Catherine A. Calder

Accounting for uncertainty in ecological analysis: the strengths and limitations of hierarchical statistical modeling

Analyses of ecological data should account for the uncertainty in the process(es) that generated the data. However, accounting for these uncertainties is a difficult task, since ecology is known for its complexity. Measurement and/or process errors are often the only sources of uncertainty modeled when addressing complex ecological problems, yet analyses should also account for uncertainty in sampling design, in model specification, in parameters governing the specified model, and in initial and boundary conditions. Only then can we be confident in the scientific inferences and forecasts made from an analysis. Probability and statistics provide a framework that accounts for multiple sources of uncertainty. Given the complexities of ecological studies, the hierarchical statistical model is an invaluable tool. This approach is not new in ecology, and there are many examples (both Bayesian and non-Bayesian) in the literature illustrating the benefits of this approach. In this article, we provide a baseline for concepts, notation, and methods, from which discussion on hierarchical statistical modeling in ecology can proceed. We have also planted some seeds for discussion and tried to show where the practical difficulties lie. Our thesis is that hierarchical statistical modeling is a powerful way of approaching ecological analysis in the presence of inevitable but quantifiable uncertainties, even if practical issues sometimes require pragmatic compromises.

INCORPORATING MULTIPLE SOURCES OF STOCHASTICITY INTO DYNAMIC POPULATION MODELS

Many standard statistical models used to examine population dynamics ignore significant sources of stochasticity. Usually only process error is included, and uncertainty due to errors in data collection is omitted or not directly specified in the model. We show how standard time-series models for population dynamics can be extended to include both observational and process error and how to perform inference on parameters in these models in the Bayesian setting. Using simulated data, we show how ignoring observation error can be misleading. We argue that the standard Bayesian techniques used to perform inference, including freely available software, are generally applicable to a variety of time-series models. Corresponding Editor: O. N. Bjornstad.

An assessment of coefficient accuracy in linear regression models with spatially varying coefficients

The realization in the statistical and geographical sciences that a relationship between an explanatory variable and a response variable in a linear regression model is not always constant across a study area has led to the development of regression models that allow for spatially varying coefficients. Two competing models of this type are geographically weighted regression (GWR) and Bayesian regression models with spatially varying coefficient processes (SVCP). In the application of these spatially varying coefficient models, marginal inference on the regression coefficient spatial processes is typically of primary interest. In light of this fact, there is a need to assess the validity of such marginal inferences, since these inferences may be misleading in the presence of explanatory variable collinearity. In this paper, we present the results of a simulation study designed to evaluate the sensitivity of the spatially varying coefficients in the competing models to various levels of collinearity. The simulation study results show that the Bayesian regression model produces more accurate inferences on the regression coefficients than does GWR. In addition, the Bayesian regression model is overall fairly robust in terms of marginal coefficient inference to moderate levels of collinearity, and degrades less substantially than GWR with strong collinearity.

Commercial Density, Residential Concentration, and Crime: Land Use Patterns and Violence in Neighborhood Context

Drawing on Jacobs’s (1961) and Taylor’s (1988) discussions of the social control implications of mixed land use, the authors explore the link between commercial and residential density and violent crime in urban neighborhoods. Using crime, census, and tax parcel data for Columbus, Ohio, the authors find evidence of a curvilinear association between commercial and residential density and both homicide and aggravated assault, consistent with Jacobs’s expectations. At low levels, increasing commercial and residential density is positively associated with homicide and aggravated assault. Beyond a threshold, however, increasing commercial and residential density serves to reduce the likelihood of both outcomes. In contrast, the association between commercial and residential density and robbery rates is positive and linear. The implications of these findings for understanding the sources of informal social control in urban neighborhoods are discussed.

Dynamic Factor Process Convolution Models for Multivariate Space-Time Data with Application to Air Quality Assessment

We propose a Bayesian dynamic factor process convolution model for multivariate spatial temporal processes and illustrate the utility of this approach in modeling large air quality monitoring data. Key advantages of this modeling framework are a descriptive parametrization of the cross-covariance structure of the space-time processes and dimension reduction features that allow full Bayesian inference procedures to remain computationally tractable for large data sets. These features result from modeling space-time data as realizations of linear combinations of underlying space-time fields. The underlying latent components are constructed by convolving temporally-evolving processes defined on a grid covering the spatial domain and include both trend and cyclical components. We argue that mixtures of such components can realistically describe a variety of space-time environmental processes and are especially applicable to air pollution processes that have complex space–time dependencies. In addition to the computational benefits that arise from the dimension reduction features of the model, the process convolution structure permits misaligned and missing data without the need for imputation when fitting the model. This advantage is especially useful when constructing models for data collected at monitoring stations that have misaligned sampling schedules and that are frequently out of service for long stretches of time. We illustrate the modeling approach using a multivariate pollution dataset taken from the EPA’s CASTNet database.

