A Modification of Stone's Test for Trend for Binary Outcome

Stone (1988) suggested the “first isotonic regression estimator” as a tool for drawing inferences on possibly increased cancer case counts among several subregions around a putative source. He assumed the case counts to be Poisson distributed and therefore introduced a rare disease assumption into his approach. However, when analyzing cross sectional data one would rather refer to prevalence estimates among these subregions around a point risk source (for example the origin of chemical fallout). Therefore we applied antitonic regression estimation in Binomial distributions to derive a test statistic and a p value to test for a possible trend in the observed prevalence data around the putative source. The computation of this p value will be illustrated as well as severe difficulties concerning its interpretation. Further the Maximum Likelihood Ratio approach will be used to derive an alternative test statistic.     

A Field Evaluation of Distance Measurement Error in Auditory Avian Point Count Surveys

ABSTRACT Detection distance is an important and common auxiliary variable measured during avian point count surveys. Distance data are used to determine the area sampled and to model the detection process using distance sampling theory. In densely forested habitats, visual detections of birds are rare, and most estimates of detection distance are based on auditory cues. Distance sampling theory assumes detection distances are measured accurately, but empirical validation of this assumption for auditory detections is lacking. We used a song playback system to simulate avian point counts with known distances in a forested habitat to determine the error structure of distance estimates based on auditory detections. We conducted field evaluations with 6 experienced observers both before and after distance estimation training. We conducted additional studies to determine the effect of height and speaker orientation (toward or away from observers) on distance estimation error. Distance estimation errors for all evaluations were substantial, although training reduced errors and bias in distance estimates by approximately 15%. Measurement errors showed a nonlinear relationship to distance. Our results suggest observers were not able to differentiate distances beyond 65 m. The height from which we played songs had no effect on distance estimation errors in this habitat. The orientation of the song source did have a large effect on distance estimation errors; observers generally doubled their distance estimates for songs played away from them compared with distance estimates for songs played directly toward them. These findings, which we based on realistic field conditions, suggest measures of uncertainty in distance estimates to auditory detections are substantially higher than assumed by most researchers. This means aural point count estimates of avian abundance based on distance methods deserve careful scrutiny because they are likely biased.     

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.     

Covariate Adjustment and Ranking Methods to Identify Regions with High and Low Mortality Rates

Summary Identifying regions with the highest and lowest mortality rates and producing the corresponding color‐coded maps help epidemiologists identify promising areas for analytic etiological studies. Based on a two‐stage Poisson–Gamma model with covariates, we use information on known risk factors, such as smoking prevalence, to adjust mortality rates and reveal residual variation in relative risks that may reflect previously masked etiological associations. In addition to covariate adjustment, we study rankings based on standardized mortality ratios (SMRs), empirical Bayes (EB) estimates, and a posterior percentile ranking (PPR) method and indicate circumstances that warrant the more complex procedures in order to obtain a high probability of correctly classifying the regions with the upper 100γ% and lower 100γ% of relative risks for γ= 0.05, 0.1 , and 0.2. We also give analytic approximations to the probabilities of correctly classifying regions in the upper 100γ% of relative risks for these three ranking methods. Using data on mortality from heart disease, we found that adjustment for smoking prevalence has an important impact on which regions are classified as high and low risk. With such a common disease, all three ranking methods performed comparably. However, for diseases with smaller event counts, such as cancers, and wide variation in event counts among regions, EB and PPR methods outperform ranking based on SMRs.     

Inferring Models of Bacterial Dynamics toward Point Sources

Experiments have shown that bacteria can be sensitive to small variations in chemoattractant (CA) concentrations. Motivated by these findings, our focus here is on a regime rarely studied in experiments: bacteria tracking point CA sources (such as food patches or even prey). In tracking point sources, the CA detected by bacteria may show very large spatiotemporal fluctuations which vary with distance from the source. We present a general statistical model to describe how bacteria locate point sources of food on the basis of stochastic event detection, rather than CA gradient information. We show how all model parameters can be directly inferred from single cell tracking data even in the limit of high detection noise. Once parameterized, our model recapitulates bacterial behavior around point sources such as the “volcano effect”. In addition, while the search by bacteria for point sources such as prey may appear random, our model identifies key statistical signatures of a targeted search for a point source given any arbitrary source configuration.     

