efficient regression modeling for correlated and overdispersed count data

Abstract

The objective of this paper is to propose an efficient estimation procedure in a marginal mean regression model for longitudinal count data and to develop a hypothesis test for detecting the presence of overdispersion. We extend the matrix expansion idea of quadratic inference functions to the negative binomial regression framework that entails accommodating both the within-subject correlation and overdispersion issue. Theoretical and numerical results show that the proposed procedure yields a more efficient estimator asymptotically than the one ignoring either the within-subject correlation or overdispersion. When the overdispersion is absent in data, the proposed method might hinder the estimation efficiency in practice, yet the Poisson regression based regression model is fitted to the data sufficiently well. Therefore, we construct the hypothesis test that recommends an appropriate model for the analysis of the correlated count data. Extensive simulation studies indicate that the proposed test can identify the effective model consistently. The proposed procedure is also applied to a transportation safety study and recommends the proposed negative binomial regression model.