Space–time clustering of case–control data with residential histories: insights into empirical induction periods, age-specific susceptibility, and calendar year-specific effects

Our research group recently developed Q-statistics for evaluating space–time clustering in case–control studies with residential histories. This technique relies on time-dependent nearest-neighbor relationships to examine clustering at any moment in the life-course of the residential histories of cases relative to that of controls. In addition, in place of the widely used null hypothesis of spatial randomness, each individual’s probability of being a case is based instead on his/her risk factors and covariates. In this paper, we extend this approach to illustrate how alternative temporal orientations (e.g., years prior to diagnosis/recruitment, participant’s age, and calendar year) influence a spatial clustering pattern. These temporal orientations are valuable for shedding light on the duration of time between clustering and subsequent disease development (known as the empirical induction period), and for revealing age-specific susceptibility windows and calendar year-specific effects. An ongoing population-based bladder cancer case–control study is used to demonstrate this approach. Data collection is currently incomplete and therefore no inferences should be drawn; we analyze these data to demonstrate these novel methods. Maps of space–time clustering of bladder cancer cases are presented using different temporal orientations while accounting for covariates and known risk factors. This systematic approach for evaluating space–time clustering has the potential to generate novel hypotheses about environmental risk factors and provides insights into empirical induction periods, age-specific susceptibility, and calendar year-specific effects.     

A comparison of methods for spatial relative risk mapping of human neural tube defects

Birth defects are a major cause of infant mortality and disability in many parts of the world. Yet the etiology of neural tube defects (NTDs), the most common types of birth defects, is still unknown. The construction and analysis of maps of disease incidence data can help explain the geographical distribution of NTDs and can point to possible environmental causes of these birth defects. We compared two methods of mapping spatial relative risk of NTDs: (1) hierarchical Bayesian model, and (2) Spatial filtering method. Heshun county, which has the highest rate of NTDs in China, was selected as the region of interest. Both methods were used to produce a risk map of NTDs for rural Heshun for 1998–2001. Hierarchical Bayesian model estimated the relative risk for any given village in Heshun by “borrowing” strength from other villages in the study region. It did not remove all the random spatial noise in the rude disease rate. There were several areas of high incidence scattered around its risk map with no readily apparent pattern. The spatial filtering method calculated the relative risk for all villages based on a series of circulars. The risk map from the spatial filtering method revealed some spatial clusters of NTDs in Heshun. These two methods differed in their ability to map the spatial relative risk of NTDs. Distributional assumption of relative risk and the target of the risk assessment should be taken into consideration when choosing which method to use.     

Integration of principal components analysis and cellular automata for spatial decisionmaking and urban simulation

This paper discusses the issues about the correlation of spatial variables during spatial decisionmaking using multicriteria evaluation (MCE) and cellular automata (CA). The correlation of spatial variables can cause the malfunction of MCE. In urban simulation, spatial factors often exhibit a high degree of correlation which is considered as an undesirable property for MCE. This study uses principal components analysis (PCA) to remove data redundancy among a large set of spatial variables and determine ‘ideal points’ for land development. PCA is integrated with cellular automata and geographical information systems (GIS) for the simulation of idealized urban forms for planning purposes.     

Stochastic multi-site generation of daily rainfall occurrence in south Florida

This paper presents a stochastic model to generate daily rainfall occurrences at multiple gauging stations in south Florida. The model developed in this study is a space–time model that takes into account the spatial as well as temporal dependences of daily rainfall occurrence based on a chain-dependent process. In the model, a Markovian method was used to represent the temporal dependence of daily rainfall occurrence and a direct acyclic graph (DAG) method was introduced to encode the spatial dependence of daily rainfall occurrences among gauging stations. The DAG method provides an optimal sequence of generation by maximizing the spatial dependence index of daily rainfall occurrences over the region. The proposed space–time model shows more promising performance in generating rainfall occurrences in time and space than the conventional Markov type model. The space–time model well represents the temporal as well as the spatial dependence of daily rainfall occurrences, which can reduce the complexity in the generation of daily rainfall amounts.     

