Spatial Random Effects Research Articles

Bayesian Shared Component Spatial Modeling for Assessing the Shared and Age Group-Specific Mental Health Disorder Risk of Young and Old Age Groups: A Case Study of Toronto Neighborhoods, Canada

Mental health disorder risks of young and old age groups hold considerable importance for understanding present and future risk burdens. However, assessing mental health risks is significantly constrained by the influence of shared and age group-specific spatial processes and risk factors. Therefore, this study employed Bayesian shared component spatial modeling (BSCSM) to analyze mental health disorder data obtained from young (20–44 years) and old (65+ years) age groups in Toronto. BSCSM was employed to model the shared and age group-specific disorder risk and to identify hotspot areas. The unmeasured covariates, overdispersion, and latent spatial processes were adjusted using spatial and non-spatial random effect terms. The findings from BSCSM were finally compared with non-shared component modeling approaches. The results suggest that over 60% of variations in mental health disorder risk for both age groups could be explained by the shared component. The high-risk neighborhoods were mainly localized in southern and north-central Toronto for the young and old age groups. Deviance information criterion values suggested that models from BSCSM outperformed non-BSCSM models. BSCSM risk maps were also better at identifying high-risk areas. This work demonstrated that both shared and age group-specific risks are essential for assessing mental health disorder risk and devising targeted interventions.

Lung Cancer Prevalence in Virginia: A Spatial Zipcode-Level Analysis via INLA.

Examining lung cancer (LC) cases in Virginia (VA) is essential due to its significant public health implications. By studying demographic, environmental, and socioeconomic variables, this paper aims to provide insights into the underlying drivers of LC prevalence in the state adjusted for spatial associations at the zipcode level. We model the available VA zipcode-level LC counts via (spatial) Poisson and negative binomial regression models, taking into account missing covariate data, zipcode-level spatial association and allow for overdispersion. Under latent Gaussian Markov Random Field (GMRF) assumptions, our Bayesian hierarchical model powered by Integrated Nested Laplace Approximation (INLA) considers simultaneous (spatial) imputation of all missing covariates through elegant prediction. The spatial random effect across zip codes follows a Conditional Autoregressive (CAR) prior. Zip codes with elevated smoking indices demonstrated a corresponding increase in LC counts, underscoring the well-established connection between smoking and LC. Additionally, we observed a notable correlation between higher Social Deprivation Index (SDI) scores and increased LC counts, aligning with the prevalent pattern of heightened LC prevalence in regions characterized by lower income and education levels. On the demographic level, our findings indicated higher LC counts in zip codes with larger White and Black populations (with Whites having higher prevalence than Blacks), lower counts in zip codes with higher Hispanic populations (compared to non-Hispanics), and higher prevalence among women compared to men. Furthermore, zip codes with a larger population of elderly people (age ≥ 65 years) exhibited higher LC prevalence, consistent with established national patterns. This comprehensive analysis contributes to our understanding of the complex interplay of demographic and socioeconomic factors influencing LC disparities in VA at the zip code level, providing valuable information for targeted public health interventions and resource allocation. Implementation code is available at GitHub.

Modelling spatiotemporal patterns of visceral leishmaniasis incidence in two endemic states in India using environment, bioclimatic and demographic data, 2013-2022.

