Efficient measurement error correction with spatially misaligned data

Association studies in environmental statistics often involve exposure and outcome data that are misaligned in space. A common strategy is to employ a spatial model such as universal kriging to predict exposures at locations with outcome data and then estimate a regression parameter of interest using the predicted exposures. This results in measurement error because the predicted exposures do not correspond exactly to the true values. We characterize the measurement error by decomposing it into Berkson-like and classical-like components. One correction approach is the parametric bootstrap, which is effective but computationally intensive since it requires solving a nonlinear optimization problem for the exposure model parameters in each bootstrap sample. We propose a less computationally intensive alternative termed the "parameter bootstrap" that only requires solving one nonlinear optimization problem, and we also compare bootstrap methods to other recently proposed methods. We illustrate our methodology in simulations and with publicly available data from the Environmental Protection Agency.     

On estimation and prediction for spatial generalized linear mixed models

We use spatial generalized linear mixed models (GLMM) to model non-Gaussian spatial variables that are observed at sampling locations in a continuous area. In many applications, prediction of random effects in a spatial GLMM is of great practical interest. We show that the minimum mean-squared error (MMSE) prediction can be done in a linear fashion in spatial GLMMs analogous to linear kriging. We develop a Monte Carlo version of the EM gradient algorithm for maximum likelihood estimation of model parameters. A by-product of this approach is that it also produces the MMSE estimates for the realized random effects at the sampled sites. This method is illustrated through a simulation study and is also applied to a real data set on plant root diseases to obtain a map of disease severity that can facilitate the practice of precision agriculture.     

内蒙古锡林河流域草原土壤有机碳及氮素的空间异质性分析

生态系统中生物或非生物因子在空间分布上普遍具有空间异质性。本文应用地统计学的基本原理与方法（半方差分析和无偏插值）对研究区域内的草原土壤有机碳（ＳＯＣ）和全氮（ＴＮ）空间异质性进行了分析。研究结果表明：ＳＯＣ和ＴＮ的平均含量分别为１．５５５％和０．１３３３％,平均变异系数分别为１１．２％和１２．４％,二者在空间分布方面均具有明显的空间相关性,其空间相关尺度分别为８．１９ｍ和８．６９ｍ。在此基础上,应用空间局部内插法,绘制了两个因子的空间等值分布图。     

Survival Analysis with Error‐Prone Time‐Varying Covariates: A Risk Set Calibration Approach

Summary Occupational, environmental, and nutritional epidemiologists are often interested in estimating the prospective effect of time‐varying exposure variables such as cumulative exposure or cumulative updated average exposure, in relation to chronic disease endpoints such as cancer incidence and mortality. From exposure validation studies, it is apparent that many of the variables of interest are measured with moderate to substantial error. Although the ordinary regression calibration (ORC) approach is approximately valid and efficient for measurement error correction of relative risk estimates from the Cox model with time‐independent point exposures when the disease is rare, it is not adaptable for use with time‐varying exposures. By recalibrating the measurement error model within each risk set, a risk set regression calibration (RRC) method is proposed for this setting. An algorithm for a bias‐corrected point estimate of the relative risk using an RRC approach is presented, followed by the derivation of an estimate of its variance, resulting in a sandwich estimator. Emphasis is on methods applicable to the main study/external validation study design, which arises in important applications. Simulation studies under several assumptions about the error model were carried out, which demonstrated the validity and efficiency of the method in finite samples. The method was applied to a study of diet and cancer from Harvard's Health Professionals Follow‐up Study (HPFS).     

Nonparametric One‐way Analysis of Variance of Replicated Bivariate Spatial Point Patterns

A common problem in neuropathological studies is to assess the spatial patterning of cells on tissue sections and to compare spatial patterning between disorder groups. For a single cell type, the cell positions constitute a univariate point process and interest focuses on the degree of spatial aggregation. For two different cell types, the cell positions constitute a bivariate point process and the degree of spatial interaction between the cell types is of interest. We discuss the problem of analysing univariate and bivariate spatial point patterns in the one‐way design where cell patterns have been obtained for groups of subjects. A bootstrapping procedure to perform a nonparametric one‐way analysis of variance of the spatial aggregation of a univariate point process has been suggested by Diggle, Lange and Bene? (1991). We extend their replication‐based approach to allow the comparison of the spatial interaction of two cell types between groups, to include planned comparisons (contrasts) and to assess whole groups against complete spatial randomness and spatial independence. We also accommodate several replicate tissue sections per subject. An advantage of our approach is that it can be applied when processes are not stationary, a common problem in brain tissue sections since neurons are arranged in cortical layers. We illustrate our methods by applying them to a neuropathological study to investigate abnormalities in the functional relationship between neurons and astrocytes in HIV associated dementia. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On dynamics underlying variance of mass balance estimation in Chilean glaciers

