Bayesian measures of explained variance and pooling in multilevel (hierarchical) models

Explained variance (R^2) is a familiar summary of the fit of a linear regression and has been generalized in various ways to multilevel (hierarchical) models. The multilevel models we consider in this paper are characterized by hierarchical data structures in which individuals are grouped into units (which themselves might be further grouped into larger units), and there are variables measured on individuals and each grouping unit. The models are based on regression relationships at different levels, with the first level corresponding to the individual data, and subsequent levels corresponding to between-group regressions of individual predictor effects on grouping unit variables. We present an approach to defining R^2 at each level of the multilevel model, rather than attempting to create a single summary measure of fit. Our method is based on comparing variances in a single fitted model rather than comparing to a null model. In simple regression, our measure generalizes the classical adjusted R^2. We also discuss a related variance comparison to summarize the degree to which estimates at each level of the model are pooled together based on the level-specific regression relationship, rather than estimated separately. This pooling factor is related to the concept of shrinkage in simple hierarchical models. We illustrate the methods on a dataset of radon in houses within counties using a series of models ranging from a simple linear regression model to a multilevel varying-intercept, varying-slope model.adjusted R-squared, Bayesian inference, hierarchical model, multilevel regression, partial pooling, shrinkage

Model Choice and Diagnostics for Linear Mixed-Effects Models Using Statistics on Street Corners

The complexity of linear mixed-effects (LME) models means that traditional diagnostics are rendered less effective. This is due to a breakdown of asymptotic results, boundary issues, and visible patterns in residual plots that are introduced by the model fitting process. Some of these issues are well known and adjustments have been proposed. Working with LME models typically requires that the analyst keeps track of all the special circumstances that may arise. In this paper we illustrate a simpler but generally applicable approach to diagnosing LME models. We explain how to use new visual inference methods for these purposes. The approach provides a unified framework for diagnosing LME fits and for model selection. We illustrate the use of this approach on several commonly available data sets. A large-scale Amazon Turk study was used to validate the methods. R code is provided for the analyses.Comment: 52 pages, 15 figures, 3 table

Upper Midwest Climate Variations: Farmer Responses to Excess Water Risks

Persistent above average precipitation and runoff and associated increased sediment transfers from cultivated ecosystems to rivers and oceans are due to changes in climate and human action. The US Upper Midwest has experienced a 37% increase in precipitation (1958–2012), leading to increased crop damage from excess water and off-farm loss of soil and nutrients. Farmer adaptive management responses to changing weather patterns have potential to reduce crop losses and address degrading soil and water resources. This research used farmer survey (n = 4778) and climate data (1971–2011) to model influences of geophysical context, past weather, on-farm flood and saturated soils experiences, and risk and vulnerability perceptions on management practices. Seasonal precipitation varied across six Upper Midwest subregions and was significantly associated with variations in management. Increased warm-season precipitation (2007–2011) relative to the past 40 yr was positively associated with no-till, drainage, and increased planting on highly erodible land (HEL). Experience with saturated soils was significantly associated with increased use of drainage and less use of no-till, cover crops, and planting on HEL. Farmers in counties with a higher percentage of soils considered marginal for row crops were more likely to use no-till, cover crops, and plant on HEL. Respondents who sell corn through multiple markets were more likely to have planted cover crops and planted on HEL in 2011.This suggests that regional climate conditions may not well represent individual farmers’ actual and perceived experiences with changing climate conditions. Accurate climate information downscaled to localized conditions has potential to influence specific adaptation strategies

Revised estimates of influenza-associated excess mortality, United States, 1995 through 2005