Effects of branch length uncertainty on Bayesian posterior probabilities for phylogenetic hypotheses

In Bayesian phylogenetics, confidence in evolutionary relationships is expressed as posterior probability--the probability that a tree or clade is true given the data, evolutionary model, and prior assumptions about model parameters. Model parameters, such as branch lengths, are never known in advance; Bayesian methods incorporate this uncertainty by integrating over a range of plausible values given an assumed prior probability distribution for each parameter. Little is known about the effects of integrating over branch length uncertainty on posterior probabilities when different priors are assumed. Here, we show that integrating over uncertainty using a wide range of typical prior assumptions strongly affects posterior probabilities, causing them to deviate from those that would be inferred if branch lengths were known in advance; only when there is no uncertainty to integrate over does the average posterior probability of a group of trees accurately predict the proportion of correct trees in the group. The pattern of branch lengths on the true tree determines whether integrating over uncertainty pushes posterior probabilities upward or downward. The magnitude of the effect depends on the specific prior distributions used and the length of the sequences analyzed. Under realistic conditions, however, even extraordinarily long sequences are not enough to prevent frequent inference of incorrect clades with strong support. We found that across a range of conditions, diffuse priors--either flat or exponential distributions with moderate to large means--provide more reliable inferences than small-mean exponential priors. An empirical Bayes approach that fixes branch lengths at their maximum likelihood estimates yields posterior probabilities that more closely match those that would be inferred if the true branch lengths were known in advance and reduces the rate of strongly supported false inferences compared with fully Bayesian integration.     

Point Processes for Fine‐Scale Spatial Genetics and Molecular Ecology

This paper introduces point processes into fine‐scale spatial genetics and molecular ecology. Datasets given in the form of a complete map of individuals and their genotypes can be analyzed by means of the theory of marked or multivariate point processes. Beginning with reformulation of conventional spatial autocorrelation statistics in genetics by the language of point processes, this paper first shows an example of point process models that describe spatial patterns of both tree locations and their genotypes, on the assumption of limited seed dispersal and long pollen movement. The results show that isolation‐by‐distance slightly occurs from the assumption above, and more importantly, an increment of the degree of clustering of trees reduces the degree of genetic clustering. Next, the point process model is applied to field data of secondary forest regenerated after seed tree harvesting, and tests the hypothesis that the current population was formed only from a small number of seed trees. The hypothesis was not acceptable, instead, the alternative assuming advance reproduction conducted prior to the harvesting is supported. The results of this first trial of point process models suggest that point processes can provide a useful mathematical methodology in fine‐scale spatial genetics and molecular ecology.     

Estimation of effective cleanup radius for soil‐vapor extraction systems

Soil‐vapor extraction (SVE) is a standard and effective in situ treatment for the removal of volatile contaminants from vadose‐zone soil. The duration of SVE operation required to reach site closure is quite variable, however, ranging up to several years or more. An understanding of the contaminant recovery rate as a function of distance from each vapor‐extraction well allows SVE systems to be designed so that cleanup goals can be achieved within a specified time frame.

A simple one‐dimensional model has been developed that provides a rough estimate of the effective cleanup radius (defined as “the maximum distance from a vapor extraction point through which sufficient air is drawn to remove the required fraction of contamination in the desired time") for SVE systems. Because the model uses analytical rather than numerical methods, it has advantages over more sophisticated, multidimensional models, including simplicity, speed, versatility, and robustness.

The contaminant removal rate at a given distance from the vapor‐extraction point is assumed to be a function of the local rate of soil‐gas flow, the contaminant soil concentration, and the contaminant volatility. Soil‐gas flow rate as a function of distance from the vapor‐extraction point is estimated from pilot test data by assuming that the infiltration of atmospheric air through the soil surface is related to the vacuum in the soil. Although widely applicable, the model should be used with some caution when the vadose zone is highly stratified or when venting contaminated soil greater than 30 ft below grade. Since 1992, Groundwater Technology, Inc. has been using this model routinely as a design tool for SVE systems.     