As of 2021, the National Kala-azar Elimination Programme (NKAEP) in India has achieved visceral leishmaniasis (VL) elimination (<1 case / 10,000 population/year per block) in 625 of the 633 endemic blocks (subdistricts) in four states. The programme needs to sustain this achievement and target interventions in the remaining blocks to achieve the WHO 2030 target of VL elimination as a public health problem. An effective tool to analyse programme data and predict/ forecast the spatial and temporal trends of VL incidence, elimination threshold, and risk of resurgence will be of use to the programme management at this juncture. We employed spatiotemporal models incorporating environment, climatic and demographic factors as covariates to describe monthly VL cases for 8-years (2013-2020) in 491 and 27 endemic and non-endemic blocks of Bihar and Jharkhand states. We fitted 37 models of spatial, temporal, and spatiotemporal interaction random effects with covariates to monthly VL cases for 6-years (2013-2018, training data) using Bayesian inference via Integrated Nested Laplace Approximation (INLA) approach. The best-fitting model was selected based on deviance information criterion (DIC) and Watanabe-Akaike Information Criterion (WAIC) and was validated with monthly cases for 2019-2020 (test data). The model could describe observed spatial and temporal patterns of VL incidence in the two states having widely differing incidence trajectories, with >93% and 99% coverage probability (proportion of observations falling inside 95% Bayesian credible interval for the predicted number of VL cases per month) during the training and testing periods. PIT (probability integral transform) histograms confirmed consistency between prediction and observation for the test period. Forecasting for 2021-2023 showed that the annual VL incidence is likely to exceed elimination threshold in 16-18 blocks in 4 districts of Jharkhand and 33-38 blocks in 10 districts of Bihar. The risk of VL in non-endemic neighbouring blocks of both Bihar and Jharkhand are less than 0.5 during the training and test periods, and for 2021-2023, the probability that the risk greater than 1 is negligible (P<0.1). Fitted model showed that VL occurrence was positively associated with mean temperature, minimum temperature, enhanced vegetation index, precipitation, and isothermality, and negatively with maximum temperature, land surface temperature, soil moisture and population density. The spatiotemporal model incorporating environmental, bioclimatic, and demographic factors demonstrated that the KAMIS database of the national programmme can be used for block level predictions of long-term spatial and temporal trends in VL incidence and risk of outbreak / resurgence in endemic and non-endemic settings. The database integrated with the modelling framework and a dashboard facility can facilitate such analysis and predictions. This could aid the programme to monitor progress of VL elimination at least one-year ahead, assess risk of resurgence or outbreak in post-elimination settings, and implement timely and targeted interventions or preventive measures so that the NKAEP meet the target of achieving elimination by 2030.

A Bayesian spatially explicit estimation of daily egg production: application to anchovy in the Bay of Biscay

Biomass estimates of fish resources by the daily egg production method (DEPM) are sensitive to the high variability of the daily egg production ( P0) and egg mortality (Z) in space. This work presents a Bayesian approach to estimate these parameters. A prior distribution of Z based on literature serves to overcome the biologically implausible Z estimates that can result from frequentist approaches. In addition to the classical estimation of a single P0 over the spawning area, the Bayesian framework allows also the modelling of egg densities in space, by including either spatial random effects, smoothing functions, or kriging like models, providing insights into the spatial variability of P0. The Bayesian approach was applied to the Bay of Biscay anchovy DEPM surveys. Results showed that this Bayesian approximation solved the implausible Z problem resulting in tighter credible intervals of both P0 and Z. Overall, spatial models outperformed the non-spatial model in terms of goodness of fit and resulted in slightly different total production estimates across models for each year, with a moderate decrease on uncertainty estimates.

Characterizing opioid overdose hotspots for place-based overdose prevention and treatment interventions: A geo-spatial analysis of Rhode Island, USA

ObjectiveExamine differences in neighborhood characteristics and services between overdose hotspot and non-hotspot neighborhoods and identify neighborhood-level population factors associated with increased overdose incidence. MethodsWe conducted a population-based retrospective analysis of Rhode Island, USA residents who had a fatal or non-fatal overdose from 2016 to 2020 using an environmental scan and data from Rhode Island emergency medical services, State Unintentional Drug Overdose Reporting System, and the American Community Survey. We conducted a spatial scan via SaTScan to identify non-fatal and fatal overdose hotspots and compared the characteristics of hotspot and non-hotspot neighborhoods. We identified associations between census block group-level characteristics using a Besag-York-Mollié model specification with a conditional autoregressive spatial random effect. ResultsWe identified 7 non-fatal and 3 fatal overdose hotspots in Rhode Island during the study period. Hotspot neighborhoods had higher proportions of Black and Latino/a residents, renter-occupied housing, vacant housing, unemployment, and cost-burdened households. A higher proportion of hotspot neighborhoods had a religious organization, a health center, or a police station. Non-fatal overdose risk increased in a dose responsive manner with increasing proportions of residents living in poverty. There was increased relative risk of non-fatal and fatal overdoses in neighborhoods with crowded housing above the mean (RR 1.19 [95 % CI 1.05, 1.34]; RR 1.21 [95 % CI 1.18, 1.38], respectively). ConclusionNeighborhoods with increased prevalence of housing instability and poverty are at highest risk of overdose. The high availability of social services in overdose hotspots presents an opportunity to work with established organizations to prevent overdose deaths.