Mass balance of a glacier is an accepted measure of how much mass a glacier gains or loses. In theory, it is typically computed by integral functional and empirically, it is approximated by arithmetic mean. However, the variability of such an approach was not studied satisfactory yet. In this paper we provide a dynamical system of mass balance measurements under the constrains of 2nd order model with exponentially decreasing covariance. We also provide locations of optimal measurements, so called designs. We study Ornstein–Uhlenbeck (OU) processes and sheets with linear drifts and introduce K optimal designs in the correlated processes setup. We provide a thorough comparison of equidistant, Latin Hypercube Samples (LHS), and factorial designs for D- and K-optimality as well as the variance. We show differences between these criteria and discuss the role of equidistant designs for the correlated process. In particular, applications to estimation of mass balance of Olivares Alfa and Beta glaciers in Chile is investigated showing that simple application of full raster design and kriging based on inter- and extrapolation of points can lead to increased variance. We also show how the removal of certain measurement points may increase the quality of the melting assessment while decreasing costs. Blow-ups of solutions of dynamical systems underline the empirically observed fact that in a homogenous glaciers around 11 well-positioned stakes suffices for mass balance measurement.     

Estimating the variation,autocorrelation, and environmental sensitivity of phenotypic selection

Despite considerable interest in temporal and spatial variation of phenotypic selection, very few methods allow quantifying this variation while correctly accounting for the error variance of each individual estimate. Furthermore, the available methods do not estimate the autocorrelation of phenotypic selection, which is a major determinant of eco‐evolutionary dynamics in changing environments. We introduce a new method for measuring variable phenotypic selection using random regression. We rely on model selection to assess the support for stabilizing selection, and for a moving optimum that may include a trend plus (possibly autocorrelated) fluctuations. The environmental sensitivity of selection also can be estimated by including an environmental covariate. After testing our method on extensive simulations, we apply it to breeding time in a great tit population in the Netherlands. Our analysis finds support for an optimum that is well predicted by spring temperature, and occurs about 33 days before a peak in food biomass, consistent with what is known from the biology of this species. We also detect autocorrelated fluctuations in the optimum, beyond those caused by temperature and the food peak. Because our approach directly estimates parameters that appear in theoretical models, it should be particularly useful for predicting eco‐evolutionary responses to environmental change.     

Estimation of Spatial Variation in Risk Using Matched Case‐control Data

A common problem in environmental epidemiology is to estimate spatial variation in disease risk after accounting for known risk factors. In this paper we consider this problem in the context of matched case‐control studies. We extend the generalised additive model approach of Kelsall and Diggle (1998) to studies in which each case has been individually matched to a set of controls. We discuss a method for fitting this model to data, apply the method to a matched study on perinatal death in the North West Thames region of England and explain why, if spatial variation is of particular scientific interest, matching is undesirable.     

Iterative kriging for removing spatial autocorrelation in analysis of forest genetic trials

Conventional analysis of spatially correlated data in inadequately blocked field genetic trials may give erroneous results that would seriously affect breeding decisions. Forest genetic trials are commonly very large and strongly heterogeneous, so adjustments for micro-environmental heterogeneity become indispensable. This study explores the use of geostatistics to account for the spatial autocorrelation in four Pinus pinaster Ait. progeny trials established on hilly and irregular terrains with a randomized complete block design and large blocks. Data of five different traits assessed at age 8 were adjusted using an iterative method based on semivariograms and kriging, and the effects on estimates of variance components, heritability, and family effects were evaluated in relation to conventional analysis. Almost all studied traits showed nonrandom spatial structures. Therefore, after the adjustments for spatial autocorrelation, the block and family × block variance components, which were extremely high in the conventional analysis, almost disappeared. The reduction of the interaction variance was recovered by the family variance component, resulting in higher heritability estimates. The removal of the spatial autocorrelation also affected the estimation of family effects, resulting in important changes in family ranks after the spatial adjustments. Comparison among families was also greatly improved due to higher accuracy of the family effect estimations. The analysis improvement was larger for growth traits, which showed the strongest spatial heterogeneity, but was also evident for other traits such as straightness or number of whorls. The present paper demonstrates how spatial autocorrelation can drastically affect the analysis of forest genetic trials with large blocks. The iterative kriging procedure presented in this paper is a promising tool to account for this spatial heterogeneity.     