<p>Abstract</p> <p>Background</p> <p>Excess mortality due to seasonal influenza is thought to be substantial. However, influenza may often not be recognized as cause of death. Imputation methods are therefore required to assess the public health impact of influenza. The purpose of this study was to obtain estimates of monthly excess mortality due to influenza that are based on an epidemiologically meaningful model.</p> <p>Methods and Results</p> <p>U.S. monthly all-cause mortality, 1995 through 2005, was hierarchically modeled as Poisson variable with a mean that linearly depends both on seasonal covariates and on influenza-certified mortality. It also allowed for overdispersion to account for extra variation that is not captured by the Poisson error. The coefficient associated with influenza-certified mortality was interpreted as ratio of total influenza mortality to influenza-certified mortality. Separate models were fitted for four age categories (<18, 18–49, 50–64, 65+). Bayesian parameter estimation was performed using Markov Chain Monte Carlo methods. For the eleven year study period, a total of 260,814 (95% CI: 201,011–290,556) deaths was attributed to influenza, corresponding to an annual average of 23,710, or 0.91% of all deaths.</p> <p>Conclusion</p> <p>Annual estimates for influenza mortality were highly variable from year to year, but they were systematically lower than previously published estimates. The excellent fit of our model with the data suggest validity of our estimates.</p

Relative Importance of Predictors in Multilevel Modeling

The Pratt index is a useful and practical strategy for day-to-day researchers when ordering predictors in a multiple regression analysis. The purposes of this study are to introduce and demonstrate the use of the Pratt index to assess the relative importance of predictors for a random intercept multilevel model

The impact of West Nile virus on the abundance of selected North American birds

<p>Abstract</p> <p>Background</p> <p>The emergence of West Nile virus (WNV) in North America has been associated with high mortality in the native avifauna and has raised concerns about the long-term impact of WNV on bird populations. Here, we present results from a longitudinal analysis of annual counts of six bird species, using North American Breeding Bird Survey data from ten states (1994 to 2010). We fit overdispersed Poisson models to annual counts. Counts from successive years were linked by an autoregressive process that depended on WNV transmission intensity (annual West Nile neuroinvasive disease reports) and was adjusted by El Niño Southern Oscillation events. These models were fit using a Markov chain Monte Carlo algorithm.</p> <p>Results</p> <p>Model fit was mostly excellent, especially for American Crows, for which our models explained between 26% and 81% of the observed variance. The impact of WNV on bird populations was quantitatively evaluated by contrasting hypothetical count trajectories (omission of WNV) with observed counts. Populations of American crows were most consistently affected with a substantial cumulative impact in six of ten states. The largest negative impact, almost 60%, was found in Illinois. A regionally substantial decline was also seen for American Robins and House Sparrows, while the other species appeared unaffected.</p> <p>Conclusions</p> <p>Our results confirm findings from previous studies that single out American Crows as the species most vulnerable to WNV infection. We discuss strengths and limitations of this and other methods for quantifying the impact of WNV on bird populations.</p

Global patterns and drivers of ecosystem functioning in rivers and riparian zones

River ecosystems receive and process vast quantities of terrestrial organic carbon, the fate of which depends strongly on microbial activity. Variation in and controls of processing rates, however, are poorly characterized at the global scale. In response, we used a peer-sourced research network and a highly standardized carbon processing assay to conduct a global-scale field experiment in greater than 1000 river and riparian sites. We found that Earth’s biomes have distinct carbon processing signatures. Slow processing is evident across latitudes, whereas rapid rates are restricted to lower latitudes. Both the mean rate and variability decline with latitude, suggesting temperature constraints toward the poles and greater roles for other environmental drivers (e.g., nutrient loading) toward the equator. These results and data set the stage for unprecedented “next-generation biomonitoring” by establishing baselines to help quantify environmental impacts to the functioning of ecosystems at a global scale

Global patterns and drivers of ecosystem functioning in rivers and riparian zones

River ecosystems receive and process vast quantities of terrestrial organic carbon, the fate of which depends strongly on microbial activity. Variation in and controls of processing rates, however, are poorly characterized at the global scale. In response, we used a peer-sourced research network and a highly standardized carbon processing assay to conduct a global-scale field experiment in greater than 1000 river and riparian sites. We found that Earth's biomes have distinct carbon processing signatures. Slow processing is evident across latitudes, whereas rapid rates are restricted to lower latitudes. Both the mean rate and variability decline with latitude, suggesting temperature constraints toward the poles and greater roles for other environmental drivers (e.g., nutrient loading) toward the equator. These results and data set the stage for unprecedented "next-generation biomonitoring" by establishing baselines to help quantify environmental impacts to the functioning of ecosystems at a global scale.peerReviewe