Multivariate nearest‐neighbors Gaussian processes with random covariance matrices

AbstractWe propose a non‐stationary spatial model based on a normal‐inverse‐Wishart framework, conditioning on a set of nearest‐neighbors. The model, called nearest‐neighbor Gaussian process with random covariance matrices is developed for both univariate and multivariate spatial settings and allows for fully flexible covariance structures that impose no stationarity or isotropic restrictions. In addition, the model can handle duplicate observations and missing data. We consider an approach based on integrating out the spatial random effects that allows fast inference for the model parameters. We also consider a full hierarchical approach that leverages the sparse structures induced by the model to perform fast Monte Carlo computations. Strong computational efficiency is achieved by leveraging the adaptive localized structure of the model that allows for a high level of parallelization. We illustrate the performance of the model with univariate and bivariate simulations, as well as with observations from two stationary satellites consisting of albedo measurements.

Spatiotemporal analysis of suicide attempts in Colombia from 2018 to 2020.

Suicide is one of the leading death causes worldwide, mainly among young adults, and Colombia has experienced an increase during the XXI century. The suicide impact has diverged between age groups and locations in Colombia, where young adults have taken higher incidences than the other age groups. The COVID-19 lockdown induced changes in mental health, affecting the previous suicide trends in the country. We conducted a spatiotemporal analysis of suicide attempts in Colombia per age group, adopting Bayesian models that represent 85,526 individual records in 1,121 municipalities from 2018 to 2020 using R-INLA. We found that Colombia exhibited an increase in suicide-attempt incidence from 2018 to 2019, and suddenly, the incidence fell in the first semester of 2020. The fixed effect of the models evidenced the highest risk in overall municipalities per trimester in the age group between 15-19 years old. The spatial random effect per model evidenced municipalities with the highest risk in the age groups between 10 to 59 years, mainly in the states in the Andean region of Colombia, and other states such as Putumayo, Vaupés, Arauca, Córdoba, Amazonas, and Meta. The temporal random effect evidenced a decay in suicide trends from the fourth trimester of 2019 to 2020, except in the age group > 59 years old. Geographically, our study pinpointed specific regions in Colombia, particularly in the central, southwest, and southeast areas, where the incidence of suicide attempts exceeded 100 cases per 100,000 inhabitants. The nuanced breakdown of incidence across different age groups further underscores the importance of tailoring preventive strategies based on age-specific and regional risk factors.

Habitat occupancy of the threatened Diademed Plover (Phegornis mitchellii) is not affected by llama grazing or peatland size, but declines with peatland humidity.