On the change of support problem for spatio-temporal data

In practice, spatial data are sometimes collected at points (i.e. point-referenced data) and at other times are associated with areal units (i.e. block data). The change of support problem is concerned with inference about the values of a variable at points or blocks different from those at which it has been observed. In the context of block data which can be sensibly viewed as averaging over point data, we propose a unifying approach for prediction from points to points, points to blocks, blocks to points, and blocks to blocks. The approach includes fully Bayesian kriging. We also extend our approach to the the case of spatio-temporal data, wherein a judicious specification of spatio-temporal association enables manageable computation. Exemplification of the static spatial case is provided using a dataset of point-level ozone measurements in the Atlanta, Georgia metropolitan area. The dynamic spatial case is illustrated using a temporally extended version of this dataset, enabling comparison at the common time point.*To whom correspondence should be addressed.     

绿洲-荒漠过渡带景观的植被与土壤特征要素的空间异质性分析

采用变异函数的理论和方法,通过空间异质性特征比较,研究了阜康绿洲-荒漠过渡带景观中的植被盖度和土壤环境因子的空间异质性、空间格局及其关系．在此基础上,应用Kriging局部块段插值法,以三维图形显示出各个要素的空间分布格局．结果表明,除土壤全盐量空间相关性较弱外,其他各要素具有中等程度以上的空间相关性,且空间变异主要发生在较小尺度上．其中,植被盖度的空间异质性程度较高,在3km的尺度内具有明显的空间格局．从空间分布格局看,植被盖度、土壤表层含水量和土壤pH值沿样带相邻点差异对比明显,高值区和低值区交错．草本盖度与土壤表层含水量、土壤pH间有相关关系,灌木平均盖度主要依靠深层次地下水维系．灌木和草本盖度的块金值具有明显差异,可能因为二者要求不同的生态条件,因而在同等尺度上具有不同的生态学过程．     

Geostatistical approaches and optimal additional sampling schemes for spatial patterns and future sampling of bird diversity

Aim To evaluate geostatistical approaches, namely kriging, co‐kriging and geostatistical simulation, and to develop an optimal sampling design for mapping the spatial patterns of bird diversity, estimating their spatial autocorrelations and selecting additional samples of bird diversity in a 2450 km² basin. Location Taiwan. Methods Kriging, co‐kriging and simulated annealing are applied to estimate and simulate the spatial patterns of bird diversity. In addition, kriging and co‐kriging with a genetic algorithm are used to optimally select further samples to improve the kriging and co‐kriging estimations. The association between bird diversity and elevation, and bird diversity and land cover, is analysed with estimated and simulated maps. Results The Simpson index correlates spatially with the normalized difference vegetation index (NDVI) within the micro‐scale and the macro‐scale in the study basin, but the Shannon diversity index only correlates spatially with NDVI within the micro‐scale. Co‐kriging and simulated annealing simulation accurately simulate the statistical and spatial patterns of bird diversity. The mean estimated diversity and the simulated diversity increase with elevation and decrease with increasing urbanization. The proposed optimal sampling approach selects 43 additional sampling sites with a high spatial estimation variance in bird diversity. Main conclusions Small‐scale variations dominate the total spatial variation of the observed diversity due to a lack of spatial information and insufficient sampling. However, simulations of bird diversity consistently capture the sampling statistics and spatial patterns of the observed bird diversity. The data thus accumulated can be used to understand the spatial patterns of bird diversity associated with different types of land cover and elevation, and to optimize sample selection. Co‐kriging combined with a genetic algorithm yields additional optimal sampling sites, which can be used to augment existing sampling points in future studies of bird diversity.     