西藏高原拉萨河流域生态风险评估

随着我国西部崛起战略的实施,社会经济发展所引发的生态风险逐渐引起生态学者和政府管理部门的高度关注。拉萨河流域作为高原生物多样性的维持基地,也是我国及东亚的重要生态安全屏障区,为有效管控其社会经济发展中产生的生态环境风险,通过适合高寒流域特征的风险评估方法,识别了研究区风险源,选择风险受体,采用相对风险模型计算了各研究单元的风险等级。结果表明:城关区的生态风险最高,主要风险来源于城镇扩张和旅游娱乐;当雄县生态风险最低,风险来源于畜牧养殖;林周县、曲水县、达孜县、堆龙德庆区以及墨竹工卡县的首要风险源是农业污染。水体和湿地的风险值明显高于其他生态系统。风险值的空间分布表现为南高北低的特征,且风险等级以城关区为中心向外围逐渐降低。     

Cut‐off Point of BMI and Obesity‐Related Comorbidities and Mortality in Middle‐Aged Koreans

Objective: The need for a lower BMI to classify overweight in Asian populations has been controversial. Using both disease and mortality outcomes, we investigated whether lower BMI cut‐off points are appropriate for identifying increased health risk in Koreans. Research Methods and Procedures: We conducted a cohort study among 773, 915 men and women from 30 to 59 years old with 8‐ to 10‐year follow‐up periods. Primary outcomes were change of obesity prevalence, obesity‐related disease incidence, and all‐cause mortality. Results: Prevalence of overweight (BMI of 25.0‐29.9) has steadily increased (1.3% annually), whereas obesity (BMI ≥ 30) showed a lower prevalence and only a slight increase (0.1%‐0.2% annually). Our study revealed that dose‐response relationships exist between obesity and related disease incidences (hypertension, type 2 diabetes, and hypercholesterolemia) beginning at lower BMI levels than previously reported. Compared with those in the healthy weight range, Koreans with a BMI ≥ 25 were not at greater risk of hypertension, type 2 diabetes, or hypercholesterolemia than has been reported for whites in similar studies. Obesity‐related all‐cause mortality also did not seem so different from that of whites. Discussion: Our findings did not support the use of a lower BMI cut‐off point for defining overweight in Koreans compared with whites for the purpose of identifying different risks. However, populations with BMI ≥ 25 are rapidly increasing and have substantial risks of diseases. To preempt the rapid increases in obesity and related health problems that are occurring in Western countries, Korea should consider using a BMI of 25 as an action point for obesity prevention and control interventions.     

Seasonality and Coronary Heart Disease Deaths in United States Firefighters

United States firefighters have a high on‐duty fatality rate, and coronary heart disease is the leading cause. Seasonality affects the incidence of cardiovascular events in the general population, but its effects on firefighters are unknown. This study statistically examined the seasonal and annual variation of all on‐duty coronary heart disease deaths among US firefighters between 1994 and 2004 using the chi‐square distribution and Poisson regression model of the monthly fatality counts. It also examined the effect of ambient temperature (apparent as well as wind chill temperature) on coronary heart disease fatalities during the study span using a time‐stratified, case‐crossover study design. When grouped by season, we observed the distribution of the 449 coronary heart disease fatalities to show a relative peak in winter (32%) and relative nadir in spring (21%). This pattern was significantly different (p=0.005) from the expected distribution under the null hypothesis of season having no effect. The pattern persisted in additional analyses, stratifying the deaths by the type of duty in which the firefighters were engaged at the time of their deaths. In the Poisson regression model of the monthly fatality counts, the overall goodness‐of‐fit between the actual and predicted case counts was excellent (χ₄²=16.63; p=0.002). Two distinct peaks were detected: one in January–February and the other in August–September. Overall temperature was not associated with increased risk of on‐duty death. After allowing for different effects of temperature in mild/hot versus cold periods, a 1°C increase was not protective in cold weather; nor did it increase the risk of death in warmer weather. The findings of this study reveal statistical evidence for excess coronary heart disease deaths among firefighters during winter; however, the temporal pattern of coronary heart disease deaths was not linked to temperature variation. The seasonal pattern was also found to be independent of duty‐related risks.     

Spatio-temporal dynamics of corn rootworm, Diabrotica spp., adults and their spatial association with environment

Multiyear and multilocation studies were conducted to investigate the within‐ and the between‐year spatial dynamics of corn rootworms Diabrotica spp. (Coleoptera: Chrysomelidae), adults and their spatial associations with environmental factors in cornfields. Grid‐based spatial sampling was conducted by trapping adults emerging from the soil (i.e., ‘emergence‐trap counts’) and by counting the number of adults in the ear zone of corn plant (i.e., ‘ear‐zone count’). Spatial analysis with distance indices (SADIE) was used to determine spatial distribution patterns and to investigate spatial associations. Ear‐zone counts and emergence‐trap counts were spatially dynamic within a year and more frequently aggregated in the middle of growing seasons and random early and late in the season. However, during the peak population periods, spatial distribution of ear‐zone counts and emergence‐trap counts were significantly consistent between years, indicating predictability of future spatial distributions. Spatial distribution of emergence‐trap counts and ear‐zone counts were positively associated with soil moisture and presence of corn plants with silks, respectively. This study demonstrated that within‐year spatial distribution patterns were dynamic and that there was between‐year spatial consistency of corn rootworm adult distributions. Such information can be used to improve preventative management of corn rootworms.     