Many habitat-specialist organisms occur in distinct, patchy habitat, yet do not occupy all patches, and an important question is why apparently suitable habitat remains unoccupied. We examined factors influencing patch occupancy in near-threatened, little-known Diademed Plovers (Phegornis mitchellii), arguably the bird most specialized to life in High Andean peatlands. Andean peatlands are well-suited to occupancy modelling because they are discrete patches of humid habitat within a matrix of high-altitude steppe. We hypothesized that Diademed Plovers occupy preferably larger and more humid peatlands, and avoid peatlands used for grazing by llamas and vicuñas, which may trample vegetation and nests. From December 2021 to February 2022 (breeding season), we conducted plover occupancy surveys (2-4) on 40 peatlands at Lagunas de Vilama, a landscape of arid steppe and wetlands above 4,500 m in NW Argentina. We measured peatland size, grazing pressure, topographic and remotely-sensed variables that correlate with humidity, and incorporated these as covariates in occupancy models. Occupancy models showed that more than 50% of the studied peatlands were used by Diademed Plovers and most showed signs of reproduction, highlighting the importance of the Vilama Wetlands for Diademed Plover conservation. Within peatlands, Diademed Plovers were most often associated with headwaters. The top ranked occupancy model included constant detection, random spatial effects, and a single occupancy covariate: mean NDWI (Normalized Difference Water Index, an index correlated with water content and humidity) over the previous three years. Contrary to our prediction, Diademed Plovers preferred less water-saturated peatlands (lower NDWI), possibly to avoid nest flooding. This may be especially important in wet years, like the year when we conducted our surveys. Neither peatland size nor grazing by llamas and vicuñas affected peatland use by Diademed Plovers, suggesting that llama grazing at current levels may be compatible with plover conservation. For organisms that specialize on humid habitats, such as peatlands, factors affecting occupancy may vary temporally with variation in climate, and we recommend follow-up surveys across multi-year timescales to untangle the impact of climate on animals' use of humid habitats.

A Comparison of Yield Prediction Approaches Using Long-Term Multi-Crop Site-Specific Data

Highlights Long-term site-specific multiple-crop yield data is used in two modeling approaches. Different sources of data, including soil properties, topography, weather, and multispectral data, are tested. Linear modeling and Bayesian hierarchical modeling (BHM) are evaluated by examining predictions of relative yield. BHM with spatio-temporal effects provides the best estimates, handles missing data, and provides uncertainty estimates in time and space. Abstract. Growers in the inland Pacific Northwest face numerous challenges in managing cropping systems. Climate variability, soil degradation, and topography all lead to significant spatial and temporal variability in yield. Often, yield modeling approaches such as deep learning can be “black boxes” or suffer from parameter uncertainty and instability, as with process-based crop models. To explore an alternative, we examined nearly two decades of crop rotation data from the R.J. Cook Agronomy Farm, along with soil properties, topography, weather, and multispectral data. We then tested two modeling approaches to estimate yield: linear modeling (LM) and Bayesian hierarchical modeling (BHM). We found BHM with spatial and temporal random effects performed best in predicting relative yield, both using soil variables as predictors or remotely sensed data. Since the BHM approach handles missing data, offers the possibility of farmer knowledge to be incorporated into prior probabilities, and gives uncertainty, this methodology lends itself well to decision support tools and on-farm study design. Keywords: Bayesian, Long-term, Modeling, Remote sensing, Yield.

The Contribution of Road Construction on Regional Economic Development in Indonesia

The development process in Indonesia's provinces is expected to contribute to regional economic growth. This research aims to evaluate road construction's impact on regional economic development and the spatial spillover effect in 34 Indonesian provinces. This paper constructs Spatial Autoregressive Model (SAR), Spatial Durbin Model (SDM), Spatial Autocorrelation Model (SAC), Spatial Error Model (SEM), and Generalised Spatial Random Effect Model (GSPRE) for 34 provinces in Indonesia from 2010 to 2020 to capture the effect of road construction on regional economic development and spatial spillover effect. Compared to other models, the spatial Durbin model has the minimum Akaike information criterion (AIC) value, indicating that it is the most suited model. The result indicates that road infrastructure has little impact on regional economic development. Moreover, road development has a negative value and negligible spatial spillover effect.

A simplified spatial+ approach to mitigate spatial confounding in multivariate spatial areal models

Spatial areal models encounter the well-known and challenging problem of spatial confounding. This issue makes it arduous to distinguish between the impacts of observed covariates and spatial random effects. Despite previous research and various proposed methods to tackle this problem, finding a definitive solution remains elusive. In this paper, we propose a simplified version of the spatial+ approach that involves dividing the covariate into two components. One component captures large-scale spatial dependence, while the other accounts for short-scale dependence. This approach eliminates the need to separately fit spatial models for the covariates. We apply this method to analyse two forms of crimes against women, namely rapes and dowry deaths, in Uttar Pradesh, India, exploring their relationship with socio-demographic covariates. To evaluate the performance of the new approach, we conduct extensive simulation studies under different spatial confounding scenarios. The results demonstrate that the proposed method provides reliable estimates of fixed effects and posterior correlations between different responses.