Maximum likelihood estimation of marginal pairwise associations with multiple source predictors

Researchers interested in the association of a predictor with an outcome will often collect information about that predictor from more than one source. Standard multiple regression methods allow estimation of the effect of each predictor on the outcome while controlling for the remaining predictors. The resulting regression coefficient for each predictor has an interpretation that is conditional on all other predictors. In settings in which interest is in comparison of the marginal pairwise relationships between each predictor and the outcome separately (e.g., studies in psychiatry with multiple informants or comparison of the predictive values of diagnostic tests), standard regression methods are not appropriate. Instead, the generalized estimating equations (GEE) approach can be used to simultaneously estimate, and make comparisons among, the separate pairwise marginal associations. In this paper, we consider maximum likelihood (ML) estimation of these marginal relationships when the outcome is binary. ML enjoys benefits over GEE methods in that it is asymptotically efficient, can accommodate missing data that are ignorable, and allows likelihood-based inferences about the pairwise marginal relationships. We also explore the asymptotic relative efficiency of ML and GEE methods in this setting.     

田块尺度下土壤磷素的空间变异性

采用经典统计学与地统计学相结合的方法,对中国科学院沈阳生态实验站30 m×42 m样地进行网格法分层(0～10和10～20 cm)取样,研究了田块尺度下土壤全P和Olsen-P的空间变异特征.结果表明,49对样本土壤Olsen-P的变异系数(4.5%～5.42%)远高于全P(11.8%～13.33%);全P和Olsen-P具有较好的空间结构且具有较相近的空间相关距离.最佳理论模型的参数显示各变量空间变异主要受结构性因素的影响,各变量半方差变异函数的C/(C₀+C)均高于%.全P和Olsen-P之间及在2个土层之间均具有较相似的空间分布格局.变异系数结合空间格局分析可以大大降低试验取样的数量.     

A repeated measures approach to pooled and calibrated biomarker data

Participant-level meta-analysis across multiple studies increases the sample size for pooled analyses, thereby improving precision in effect estimates and enabling subgroup analyses. For analyses involving biomarker measurements as an exposure of interest, investigators must first calibrate the data to address measurement variability arising from usage of different laboratories and/or assays. In practice, the calibration process involves reassaying a random subset of biospecimens from each study at a central laboratory and fitting models that relate the study-specific “local” and central laboratory measurements. Previous work in this area treats the calibration process from the perspective of measurement error techniques and imputes the estimated central laboratory value among individuals with only a local laboratory measurement. In this work, we propose a repeated measures method to calibrate biomarker measurements pooled from multiple studies with study-specific calibration subsets. We account for correlation between measurements made on the same person and between measurements made at the same laboratory. We demonstrate that the repeated measures approach provides valid inference, and compare it to existing calibration approaches grounded in measurement error techniques in an example describing the association between circulating vitamin D and stroke.     

Comparing the effects of continuous and discrete covariate mismeasurement,with emphasis on the dichotomization of mismeasured predictors

It is well known that imprecision in the measurement of predictor variables typically leads to bias in estimated regression coefficients. We compare the bias induced by measurement error in a continuous predictor with that induced by misclassification of a binary predictor in the contexts of linear and logistic regression. To make the comparison fair, we consider misclassification probabilities for a binary predictor that correspond to dichotomizing an imprecise continuous predictor in lieu of its precise counterpart. On this basis, nondifferential binary misclassification is seen to yield more bias than nondifferential continuous measurement error. However, it is known that differential misclassification results if a binary predictor is actually formed by dichotomizing a continuous predictor subject to nondifferential measurement error. When the postulated model linking the response and precise continuous predictor is correct, this differential misclassification is found to yield less bias than continuous measurement error, in contrast with nondifferential misclassification, i.e., dichotomization reduces the bias due to mismeasurement. This finding, however, is sensitive to the form of the underlying relationship between the response and the continuous predictor. In particular, we give a scenario where dichotomization involves a trade-off between model fit and misclassification bias. We also examine how the bias depends on the choice of threshold in the dichotomization process and on the correlation between the imprecise predictor and a second precise predictor.     