Non-stationary spatial covariance structure estimation in oversampled domains by cluster differences scaling with spatial constraints

In the analysis of spatiotemporal processes underlying environmental studies, the estimation of the non-stationary spatial covariance structure is a well known issue in which multidimensional scaling (MDS) provides an important methodological approach (Sampson and Guttorp in J Am Stat Assoc 87:108–119, 1992). It is also well known that approximating dispersion by a non-metric MDS procedure offers, in general, low precision when accurate differences in spatial dispersion are needed for interpolation purposes, specially if a low dimensional configuration is employed besides a high number of stations in oversampled domains. This paper presents a modification, consisting of including geographical spatial constraints, of Heiser and Groenen’s (Psychometrika 62:63–83, 1997) cluster differences scaling algorithm by which not the original stations but the cluster centres can be represented, while the stations and clusters retain their spatial relationships. A decomposition of the sum of squared dissimilarities into contributions from several sources of variation can be employed for an exploratory diagnosis of the model. Real data are analyzed and differences between several cluster-MDS strategies are discussed.     

Climate-driven shifts in diatom assemblages recorded in annually laminated sediments of Sacrower See (NE Germany)

Sacrower See is a eutrophic lake with annually laminated sediments extending back to A.D. 1868. Analysis of annual layers revealed multi-decadal periods of distinct diatom assemblages at A.D. 1868–1875, 1876–1940, 1941–1978, and 1979–2000. Detrended correspondence analysis performed on individual seasonal sediment layers showed decadal-scale patterns of turnover in the diatom flora. The spring–summer layers showed higher sample scores until the early 1960s, after which the differences with the autumn–winter layers became smaller. Rates-of-change analysis revealed that the seasonal variability in diatom assemblages was higher than the annual changes. Summer diatom rates of change over the period A.D. 1894–1960 was on average higher than for winter, whereas between the 1960s and 1970s the winter rates of change became higher than the summer ones. Redundancy Analyses showed that seasonal temperatures and wind strength were significant explanatory variables for diatom assemblages in both annual and seasonal layers. These results suggest that meteorological changes indirectly affected diatom assemblages via the mixing regime of the lake. A comparison of the diatom rates of change with the amplitude of inter-annual climate change shows a statistically significant correlation for the spring-summer layers in the period of A.D. 1963–2000, showing that the sensitivity of diatom assemblages to meteorological changes has varied over the past century, with a stronger effect on diatoms registered during the past 40 years.     

Partitioned multiobjective risk modeling of carcinogenic compounds in groundwater

Quantifying human cancer risk arising from exposure to contaminated groundwater is complicated by the many hydrogeological, environmental, and toxicological uncertainties involved. In this study, we used Monte Carlo simulation to estimate cancer risk associated with tetrachloroethene (PCE) dissolved in groundwater by linking three separate models for: (1) reactive contaminant transport; (2) human exposure pathways; and (3) the PCE cancer potency factor. The hydrogeologic model incorporates an analytical solution for a one-dimensional advective–dispersive–reactive transport equation to determine the PCE concentration in a water supply well located at a fixed distance from a continuous source. The pathway model incorporates PCE exposure through ingestion, inhalation, and dermal contact. The toxicological model combines epidemiological data from eight rodent bioassays of PCE exposure in the form of a composite cumulative distribution frequency curve for the human PCE cancer potency factor. We assessed the relative importance of individual model variables through their correlation with expected cancer risk calculated in an ensemble of Monte Carlo simulations with 20,000 trials. For the scenarios evaluated, three factors were most highly correlated with cancer risk: (1) the microbiological decay constant for PCE in groundwater, (2) the linear groundwater pore velocity, and (3) the cancer potency factor. We then extended our analysis beyond conventional expected value risk assessment using the partitioned multiobjective risk method (PMRM) to generate expected-value functions conditional to a 1 in 100,000 increased cancer risk threshold. This approach accounts for low probability/high impact outcomes separately from the conventional unconditional expected values. Thus, information on potential worst-case outcomes can be quantified for decision makers. Using PMRM, we evaluated the cost-benefit relationship of implementing several postulated risk management alternatives intended to mitigate the expected and conditional cancer risk. Our results emphasize the importance of hydrogeologic models in risk assessment, but also illustrate the importance of integrating environmental and toxicological uncertainty. When coupled with the PMRM, models integrating uncertainty in transport, exposure, and potency constitute an effective risk assessment tool for use within a risk-based corrective action (RBCA) framework.     