Generative spatial generalized dissimilarity mixed modelling (spGDMM): An enhanced approach to modelling beta diversity

Abstract Turnover, or change in the composition of species over space and time, is one of the primary ways to define beta diversity. Inferring what factors impact beta diversity is not only important for understanding biodiversity processes but also for conservation planning. At present, a popular approach to understanding the drivers of compositional turnover is through generalized dissimilarity modelling (GDM). We argue that the current GDM approach suffers several limitations and provide an alternative modelling approach that remedies these issues. We propose using generative spatial random effects models implemented in a Bayesian framework. We offer hierarchical specifications to yield full regression and spatial predictive inference, both with associated full uncertainties. The approach is illustrated by examining dissimilarity in three datasets: tree survey data from Panama's Barro Colorado Island (BCI), plant occurrence data from southwest Australia and plant abundance surveys from the Greater Cape Floristic Region (GCFR) of South Africa. We select a best model using out‐of‐sample predictive performance. We find that the form of the best model differs across the three datasets, but our models provide performance ranging from comparable to significant improvement over GDMs. Within the GCFR, the spatial random effects play a more important role in the modelling than all the environmental variables. We have proposed a model that provides several improvements to the current GDM framework. This includes advantages such as a flexible spatially varying mean function, spatial random effects that capture dependence unaccounted for by explanatory variables, and spatially heterogeneous variance structure. All these features are offered in a model that can adequately handle a large incidence of total dissimilarity through ‘one‐inflation’, as would be expected from highly biodiverse areas with steep turnover gradients.

Spatial Confounding and Spatial+ for Nonlinear Covariate Effects

Regression models for spatially varying data use spatial random effects to reflect spatial correlation structure. Such random effects, however, may interfere with the covariate effect estimates and make them unreliable. This problem, known as spatial confounding, is complex and has only been studied for models with linear covariate effects. However, as illustrated by a forestry example in which we assess the effect of soil, climate, and topography variables on tree health, the covariate effects of interest are in practice often unknown and nonlinear. We consider, for the first time, spatial confounding in spatial models with nonlinear effects implemented in the generalised additive models (GAMs) framework. We show that spatial+, a recently developed method for alleviating confounding in the linear case, can be adapted to this setting. In practice, spatial+ can then be used both as a diagnostic tool for investigating whether covariate effect estimates are affected by spatial confounding and for correcting the estimates for the resulting bias when it is present. Supplementary materials accompanying this paper appear online.

A fast method for fitting integrated species distribution models

Abstract Integrated distribution models (IDMs) predict where species might occur using data from multiple sources, a technique thought to be especially useful when data from any individual source are scarce. Recent advances allow us to fit such models with latent terms to account for dependence within and between data sources, but they are computationally challenging to fit. We propose a fast new methodology for fitting integrated distribution models using presence/absence and presence‐only data, via a spatial random effects approach combined with automatic differentiation. We have written an R package (called scampr) for straightforward implementation of our approach. We use simulation to demonstrate that our approach has comparable performance to INLA—a common framework for fitting IDMs—but with computation times up to an order of magnitude faster. We also use simulation to look at when IDMs can be expected to outperform models fitted to a single data source, and find that the amount of benefit gained from using an IDM is a function of the relative amount of additional information available from incorporating a second data source into the model. We apply our method to predict 29 plant species in NSW, Australia, and find particular benefit in predictive performance when data from a single source are scarce and when compared to models for presence‐only data. Our faster methods of fitting IDMs make it feasible to more deeply explore the model space (e.g. comparing different ways to model latent terms), and in future work, to consider extensions to more complex models, for example the multi‐species setting.