Bias-corrected variance estimation and hypothesis testing for spatial point and marked point processes using subsampling

Summary We introduce novel regression extrapolation based methods to correct the often large bias in subsampling variance estimation as well as hypothesis testing for spatial point and marked point processes. For variance estimation, our proposed estimators are linear combinations of the usual subsampling variance estimator based on subblock sizes in a continuous interval. We show that they can achieve better rates in mean squared error than the usual subsampling variance estimator. In particular, for n×n observation windows, the optimal rate of n^?2 can be achieved if the data have a finite dependence range. For hypothesis testing, we apply the proposed regression extrapolation directly to the test statistics based on different subblock sizes, and therefore avoid the need to conduct bias correction for each element in the covariance matrix used to set up the test statistics. We assess the numerical performance of the proposed methods through simulation, and apply them to analyze a tropical forest data set.     

Bayesian Wombling for Spatial Point Processes

Summary In many applications involving geographically indexed data, interest focuses on identifying regions of rapid change in the spatial surface, or the related problem of the construction or testing of boundaries separating regions with markedly different observed values of the spatial variable. This process is often referred to in the literature as boundary analysis or wombling. Recent developments in hierarchical models for point‐referenced (geostatistical) and areal (lattice) data have led to corresponding statistical wombling methods, but there does not appear to be any literature on the subject in the point‐process case, where the locations themselves are assumed to be random and likelihood evaluation is notoriously difficult. We extend existing point‐level and areal wombling tools to this case, obtaining full posterior inference for multivariate spatial random effects that, when mapped, can help suggest spatial covariates still missing from the model. In the areal case we can also construct wombled maps showing significant boundaries in the fitted intensity surface, while the point‐referenced formulation permits testing the significance of a postulated boundary. In the computationally demanding point‐referenced case, our algorithm combines Monte Carlo approximants to the likelihood with a predictive process step to reduce the dimension of the problem to a manageable size. We apply these techniques to an analysis of colorectal and prostate cancer data from the northern half of Minnesota, where a key substantive concern is possible similarities in their spatial patterns, and whether they are affected by each patient's distance to facilities likely to offer helpful cancer screening options.     

Stochastic approach to determine spatial patterns of lizard community on a desert island

One of the principal sources of error in identifying spatial arrangements is autocorrelation, since nearby points in space tend to have more similar values than would be expected by random change. When a Markovian approach is used, spatial arrangements can be measured as a transition probability between occupied and empty spaces in samples that are spatially dependent. We applied a model that incorporates first-order Markov chains to analyse spatial arrangement of numerical dominance, richness, and abundance on a lizard community at different spatial and temporal scales. We hypothesized that if a spatial dependence on abundance and richness exists in a diurnal desert community, then the Markov chains can predict the spatial arrangement. We found that each pair of values was dependent only on its immediate predecessor segment. In this sense, we found intergeneric differences at temporal and spatial scales of recurrence estimates. Also, in desert scrub, species show higher spatial aggregation and had lower species richness than at the island level; the inverse pattern occurred on rocky hillsides. At the species level, Uta stansburiana is the most abundant species in desert scrub, while Sauromalus slevini is the most abundant species on rocky hillsides. This report attempts to understand, using Markovian spatial models, the effect of nearby samples on local abundance and richness on different scales and over several seasons.     

Correcting for measurement error in fractional polynomial models using Bayesian modelling and regression calibration,with an application to alcohol and mortality

Exposure measurement error can result in a biased estimate of the association between an exposure and outcome. When the exposure–outcome relationship is linear on the appropriate scale (e.g. linear, logistic) and the measurement error is classical, that is the result of random noise, the result is attenuation of the effect. When the relationship is non‐linear, measurement error distorts the true shape of the association. Regression calibration is a commonly used method for correcting for measurement error, in which each individual's unknown true exposure in the outcome regression model is replaced by its expectation conditional on the error‐prone measure and any fully measured covariates. Regression calibration is simple to execute when the exposure is untransformed in the linear predictor of the outcome regression model, but less straightforward when non‐linear transformations of the exposure are used. We describe a method for applying regression calibration in models in which a non‐linear association is modelled by transforming the exposure using a fractional polynomial model. It is shown that taking a Bayesian estimation approach is advantageous. By use of Markov chain Monte Carlo algorithms, one can sample from the distribution of the true exposure for each individual. Transformations of the sampled values can then be performed directly and used to find the expectation of the transformed exposure required for regression calibration. A simulation study shows that the proposed approach performs well. We apply the method to investigate the relationship between usual alcohol intake and subsequent all‐cause mortality using an error model that adjusts for the episodic nature of alcohol consumption.