Abstract. 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.

The statistic known as Moran's I is widely used to test for the presence of spatial dependence in observations taken on a lattice. Under the null hypothesis that the data are independent and identically distributed normal random variates, the distribution of Moran's I is known, and hypothesis tests based on this statistic have been shown in the literature to have various optimality properties. Given its simplicity, Moran's I is also frequently used outside of the formal hypothesis-testing setting in exploratory analyses of spatially referenced data; however, its limitations are not very well understood. To illustrate these limitations, we show that, for data generated according to the spatial autoregressive (SAR) model, Moran's I is only a good estimator of the SAR model's spatialdependence parameter when the parameter is close to 0. In this research, we develop an alternative closed-form measure of spatial autocorrelation, which we call APLE, because it is an approximate profile-likelihood estimator (APLE) of the SAR model's spatialdependence parameter. We show that APLE can be used as a test statistic for, and an estimator of, the strength of spatial autocorrelation. We include both theoretical and simulation-based motivations (including comparison with the maximum-likelihood estimator), for using APLE as an estimator. In conjunction, we propose the APLE scatterplot, an exploratory graphical tool that is analogous to the Moran scatterplot, and we demonstrate that the APLE scatterplot is a better visual tool for assessing the strength of spatial autocorrelation in the data than the Moran scatterplot. In addition, Monte Carlo tests based on both APLE and Moran's I are introduced and compared. Finally, we include an analysis of the well-known Mercer and Hall wheat-yield data to illustrate the difference between APLE and Moran's I when they are used in exploratory spatial data analysis. MotivationThe The matrix W {w ij } is a known spatial-neighborhood matrix with elements w ii 5 0 for i 5 1, . . . , n, and « («(s 1 ), . . ., «(s n )) 0 is a vector of independently and identically distributed normal random variables, each with mean zero and variance s 2 . As Z 5 (I À rW) À 1 «, it is clear that E(Z) 5 0. In practice, data will almost always have to undergo some detrending. For the rest of the article, we shall assume that this detrending has already taken place, and hence it is appropriate for Z to have mean 0.In exploratory analyses of spatial data, a formal statistical model may not be explicitly assumed. However, we argue that in these situations the informal notion of spatial dependence, or spatial autocorrelation, is often implicitly based on a SAR framework where the goal is to assess the predictive ability of neighboring values of the data. In order for informal assessments of the strength of spatial autocorrelation to translate into this implied formal statistical modeling framework, exploratory spatial data analysis (ESDA) tools should be based on estimators of r, the spatial autocorrelation parameter in ...

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. Bjørnstad.

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.