A statistical framework for ecological and aggregate studies

Inference from studies that make use of data at the level of the area, rather than at the level of the individual, is more difficult for a variety of reasons. Some of these difficulties arise because frequently exposures (including confounders) vary within areas. In the most basic form of ecological study the outcome measure is regressed against a simple area level summary of exposure. In the aggregate data approach a survey of exposures and confounders is taken within each area. An alternative approach is to assume a parametric form for the within-area exposure distribution. We provide a framework within which ecological and aggregate data studies may be viewed, and we review some approaches to inference in such studies, clarifying the assumptions on which they are based. General strategies for analysis are provided including an estimator based on Monte Carlo integration that allows inference in the case of a general risk–exposure model. We also consider the implications of the introduction of random effects, and the existence of confounding and errors in variables.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

Reducing bias in ecological studies: an evaluation of different methodologies

Summary.  Statistical methods of ecological analysis that attempt to reduce ecological bias are empirically evaluated to determine in which circumstances each method might be practicable. The method that is most successful at reducing ecological bias is stratified ecological regression. It allows individual level covariate information to be incorporated into a stratified ecological analysis, as well as the combination of disease and risk factor information from two separate data sources, e.g. outcomes from a cancer registry and risk factor information from the census sample of anonymized records data set. The aggregated individual level model compares favourably with this model but has convergence problems. In addition, it is shown that the large areas that are covered by local authority districts seem to reduce between-area variability and may therefore not be as informative as conducting a ward level analysis. This has policy implications because access to ward level data is restricted.     

Models, assumptions and model checking in ecological regressions

Ecological regression is based on assumptions that are untestable from aggregate data. However, these assumptions seem more questionable in some applications than in others. There has been some research on implicit models of individual data underlying aggregate ecological regression modelling. We discuss ways in which these implicit models can be checked from aggregate data. We also explore the differences in applications of ecological regressions in two examples: estimating the effect of radon on lung cancer in the United States and estimating voting patterns for different ethnic groups in New York City.     

On the clustering term in ecological analysis: how do different prior specifications affect results?

We study how different prior assumptions on the spatially structured heterogeneity term of the convolution hierarchical Bayesian model for spatial disease data could affect the results of an ecological analysis when response and exposure exhibit a strong spatial pattern. We show that in this case the estimate of the regression parameter could be strongly biased, both by analyzing the association between lung cancer mortality and education level on a real dataset and by a simulation experiment. The analysis is based on a hierarchical Bayesian model with a time dependent covariate in which we allow for a latency period between exposure and mortality, with time and space random terms and misaligned exposure-disease data.     

Logistic regression in meta-analysis using aggregate data

We derived two methods to estimate the logistic regression coefficients in a meta-analysis when only the 'aggregate' data (mean values) from each study are available. The estimators we proposed are the discriminant function estimator and the reverse Taylor series approximation. These two methods of estimation gave similar estimators using an example of individual data. However, when aggregate data were used, the discriminant function estimators were quite different from the other two estimators. A simulation study was then performed to evaluate the performance of these two estimators as well as the estimator obtained from the model that simply uses the aggregate data in a logistic regression model. The simulation study showed that all three estimators are biased. The bias increases as the variance of the covariate increases. The distribution type of the covariates also affects the bias. In general, the estimator from the logistic regression using the aggregate data has less bias and better coverage probabilities than the other two estimators. We concluded that analysts should be cautious in using aggregate data to estimate the parameters of the logistic regression model for the underlying individual data.     

Maximum Likelihood Estimators in Regression Models for Error‐prone Group Testing Data

Since Dorfman's seminal work on the subject, group testing has been widely adopted in epidemiological studies. In Dorfman's context of detecting syphilis, group testing entails pooling blood samples and testing the pools, as opposed to testing individual samples. A negative pool indicates all individuals in the pool free of syphilis antigen, whereas a positive pool suggests one or more individuals carry the antigen. With covariate information collected, researchers have considered regression models that allow one to estimate covariate‐adjusted disease probability. We study maximum likelihood estimators of covariate effects in these regression models when the group testing response is prone to error. We show that, when compared with inference drawn from individual testing data, inference based on group testing data can be more resilient to response misclassification in terms of bias and efficiency. We provide valuable guidance on designing the group composition to alleviate adverse effects of misclassification on statistical inference.     