Spatio-temporal modelling of hydro-meteorological derived risk using a Bayesian approach: a case study in Venezuela

Extreme environmental events have considerable impacts on society. Preparation to mitigate or forecast accurately these events is a growing concern for governments. In this regard, policy and decision makers require accurate tools for risk estimation in order to take informed decisions. This work proposes a Bayesian framework for a unified treatment and statistical modeling of the main components of risk: hazard, vulnerability and exposure. Risk is defined as the expected economic loss or population affected as a consequence of a hazard event. The vulnerability is interpreted as the loss experienced by an exposed population due to hazard events. The framework combines data of different spatial and temporal supports. It produces a sequence of temporal risk maps for the domain of interest including a measure of uncertainty for the hazard and vulnerability. In particular, the considered hazard (rainfall) is interpolated from point-based measured rainfall data using a hierarchical spatio-temporal Kriging model, whose parameters are estimated using the Bayesian paradigm. Vulnerability is modeled using zero-inflated distributions with parameters dependent on climatic variables at local and large scales. Exposure is defined as the total population settled in the spatial domain and is interpolated using census data. The proposed methodology was applied to the Vargas state of Venezuela to map the spatio-temporal risk for the period 1970–2006. The framework highlights both high and low risk areas given extreme rainfall events.     

Why do we need and how should we implement Bayesian kriging methods

The spatial prediction methodology that has become known under the heading of kriging is largely based on the assumptions that the underlying random field is Gaussian and the covariance function is exactly known. In practical applications, however, these assumptions will not hold. Beyond Gaussianity of the random field, lognormal kriging, disjunctive kriging, (generalized linear) model-based kriging and trans-Gaussian kriging have been proposed in the literature. The latter approach makes use of the Box–Cox-transform of the data. Still, all the alternatives mentioned do not take into account the uncertainty with respect to the distribution (or transformation) and the estimated covariance function of the data. The Bayesian trans-Gaussian kriging methodology proposed in the present paper is in the spirit of the “Bayesian bootstrap” idea advocated by Rubin (Ann Stat 9:130–134, 1981) and avoids the unusual specification of noninformative priors often made in the literature and is entirely based on the sample distribution of the estimators of the covariance function and of the Box–Cox parameter. After some notes on Bayesian spatial prediction, noninformative priors and developing our new methodology finally we will present an example illustrating our pragmatic approach to Bayesian prediction by means of a simulated data set.     

Applicability of the Nagao–Kadoya bivariate exponential distribution for modeling two correlated exponentially distributed variates

 The open literature reveals several types of bivariate exponential distributions. Of them only the Nagao–Kadoya distribution (Nagao and Kadoya, 1970, 1971) has a general form with marginals that are standard exponential distributions and the correlation coefficient being 0≤ρ<1. On the basis of the principle that if a theoretical probability distribution can represent statistical properties of sample data, then the computed probabilities from the theoretical model should provide a good fit to observed ones, numerical experiments are executed to investigate the applicability of the Nagao–Kadoya bivariate exponential distribution for modeling the joint distribution of two correlated random variables with exponential marginals. Results indicate that this model is suitable for analyzing the joint distribution of two exponentially distributed variables. The procedure for the use of this model to represent the joint statistical properties of two correlated exponentially distributed variables is also presented.     

Is correlation dimension a reliable proxy for the number of dominant influencing variables for modeling risk of arsenic contamination in groundwater?