The neighborhood deprivation gradient and child physical abuse and neglect: A Bayesian spatial model

BackgroundChild abuse and neglect is a public health priority due to its long-term maladaptive consequences. No study in the USA has assessed the nature and magnitude of the social deprivation effect on substantiated child maltreatment risk. ObjectivesTo examine linear and non-linear relationships between area level deprivation and the log-risk of both substantiated physical abuse and neglect while accounting for spatial and heterogeneous random effects. MethodsSubstantiated child maltreatment and population data (2008–2015) were aggregated to neighborhoods in Bernalillo County, New Mexico. The contribution of area level deprivation to the geographical variation in the log-risks of substantiated child physical abuse and neglect was modeled using Bayesian spatial regression. ResultsForty-three percent and 46.4 % of the 153 neighborhoods recorded greater risk for either substantiated physical abuse or neglect compared to the county average. The most deprived 20 % of neighborhoods had 71 % and 72 % more cases of substantiated physical abuse and neglect, respectively, than would be expected if the substantiations were randomly distributed throughout the county. Area level deprivation explained 47 % of the variation in substantiated physical abuse and 51 % of the variation in substantiated neglect after controlling for both spatial autocorrelation and heterogeneity. ConclusionsImplications from this study can be used to quantify disparities in substantiated child maltreatment attributed to regional differences in social deprivation and to identify priority areas for intervention.

Understanding the role of environmental and socioeconomic factors in the geographic variation of breast cancer risk in the US-wide Sister Study

ObjectiveTo describe the geographic pattern of breast cancer incidence in a nationwide prospective cohort and investigate whether environmental exposures and/or neighborhood socioeconomic status explain observed geographic disparities. MethodsUsing accelerated failure time models with a spatial random effect term, we mapped the health region-level association between residential location and breast cancer incidence for 44,707 participants in the Sister Study after controlling for established individual-level breast cancer risk factors. We performed a variable selection process to select environmental exposures [i.e., ambient nitrogen dioxide (NO2) and fine particulate matter (PM2.5), PM2.5 chemical composition, outdoor light at night (LAN), ambient noise, ultraviolet radiation, and greenspace] and neighborhood-level factors [i.e., population density and area deprivation index (ADI)] that predicted breast cancer incidence and quantified the spatial variation explained by the selected factors. We also considered whether the geographic pattern and predictors were similar when restricting to estrogen receptor-positive (ER+) tumors. ResultsWe observed a spatial patterning in the incidence of overall breast cancer (Moran's I = 16.7, p < 0.05) and ER+ breast cancer (Moran's I = 13.2, p < 0.05), with a lower risk observed in the South and Southeast and a greater risk in the Northwest and certain areas of the Midwest and Northeast. NO2, LAN, and ADI explained 21.4% of the spatial variation in overall breast cancer incidence whereas NO2, PM2.5 chemical composition, LAN, greenspace, and ADI together explained 63.3% of the spatial variation in ER+ breast cancer incidence. ConclusionsOur findings provide additional evidence for a role of environmental exposures in breast cancer incidence and suggest that geographic-based risk factors may vary according to breast cancer subtype. Our findings support the need for additional research to quantify the relative contributions of geographic-based risk factors for breast cancer.

Spatially Dependent Bayesian Modeling of Geostatistics Data and Its Application for Tuberculosis (TB) in China

Geostatistics data in regions always have highly spatial heterogeneous, yet the regional features of the data itself cannot be ignored. In this paper, a novel latent Bayesian spatial model is proposed, which incorporates the spatial dependence of different subjects and the spatial random effects to further analysis the influence of spatial effect. The model is verified to be compatible with the integrated nested Laplace approximation (INLA) framework and is fitted using INLA and stochastic partial differential equation (SPDE). The posterior marginal distribution of parameters is estimated with high precision. Additionally, a practical application of the model is presented using tuberculosis (TB) incidence data in China from 2015 to 2019. The results show that the fitting accuracy of our model is higher than the general Bayesian spatial model using INLA-SPDE.