Deprivation, ill-health and the ecological fallacy

The use of ecological studies in explaining the relationship between deprivation and ill-health is widespread in many health applications. However, inferences drawn from these studies about individuals are susceptible to serious bias known as the ecological fallacy. Our paper demonstrates the ecological fallacy effect in this context but also shows how it can be considerably reduced by taking into account different population structures at the aggregate level. Two regression analyses of limiting long-term illness are performed, one at the individual level and one at the electoral ward level, using the 1991 UK census sample of anonymized records and the small area statistics. The analyses compare several measures of deprivation including the standard Carstairs index, with the separate variables which make up the indices, to determine their effectiveness in explaining rates of illness. Two of the deprivation scores are constructed using latent variable modelling techniques which enable a score to be generated at the individual level as well as at the ward level. It is shown that, given the right choice of socioeconomic variables and taking into account the age structure of the population, it should be possible to construct a single aggregate deprivation index that will explain most of the variation in rates of illness across the study region.     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

Ecological Bias: Use of Maximum-Entropy Approximations

The focus of geographical studies in epidemiology has recently moved towards looking for effects of exposures based on data taken at local levels of aggregation (i.e. small areas). This paper investigates how regression coefficients measuring covariate effects at the point level are modified under aggregation. Changing the level of aggregation can lead to completely different conclusions about exposure–effect relationships, a phenomenon often referred to as ecological bias. With partial knowledge of the within‐area distribution of the exposure variable, the notion of maximum entropy can be used to approximate that part of the distribution that is unknown. From the approximation, an expression for the ecological bias is obtained; simulations and an example show that the maximum‐entropy approximation is often better than other commonly used approximations.     

Stratified proportional subdistribution hazards model with covariate-adjusted censoring weight for case-cohort studies

The case-cohort study design is widely used to reduce cost when collecting expensive covariates in large cohort studies with survival or competing risks outcomes. A case-cohort study dataset consists of two parts: (a) a random sample and (b) all cases or failures from a specific cause of interest. Clinicians often assess covariate effects on competing risks outcomes. The proportional subdistribution hazards model directly evaluates the effect of a covariate on the cumulative incidence function under the non-covariate-dependent censoring assumption for the full cohort study. However, the non-covariate-dependent censoring assumption is often violated in many biomedical studies. In this article, we propose a proportional subdistribution hazards model for case-cohort studies with stratified data with covariate-adjusted censoring weight. We further propose an efficient estimator when extra information from the other causes is available under case-cohort studies. The proposed estimators are shown to be consistent and asymptotically normal. Simulation studies show (a) the proposed estimator is unbiased when the censoring distribution depends on covariates and (b) the proposed efficient estimator gains estimation efficiency when using extra information from the other causes. We analyze a bone marrow transplant dataset and a coronary heart disease dataset using the proposed method.     

A dependent Dirichlet process model for survival data with competing risks

In this paper, we first propose a dependent Dirichlet process (DDP) model using a mixture of Weibull models with each mixture component resembling a Cox model for survival data. We then build a Dirichlet process mixture model for competing risks data without regression covariates. Next we extend this model to a DDP model for competing risks regression data by using a multiplicative covariate effect on subdistribution hazards in the mixture components. Though built on proportional hazards (or subdistribution hazards) models, the proposed nonparametric Bayesian regression models do not require the assumption of constant hazard (or subdistribution hazard) ratio. An external time-dependent covariate is also considered in the survival model. After describing the model, we discuss how both cause-specific and subdistribution hazard ratios can be estimated from the same nonparametric Bayesian model for competing risks regression. For use with the regression models proposed, we introduce an omnibus prior that is suitable when little external information is available about covariate effects. Finally we compare the models’ performance with existing methods through simulations. We also illustrate the proposed competing risks regression model with data from a breast cancer study. An R package “DPWeibull” implementing all of the proposed methods is available at CRAN.

     

A Flexible Approach to Time-varying Coefficients in the Cox Regression Setting

Research on methods for studying time-to-event data (survival analysis) has been extensive in recent years. The basic model in use today represents the hazard function for an individual through a proportional hazards model (Cox, 1972). Typically, it is assumed that a covariate's effect on the hazard function is constant throughout the course of the study. In this paper we propose a method to allow for possible deviations from the standard Cox model, by allowing the effect of a covariate to vary over time. This method is based on a dynamic linear model. We present our method in terms of a Bayesian hierarchical model. We fit the model to the data using Markov chain Monte Carlo methods. Finally, we illustrate the approach with several examples. This revised version was published online in July 2006 with corrections to the Cover Date.     

Regression analysis of competing risks data via semi-parametric additive hazard model

When the subjects in a study possess different demographic and disease characteristics and are exposed to more than one types of failure, a practical problem is to assess the covariate effects on each type of failure as well as on all-cause failure. The most widely used method is to employ the Cox models on each cause-specific hazard and the all-cause hazard. It has been pointed out that this method causes the problem of internal inconsistency. To solve such a problem, the additive hazard models have been advocated. In this paper, we model each cause-specific hazard with the additive hazard model that includes both constant and time-varying covariate effects. We illustrate that the covariate effect on all-cause failure can be estimated by the sum of the effects on all competing risks. Using data from a longitudinal study on breast cancer patients, we show that the proposed method gives simple interpretation of the final results, when the primary covariate effect is constant in the additive manner on each cause-specific hazard. Based on the given additive models on the cause-specific hazards, we derive the inferences for the adjusted survival and cumulative incidence functions.     