The correlation dimension (CD) of a time series provides information on the number of dominant variables present in the evolution of the underlying system dynamics. In this study, we explore, using logistic regression (LR), possible physical connections between the CD and the mathematical modeling of risk of arsenic contamination in groundwater. Our database comprises a large-scale arsenic survey conducted in Bangladesh. Following the recommendation by Hossain and Sivakumar (Stoch Environ Res Risk Assess 20(1–2):66–76, 2006), who reported CD values ranging from 8 to 11 for this database, 11 variables are considered herein as indicators of the aquifer’s geochemical regime with potential influence on the arsenic concentration in groundwater. A total of 2,048 possible combinations of influencing variables are considered as candidate LR risk models to delineate the impact of the number of variables on the prediction accuracy of the model. We find that the uncertainty associated with prediction of wells as safe and unsafe by LR risk model declines systematically as the total number of influencing variables increases from 7 to 11. The sensitivity of the mean predictive performance also increases noticeably for this range. The consistent reduction in predictive uncertainty coupled with the increased sensitivity of the mean predictive behavior within the universal sample space exemplify the ability of CD to function as a proxy for the number of dominant influencing variables. Such a rapid proxy, based on non-linear dynamic concepts, appears to have considerable merit for application in current management strategies on arsenic contamination in developing countries, where both time and resources are very limited.     

Probabilistic data integration for characterization of spatial distribution of residual LNAPL

The spatial distribution of residual light non-aqueous phase liquid (LNAPL) is an important factor in reactive solute transport modeling studies. There is great uncertainty associated with both the areal limits of LNAPL source zones and smaller scale variability within the areal limits. A statistical approach is proposed to construct a probabilistic model for the spatial distribution of residual NAPL and it is applied to a site characterized by ultra-violet-induced-cone-penetration testing (CPT–UVIF). The uncertainty in areal limits is explicitly addressed by a novel distance function (DF) approach. In modeling the small-scale variability within the areal limits, the CPT–UVIF data are used as primary source of information, while soil texture and distance to water table are treated as secondary data. Two widely used geostatistical techniques are applied for the data integration, namely sequential indicator simulation with locally varying means (SIS–LVM) and Bayesian updating (BU). A close match between the calibrated uncertainty band (UB) and the target probabilities shows the performance of the proposed DF technique in characterization of uncertainty in the areal limits. A cross-validation study also shows that the integration of the secondary data sources substantially improves the prediction of contaminated and uncontaminated locations and that the SIS–LVM algorithm gives a more accurate prediction of residual NAPL contamination. The proposed DF approach is useful in modeling the areal limits of the non-stationary continuous or categorical random variables, and in providing a prior probability map for source zone sizes to be used in Monte Carlo simulations of contaminant transport or Monte Carlo type inverse modeling studies.     

Representing spatial dependence and spatial discontinuity in ecological epidemiology: a scale mixture approach

Variation in disease risk underlying observed disease counts is increasingly a focus for Bayesian spatial modelling, including applications in spatial data mining. Bayesian analysis of spatial data, whether for disease or other types of event, often employs a conditionally autoregressive prior, which can express spatial dependence commonly present in underlying risks or rates. Such conditionally autoregressive priors typically assume a normal density and uniform local smoothing for underlying risks. However, normality assumptions may be affected or distorted by heteroscedasticity or spatial outliers. It is also desirable that spatial disease models represent variation that is not attributable to spatial dependence. A spatial prior representing spatial heteroscedasticity within a model accommodating both spatial and non-spatial variation is therefore proposed. Illustrative applications are to human TB incidence. A simulation example is based on mainland US states, while a real data application considers TB incidence in 326 English local authorities.     

