Many statisticians have had the experience of fitting a linear model with uncorrelated errors, then adding a spatiallycorrelated error term (random effect) and finding that the estimates of the fixed-effect coefficients have changed substantially. We show that adding a spatially-correlated error term to a linear model is equivalent to adding a saturated collection of canonical regressors, the coefficients of which are shrunk toward zero, where the spatial map determines both the canonical regressors and the relative extent of the coefficients' shrinkage. Adding a spatially-correlated error term can also be seen as inflating the error variances associated with specific contrasts of the data, where the spatial map determines the contrasts and the extent of error-variance inflation. We show how to avoid this spatial confounding by restricting the spatial random effect to the orthogonal complement (residual space) of the fixed effects, which we call restricted spatial regression. We consider five proposed interpretations of spatial confounding and draw implications about what, if anything, one should do about it. In doing so, we debunk the common belief that adding a spatially-correlated random effect adjusts fixed-effect estimates for spatially-structured missing covariates. This article has supplementary material online. These helpful people cannot be blamed for the results.

Variable selection can be challenging, particularly in situations with a large number of predictors with possibly high correlations, such as gene expression data. In this article, a new method called the OSCAR (octagonal shrinkage and clustering algorithm for regression) is proposed to simultaneously select variables while grouping them into predictive clusters. In addition to improving prediction accuracy and interpretation, these resulting groups can then be investigated further to discover what contributes to the group having a similar behavior. The technique is based on penalized least squares with a geometrically intuitive penalty function that shrinks some coefficients to exactly zero. Additionally, this penalty yields exact equality of some coefficients, encouraging correlated predictors that have a similar effect on the response to form predictive clusters represented by a single coefficient. The proposed procedure is shown to compare favorably to the existing shrinkage and variable selection techniques in terms of both prediction error and model complexity, while yielding the additional grouping information.

With the advance of methods for estimating species distribution models has come an interest in how to best combine datasets to improve estimates of species distributions. This has spurred the development of data integration methods that simultaneously harness information from multiple datasets while dealing with the specific strengths and weaknesses of each dataset. We outline the general principles that have guided data integration methods and review recent developments in the field. We then outline key areas that allow for a more general framework for integrating data and provide suggestions for improving sampling design and validation for integrated models. Key to recent advances has been using point‐process thinking to combine estimators developed for different data types. Extending this framework to new data types will further improve our inferences, as well as relaxing assumptions about how parameters are jointly estimated. These along with the better use of information regarding sampling effort and spatial autocorrelation will further improve our inferences. Recent developments form a strong foundation for implementation of data integration models. Wider adoption can improve our inferences about species distributions and the dynamic processes that lead to distributional shifts.

Abstract. The last decade has seen a dramatic increase in the use of species distribution models (SDMs) to characterize patterns of species' occurrence and abundance. Efforts to parameterize SDMs often create a tension between the quality and quantity of data available to fit models. Estimation methods that integrate both standardized and non-standardized data types offer a potential solution to the tradeoff between data quality and quantity. Recently several authors have developed approaches for jointly modeling two sources of data (one of high quality and one of lesser quality). We extend their work by allowing for explicit spatial autocorrelation in occurrence and detection error using a Multivariate Conditional Autoregressive (MVCAR) model and develop three models that share information in a less direct manner resulting in more robust performance when the auxiliary data is of lesser quality. We describe these three new approaches ("Shared," "Correlation," "Covariates") for combining data sources and show their use in a case study of the Brown-headed Nuthatch in the Southeastern U.S. and through simulations. All three of the approaches which used the second data source improved out-of-sample predictions relative to a single data source ("Single"). When information in the second data source is of high quality, the Shared model performs the best, but the Correlation and Covariates model also perform well. When the information quality in the second data source is of lesser quality, the Correlation and Covariates model performed better suggesting they are robust alternatives when little is known about auxiliary data collected opportunistically or through citizen scientists. Methods that allow for both data types to be used will maximize the useful information available for estimating species distributions.

SUMMARYSince quantile regression curves are estimated individually, the quantile curves can cross, leading to an invalid distribution for the response. A simple constrained version of quantile regression is proposed to avoid the crossing problem for both linear and nonparametric quantile curves. A simulation study and a reanalysis of tropical cyclone intensity data shows the usefulness of the procedure. Asymptotic properties of the estimator are equivalent to the typical approach under standard conditions, and the proposed estimator reduces to the classical one if there is no crossing. The performance of the constrained estimator has shown significant improvement by adding smoothing and stability across the quantile levels.