Estimating driver crash risks based on the extended Bradley–Terry model: an induced exposure method

Quantifying driver crash risks has been difficult because the exposure data are often incompatible with crash frequency data. Induced exposure methods provide a promising idea that a relative measurement of driver crash risks can be derived solely from crash frequency data. This paper describes an application of the extended Bradley–Terry model for paired preferences to estimating driver crash risks. We estimate the crash risk for driver groups defined by driver–vehicle characteristics from log-linear models in terms of a set of relative risk scores by using only crash frequency data. Illustrative examples using police-reported crash data from Hawaii are presented.     

Meta-analysis using multilevel models with an application to the study of class size effects

Meta-analysis is formulated as a special case of a multilevel (hierarchical data) model in which the highest level is that of the study and the lowest level that of an observation on an individual respondent. Studies can be combined within a single model where the responses occur at different levels of the data hierarchy and efficient estimates are obtained. An example is given from studies of class sizes and achievement in schools, where study data are available at the aggregate level in terms of overall mean values for classes of different sizes, and also at the student level.     

Hierarchical related regression for combining aggregate and individual data in studies of socio-economic disease risk factors

Summary.  To obtain information about the contribution of individual and area level factors to population health, it is desirable to use both data collected on areas, such as censuses, and on individuals, e.g. survey and cohort data. Recently developed models allow us to carry out simultaneous regressions on related data at the individual and aggregate levels. These can reduce 'ecological bias' that is caused by confounding, model misspecification or lack of information and increase power compared with analysing the data sets singly. We use these methods in an application investigating individual and area level sociodemographic predictors of the risk of hospital admissions for heart and circulatory disease in London. We discuss the practical issues that are encountered in this kind of data synthesis and demonstrate that this modelling framework is sufficiently flexible to incorporate a wide range of sources of data and to answer substantive questions. Our analysis shows that the variations that are observed are mainly attributable to individual level factors rather than the contextual effect of deprivation.     

基于STLS模型的空气质量生态价值估算——以云南省为例

文章扩展了条件价值评估方法在中国省域空气质量生态价值评估中的应用,以云南省为研究对象进行空气质量生态价值估算。为提高估计结果的可靠性,在Tobit模型估计时采用对称修剪最小二乘法;同时,考虑到个体效应,首次将AQI数据作为控制变量进行模型估计。研究结果表明,2016年云南省空气质量生态价值为550.91亿元;家庭月均收入、是否曾受空气质量影响因素对支付意愿的影响显著;教育程度、健康状况、家庭月均收入、是否曾受空气质量影响、是否为云南省常住人口因素对支付意愿金额具有显著性影响。     

Joint Modeling of Event Time and Nonignorable Missing Longitudinal Data

Survival studies usually collect on each participant, both duration until some terminal event and repeated measures of a time-dependent covariate. Such a covariate is referred to as an internal time-dependent covariate. Usually, some subjects drop out of the study before occurence of the terminal event of interest. One may then wish to evaluate the relationship between time to dropout and the internal covariate. The Cox model is a standard framework for that purpose. Here, we address this problem in situations where the value of the covariate at dropout is unobserved. We suggest a joint model which combines a first-order Markov model for the longitudinaly measured covariate with a time-dependent Cox model for the dropout process. We consider maximum likelihood estimation in this model and show how estimation can be carried out via the EM-algorithm. We state that the suggested joint model may have applications in the context of longitudinal data with nonignorable dropout. Indeed, it can be viewed as generalizing Diggle and Kenward's model (1994) to situations where dropout may occur at any point in time and may be censored. Hence we apply both models and compare their results on a data set concerning longitudinal measurements among patients in a cancer clinical trial.     

Bayesian joint modeling of correlated counts data with application to adverse birth outcomes

In disease mapping, health outcomes measured at the same spatial locations may be correlated, so one can consider joint modeling the multivariate health outcomes accounting for their dependence. The general approaches often used for joint modeling include shared component models and multivariate models. An alternative way to model the association between two health outcomes, when one outcome can naturally serve as a covariate of the other, is to use ecological regression model. For example, in our application, preterm birth (PTB) can be treated as a predictor for low birth weight (LBW) and vice versa. Therefore, we proposed to blend the ideas from joint modeling and ecological regression methods to jointly model the relative risks for LBW and PTBs over the health districts in Saskatchewan, Canada, in 2000–2010. This approach is helpful when proxy of areal-level contextual factors can be derived based on the outcomes themselves when direct information on risk factors are not readily available. Our results indicate that the proposed approach improves the model fit when compared with the conventional joint modeling methods. Further, we showed that when no strong spatial autocorrelation is present, joint outcome modeling using only independent error terms can still provide a better model fit when compared with the separate modeling.