Functional time series analysis of spatio–temporal epidemiological data

Spatio–temporal statistical models have been proposed for the analysis of the temporal evolution of the geographical pattern of mortality (or incidence) risks in disease mapping. However, as far as we know, functional approaches based on Hilbert-valued processes have not been used so far in this area. In this paper, the autoregressive Hilbertian process framework is adopted to estimate the functional temporal evolution of mortality relative risk maps. Specifically, the penalized functional estimation of log-relative risk maps is considered to smooth the classical standardized mortality ratio. The reproducing kernel Hilbert space (RKHS) norm is selected for definition of the penalty term. This RKHS-based approach is combined with the Kalman filtering algorithm for the spatio–temporal estimation of risk. Functional confidence intervals are also derived for detecting high risk areas. The proposed methodology is illustrated analyzing breast cancer mortality data in the provinces of Spain during the period 1975–2005. A simulation study is performed to compare the ARH(1) based estimation with the classical spatio–temporal conditional autoregressive approach.     

Investigation of Factors Controlling the Regional-Scale Distribution of Dried Soil Layers Under Forestland on the Loess Plateau,China

The effects of drought on plants have been extensively documented in water-limited systems. However, its effects on soil are seldom considered because of the lack of comparative data on profile soil water content (SWC). A dried soil layer (DSL) within the soil profile is a typical indication of soil drought caused by climate change and/or ill-advised human practices. The regional spatial variability, dominant factors, and predictive models of DSL under forestland were explored in the present study. SWC at 0–600 cm of 125 pre-selected sites across the entire Loess Plateau was measured, and then two evaluation indices of DSL (the thickness of DSL, DSLT; SWC within the DSL, DSL–SWC) were calculated. The corresponding soil, topography, plant, and meteorology factors (a total of 28 variables) for each site were also measured. Most of the forestlands across the Plateau had DSL formation within the soil profile (102 of 125 study sites). The DSL levels were considered to be serious, with DSLT generally exceeding 300 cm with a mean DSL–SWC of only 7.9% (field capacity (FC) = 18.1%). DSLT and DSL–SWC indicated a moderate and strong spatial dependence with ranges of 69 and 513 km, respectively. Thicker DSLs were mainly distributed in the center of the Plateau, whereas thinner DSLs were observed in the southern and southeastern parts. In contrast, DSL–SWC distributions demonstrated an obvious decreasing trend from the southeast to the northwest. Dominant factors affecting DSLT under forestlands were FC, bulk density, slope gradient, slope aspect, and capillary water content; while dominant factors for DSL–SWC were FC, aridity, sand content, altitude, vegetation coverage, and evaporation. Moreover, predictive models developed by multiple regressions were relatively accurate when predicting DSLs, especially DSL–SWC. Understanding these associations with DSLs formation in forestland is helpful for efficient water resource management, silviculture, and eco-environment restoration on the Loess Plateau and in other water-limited regions around the world.     

Spatial stochasticity and correlated effects in dose–response relationships for environmental pollutants: a case study of radiation effects

 Models of dose–response for environmental pollutants generally do not include explicit consideration of the stochastic nature of the spatial pattern of dose delivered to an organ or tissue, or the correlation between events leading to a final health endpoint (such as cancer). The result can be significant errors in risk calculations when these stochastic properties contribute as strongly to the dose–response relationship as do the dose–response relationships for individual cells. The present paper considers the issue of stochasticity of dose and events (initiation, promotion and inactivation) for the case of carcinogenicity following exposure to environmental pollutants, using the case of irradiation by high LET emitters such as radon and progeny from water or air. The model is based on the concepts of hit probabilities and effect-specific track length probabilities (probability of damage per unit track length), and is applied first to in vitro data and then to predictions in vivo. It is shown that inhomogeneity of dose throughout an irradiated tissue or organ volume, and correlation between initiation, promotion and inactivation, can lead to significant differences in predicted risk.     