Integer‐valued functional data analysis for measles forecasting

Measles presents a unique and imminent challenge for epidemiologists and public health officials: the disease is highly contagious, yet vaccination rates are declining precipitously in many localities. Consequently, the risk of a measles outbreak continues to rise. To improve preparedness, we study historical measles data both prevaccine and postvaccine, and design new methodology to forecast measles counts with uncertainty quantification. We propose to model the disease counts as an integer‐valued functional time series: measles counts are a function of time‐of‐year and time‐ordered by year. The counts are modeled using a negative‐binomial distribution conditional on a real‐valued latent process, which accounts for the overdispersion observed in the data. The latent process is decomposed using an unknown basis expansion, which is learned from the data, with dynamic basis coefficients. The resulting framework provides enhanced capability to model complex seasonality, which varies dynamically from year‐to‐year, and offers improved multimonth‐ahead point forecasts and substantially tighter forecast intervals (with correct coverage) compared to existing forecasting models. Importantly, the fully Bayesian approach provides well‐calibrated and precise uncertainty quantification for epi‐relevant features, such as the future value and time of the peak measles count in a given year. An R package is available online.     

A Bayesian model averaging approach for estimating the relative risk of mortality associated with heat waves in 105 U.S. cities

Estimating the risks heat waves pose to human health is a critical part of assessing the future impact of climate change. In this article, we propose a flexible class of time series models to estimate the relative risk of mortality associated with heat waves and conduct Bayesian model averaging (BMA) to account for the multiplicity of potential models. Applying these methods to data from 105 U.S. cities for the period 1987-2005, we identify those cities having a high posterior probability of increased mortality risk during heat waves, examine the heterogeneity of the posterior distributions of mortality risk across cities, assess sensitivity of the results to the selection of prior distributions, and compare our BMA results to a model selection approach. Our results show that no single model best predicts risk across the majority of cities, and that for some cities heat-wave risk estimation is sensitive to model choice. Although model averaging leads to posterior distributions with increased variance as compared to statistical inference conditional on a model obtained through model selection, we find that the posterior mean of heat wave mortality risk is robust to accounting for model uncertainty over a broad class of models.     

Fair-balance paradox, star-tree paradox, and Bayesian phylogenetics

The star-tree paradox refers to the conjecture that the posterior probabilities for the three unrooted trees for four species (or the three rooted trees for three species if the molecular clock is assumed) do not approach 1/3 when the data are generated using the star tree and when the amount of data approaches infinity. It reflects the more general phenomenon of high and presumably spurious posterior probabilities for trees or clades produced by the Bayesian method of phylogenetic reconstruction, and it is perceived to be a manifestation of the deeper problem of the extreme sensitivity of Bayesian model selection to the prior on parameters. Analysis of the star-tree paradox has been hampered by the intractability of the integrals involved. In this article, I use Laplacian expansion to approximate the posterior probabilities for the three rooted trees for three species using binary characters evolving at a constant rate. The approximation enables calculation of posterior tree probabilities for arbitrarily large data sets. Both theoretical analysis of the analogous fair-coin and fair-balance problems and computer simulation for the tree problem confirmed the existence of the star-tree paradox. When the data size n --> infinity, the posterior tree probabilities do not converge to 1/3 each, but they vary among data sets according to a statistical distribution. This distribution is characterized. Two strategies for resolving the star-tree paradox are explored: (1) a nonzero prior probability for the degenerate star tree and (2) an increasingly informative prior forcing the internal branch length toward zero. Both appear to be effective in resolving the paradox, but the latter is simpler to implement. The posterior tree probabilities are found to be very sensitive to the prior.     