Spatio-temporal trends of the age-at-menarche percentiles among Portuguese women since 1920

BackgroundThis work aims to study the spatio-temporal evolution of a woman’s age at menarche in the central region of Portugal. One of the concerns of the study is early or late menarches; thus, we consider percentile regression to build the respective curves as opposed to the more traditional mean regression approach.MethodsWe analysed the data from N=452,348 women born in the period 1920–1973 who attended a free breast cancer screening program between 1990 and 2019. Distributional regression models inside the package GAMLSS in R were considered. These methods allowed us not only to model the location (mean) of the specific probability distribution of the age at menarche, but also allowed for the scale (variance) parameter of this distribution to depend on covariates. Additionally, a spatial random-effect was considered in order to capture the correlation at the regional level. The obtained clustered spatial effects were analysed to assess geographical differences among the percentiles of the age at menarche by year of birth.ResultsA decreasing trend in the age at menarche (about 1.5 years in 5 decades) and regional differences for all the considered percentiles were found. Women living in the north-central areas of the central region of Portugal tend to have menarche at older ages.ConclusionWe obtained percentile estimates for the age at menarche by year of birth and region of residence and demonstrated that these two explanatory variables have an impact on the explanation about the decreasing trend in age at a woman’s first menstruation.

Gentrification, neighborhood socioeconomic factors and urban vegetation inequities: A study of greenspace and tree canopy increases in Montreal, Canada

This study investigates the relationship between increasing urban vegetation and census tract-level green inequities, as well as the role of social indicators in this relationship. We analyzed the augmentation of greenspace and tree canopy in Montreal, Canada, between 2011 and 2017, and its effect on green inequities based on material deprivation, the percentage of visible minorities, and gentrification status using Poisson spatial random effect models. Our analyses showed an increase in greenspace from 57.4% to 65.8% and tree canopy from 21.1% to 22.3% between 2011 and 2017. Census tracts (CTs) with higher levels of material deprivation or a higher percentage of visible minority population had less greenspace and tree canopy at baseline in 2011. Additionally, CTs that were not gentrified had less greenspace and tree canopy than ineligible for gentrification CTs. Furthermore, CTs with more visible minorities, higher levels of material deprivation, or those that did not gentrify had smaller increases in greenspace and tree canopy between 2011 and 2017. Among CTs with more visible minorities or higher levels of material deprivation, those with greater greenspace and tree canopy at baseline also experienced greater increases during the study time period. Conversely, among ineligible for gentrification CTs, those with less greenspace/tree canopy at baseline experienced greater increase in greenspace/tree canopy. Our analysis revealed that despite an increase in urban vegetation, inequities in urban vegetation persists. To reduce green inequities and promote social equity in this particular study area, urban planning policies should prioritize CTs with higher levels of material deprivation, more visible minorities, or those that did not gentrify, and focus on increasing urban vegetation.

Accounting for unobserved spatial variation in step selection analyses of animal movement via spatial random effects

Abstract Step selection analysis (SSA) is a common framework for understanding animal movement and resource selection using telemetry data. Such data are, however, inherently autocorrelated in space, a complication that could impact SSA‐based inference if left unaddressed. Accounting for spatial correlation is standard statistical practice when analysing spatial data, and its importance is increasingly recognized in ecological models (e.g. species distribution models). Nonetheless, no framework yet exists to account for such correlation when analysing animal movement using SSA. Here, we extend the popular method integrated step selection analysis (iSSA) by including a Gaussian field (GF) in the linear predictor to account for spatial correlation. For this, we use the Bayesian framework R‐INLA and the stochastic partial differential equations (SPDE) technique. We show through a simulation study that our method provides accurate fixed effects estimates, quantifies their uncertainty well and improves the predictions. In addition, we demonstrate the practical utility of our method by applying it to three wolverine (Gulo gulo) tracks. Our method solves the problems of assuming spatially independent residuals in the SSA framework. In addition, it offers new possibilities for making long‐term predictions of habitat usage.