Modelling sand–mud morphodynamics in the Friesche Zeegat

The objective of the study presented in this paper is to investigate the predictive capabilities of a process-based sand–mud model in a quantitative way. This recently developed sand–mud model bridges the gap between noncohesive sand models and cohesive mud models. It explicitly takes into account the interaction between these two sediment fractions and temporal and spatial bed composition changes in the sediment bed [Van Ledden (2002) 5:577–594, Van Ledden et al. (2004a) 24:1–11, Van Ledden et al. (2004b) 54:385–391]. The application of this model to idealized situations has demonstrated a good qualitative agreement between observed and computed bed levels and bed composition developments. However, in real-life situations, a realistic quantitative prediction of the magnitude and timescale of this response is important to assess the short-term and long-term impacts of human interventions and/or natural changes. For this purpose, the Friesche Zeegat in the Wadden Sea (the Netherlands) is used as a reference to hindcast the morphological response in the period 1970–1994. Due to the closure of the Lauwerszee in 1969, the tidal prism of this tidal basin was reduced by about 30%. Significant changes in the bed level and bed composition have occurred in the decades following the closure to adjust to the new hydrodynamic conditions. We modeled the long-term bed level and bed composition development in the Friesche Zeegat in the period 1970–1994 starting with the geometry of 1970 by using a research version of Delft3D, which incorporates the sand–mud formulations proposed by [Van Ledden (2002) 5:577–594].The computed total net deposition in the tidal basin in the period 1970–1994 agrees well with the observations, but the observed decrease of the import rate with time is not predicted. The model predicts net deposition in the deeper parts and at the intertidal area in the basin and net erosion in between, which resembles the observations qualitatively. Furthermore, the computed distribution of sand and mud in the basin of the Friesche Zeegat appears to be realistic. Analysis of the results shows that the absence of the decreasing import rate in the basin is caused by a poor quantitative prediction of the changes in the hypsometry of the basin. Because of this, the computed velocity asymmetry in the main channel tends toward flood dominance, whereas the observations indicate that the system is ebb-dominant in 1992. Although the sand–mud model needs to be further improved and verified, the results presented in this paper indicate that the model can be applied as a first step to estimate the effects of human interventions on the large-scale bed level and bed composition changes in tidal systems with sand and mud.     

Wavelet analysis residual kriging vs. neural network residual kriging

 This paper deals with the problem of spatial data mapping. A new method based on wavelet interpolation and geostatistical prediction (kriging) is proposed. The method – wavelet analysis residual kriging (WARK) – is developed in order to assess the problems rising for highly variable data in presence of spatial trends. In these cases stationary prediction models have very limited application. Wavelet analysis is used to model large-scale structures and kriging of the remaining residuals focuses on small-scale peculiarities. WARK is able to model spatial pattern which features multiscale structure. In the present work WARK is applied to the rainfall data and the results of validation are compared with the ones obtained from neural network residual kriging (NNRK). NNRK is also a residual-based method, which uses artificial neural network to model large-scale non-linear trends. The comparison of the results demonstrates the high quality performance of WARK in predicting hot spots, reproducing global statistical characteristics of the distribution and spatial correlation structure.     

The spatial and temporal distribution of marine geophysical surveys

We examine how bathymetric mapping coverage varies with distance from the coastline, here a proxy for the effort involved in collecting the data. Distances to the nearest coastline were evaluated on a 1′ × 1′ global grid. We evaluate the density of marine survey track lines, which falls off with increasing distance from the coastline and drops off precipitously for the most remote regions. Bathymetric coverage shows a marked asymmetry between the southern and northern hemispheres, the latter having a factor of 2–4 denser coverage. We find a rapid decrease in data acquisition for previously unexplored regions beginning in 1973–1975. This rate change may reflect a transition from serendipitous exploration to more targeted investigations as the plate tectonics hypothesis became accepted, but it could also reflect the 1970s oil shocks. Coverage of the seafloor varies logarithmically with mapping resolution. At 0.5° resolution, only ∼60% of the seafloor has been mapped; the 50% mark was reached in 1979 and coverage of unexplored seafloor has since been less rapid. For comparison, at 1′ resolution less than 10% of the seafloor has been mapped. Given rising fuel costs we predict the most remote areas will see a decline in future surveys. Better coordination of exploration among agencies and nations could mitigate this concern and improve global coverage, as could future altimetric mapping dedicated to bathymetric prediction.