Comparison of point counts and territory mapping for detecting effects of forest management on songbirds

Point counts are commonly used to assess changes in bird abundance, including analytical approaches such as distance sampling that estimate density. Point‐count methods have come under increasing scrutiny because effects of detection probability and field error are difficult to quantify. For seven forest songbirds, we compared fixed‐radii counts (50 m and 100 m) and density estimates obtained from distance sampling to known numbers of birds determined by territory mapping. We applied point‐count analytic approaches to a typical forest management question and compared results to those obtained by territory mapping. We used a before–after control impact (BACI) analysis with a data set collected across seven study areas in the central Appalachians from 2006 to 2010. Using a 50‐m fixed radius, variance in error was at least 1.5 times that of the other methods, whereas a 100‐m fixed radius underestimated actual density by >3 territories per 10 ha for the most abundant species. Distance sampling improved accuracy and precision compared to fixed‐radius counts, although estimates were affected by birds counted outside 10‐ha units. In the BACI analysis, territory mapping detected an overall treatment effect for five of the seven species, and effects were generally consistent each year. In contrast, all point‐count methods failed to detect two treatment effects due to variance and error in annual estimates. Overall, our results highlight the need for adequate sample sizes to reduce variance, and skilled observers to reduce the level of error in point‐count data. Ultimately, the advantages and disadvantages of different survey methods should be considered in the context of overall study design and objectives, allowing for trade‐offs among effort, accuracy, and power to detect treatment effects.     

A genetic and spatial Bayesian analysis of mastitis resistance

A nationwide health card recording system for dairy cattle was introduced in Norway in 1975 (the Norwegian Cattle Health Services). The data base holds information on mastitis occurrences on an individual cow basis. A reduction in mastitis frequency across the population is desired, and for this purpose risk factors are investigated. In this paper a Bayesian proportional hazards model is used for modelling the time to first veterinary treatment of clinical mastitis, including both genetic and environmental covariates. Sire effects were modelled as shared random components, and veterinary district was included as an environmental effect with prior spatial smoothing. A non-informative smoothing prior was assumed for the baseline hazard, and Markov chain Monte Carlo methods (MCMC) were used for inference. We propose a new measure of quality for sires, in terms of their posterior probability of being among the, say 10% best sires. The probability is an easily interpretable measure that can be directly used to rank sires. Estimating these complex probabilities is straightforward in an MCMC setting. The results indicate considerable differences between sires with regards to their daughters disease resistance. A regional effect was also discovered with the lowest risk of disease in the south-eastern parts of Norway.     

Species richness and abundance of forest birds in relation to radiation at Chernobyl

The effects of low-level radiation on the abundance of animals are poorly known, as are the effects on ecosystems and their functioning. Recent conclusions from the UN Chernobyl forum and reports in the popular media concerning the effects of radiation from Chernobyl on animals have left the impression that the Chernobyl exclusion zone is a thriving ecosystem, filled with an increasing number of rare species. Surprisingly, there are no standardized censuses of common animals in relation to radiation, leaving the question about the ecological effects of radiation unresolved. We conducted standardized point counts of breeding birds at forest sites around Chernobyl differing in level of background radiation by over three orders of magnitude. Species richness, abundance and population density of breeding birds decreased with increasing level of radiation, even after controlling statistically for the effects of potentially confounding factors such as soil type, habitat and height of the vegetation. This effect was differential for birds eating soil invertebrates living in the most contaminated top soil layer. These results imply that the ecological effects of Chernobyl on animals are considerably greater than previously assumed.     

Genotype-specific recurrence risks as indicators of the genetic architecture of complex diseases

A statistic is introduced that relates discoveries made in genome-wide association (GWA) studies to patterns of disease risks among relatives. The genotype-specific recurrence risk (GSR) is the genotype-specific risk to relatives of known relationship to affected probands. The GSRs can be used for three purposes. (1) They can provide an independent test of whether an allele identified in a GWA study is associated with the disease. (2) They can provide a test of whether interactions among loci affecting the disease are multiplicative. (3) They can be used by genetic counselors to incorporate information from GWA studies for predicting the risk to relatives of known genotype. Under a multiplicative model of disease causation, the GSRs for a locus are the genotypic risks in probands for that locus multiplied by lambda(R)/lambda(jR), where lambda(R) is Risch's recurrence risk ratio and lambda(jR) is the contribution to lambda(R) from the locus of interest. If there is saturation of risk with increasing numbers of causative alleles, then observed GSRs for individuals with high-risk genotypes will be lower than predicted by the multiplicative model.