Efficient models for correlated data via convolutions of intrinsic processes

Gaussian processes (GP) have proven to be useful and versatile stochastic models in a wide variety of applications including computer experiments, environmental monitoring, hydrology and climate modeling. A GP model is determined by its mean and covariance functions. In most cases, the mean is specified to be a constant, or some other simple linear function, whereas the covariance function is governed by a few parameters. A Bayesian formulation is attractive as it allows for formal incorporation of uncertainty regarding the parameters governing the GP. However, estimation of these parameters can be problematic. Large datasets, posterior correlation and inverse problems can all lead to difficulties in exploring the posterior distribution. Here, we propose an alternative model which is quite tractable computationally - even with large datasets or indirectly observed data - while still maintaining the flexibility and adaptiveness of traditional GP models. This model is based on convolving simple Markov random fields with a smoothing kernel. We consider applications in hydrology and aircraft prototype testing.

West Nile Virus Revisited: Consequences for North American Ecology

ABSTRACT It has been nine years since West Nile virus (WNV) emerged in New York, and its initial impacts on avian hosts and humans are evident across North America. The direct effects of WNV on avian hosts include documented population declines, but other, indirect ecological consequences of these changed bird communities, such as changes in seed dispersal, insect abundances, and scavenging services, are probable and demand attention. Furthermore, climate (seasonal precipitation and temperature) and land use are likely to influence the intensity and frequency of disease outbreaks, and research is needed to improve mechanistic understanding of these interacting forces. This article reviews the growing body of research describing the ecology of WNV and highlights critical knowledge gaps that must be addressed if we hope to manage disease risk, implement conservation strategies, and make forecasts in the presence of both climate change and WNV—or the next emergent pathogen.

Regional sensitivity of Greenland precipitation to NAO variability

The North Atlantic Oscillation (NAO) is a primary mode of interannual climate variability for the North Atlantic Ocean Basin and influences the climate over much of Europe and parts of North America. Knowledge of past variability of this oscillatory system is essential for efforts to understand, model, and predict future climate variability, particularly under a warming Earth scenario. As Greenland precipitation is modulated by the NAO, ice core-derived accumulation histories are incorporated into multi-proxy reconstructions. New ice core records from Greenland demonstrate that the NAOs influence on accumulation is temporally and spatially variable. The results presented indicate that (1) NAO modulation of accumulation is strongest and most persistent along the west-central side of Greenland, (2) records from central Greenland should be avoided and (3) the spatial character of the precipitation response to NAO variability has been influenced by the 20th century warming in the high Arctic.

A Bayesian Hierarchical Approach for Relating PM2.5 Exposure to Cardiovascular Mortality in North Carolina

Considerable attention has been given to the relationship between levels of fine particulate matter (particulate matter ≤ 2.5 μm in aerodynamic diameter; PM2.5) in the atmosphere and health effects in human populations. Since the U.S. Environmental Protection Agency began widespread monitoring of PM2.5 levels in 1999, the epidemiologic community has performed numerous observational studies modeling mortality and morbidity responses to PM2.5 levels using Poisson generalized additive models (GAMs). Although these models are useful for relating ambient PM2.5 levels to mortality, they cannot directly measure the strength of the effect of exposure to PM2.5 on mortality. In order to assess this effect, we propose a three-stage Bayesian hierarchical model as an alternative to the classical Poisson GAM. Fitting our model to data collected in seven North Carolina counties from 1999 through 2001, we found that an increase in PM2.5 exposure is linked to increased risk of cardiovascular mortality in the same day and next 2 days. Specifically, a 10-μg/m3 increase in average PM2.5 exposure is associated with a 2.5% increase in the relative risk of current-day cardiovascular mortality, a 4.0% increase in the relative risk of cardiovascular mortality the next day, and an 11.4% increase in the relative risk of cardiovascular mortality 2 days later. Because of the small sample size of our study, only the third effect was found to have > 95% posterior probability of being > 0. In addition, we compared the results obtained from our model to those obtained by applying frequentist (or classical, repeated sampling-based) and Bayesian versions of the classical Poisson GAM to our study population.

Exploring Space-Time Structure in Ozone Concentration Using a Dynamic Process Convolution Model

Given daily ozone readings from 512 weather stations in the Eastern United States, we are interested in both predicting future ozone concentrations and in gaining insight into the space-time dependence structure of the data. We model ozone concentration as a process that moves across the region over time and exhibits spatial dependence locally in time. Our hope is to better understand the space-time dependence in ozone, so that this information can be used to assess the effectiveness of new monitoring network configurations. Process convolutions not only provide a framework for incorporating time dependence in spatial modeling, but also remain computationally tractable with large datasets. Standard dynamic linear modeling methods can be used to specify the time dependence allowing efficient posterior exploration. We consider a few variations of these space-time process convolution models that incorporate different levels of space-time dependence.