## Introduction

Water managers increasingly have a need for model-based information to address eutrophication problems in large watersheds and their coastal receiving waters (e.g., Chesapeake Bay, Connecticut River Basin and Long Island Sound, Mississippi River Basin and Gulf of Mexico) where the development of effective solutions is complicated by the diversity of nutrient contributions from upstream sources and catchments (New York State Department of Environmental Conservation and Connecticut Department of Environmental Protection, 2000; Koroncai *et al.*, 2003; Mississippi River/Gulf of Mexico Watershed Nutrient Task Force, 2008). Managing the watershed response at these large scales places multiple demands on the water-quality model. The model must provide an accurate assessment of water-quality conditions for locations that are deficient in monitoring data; the model must make an accurate determination of the relative importance of an extensive list of nutrient sources contributing to a particular water-quality impairment, many of these sources being physically distant from the impaired area; and the model must adequately describe how the aquatic system will respond to hypothetical changes in nutrient sources or factors affecting their transport.

The U.S. Geological Survey’s (USGS) SPAtially Referenced Regressions On Watershed attributes (SPARROW) model – a hybrid statistical and process-based mass balance model – was developed to explain spatial variability in mean water-quality conditions and predict source contributions in the streams of large, heterogeneous basins (Schwarz *et al.*, 2006; Preston *et al.*, 2009) (see also the brief description of the SPARROW methodology contained in the Supporting Information for this article). SPARROW has been used previously to assess nutrient loadings in large watersheds in the United States (U.S.) (e.g., Smith *et al.*, 1997; Alexander *et al.*, 2008) and other countries (Alexander *et al.*, 2002). An important question in applying SPARROW and other water-quality models in large watersheds is how the governing material transport equations should differ spatially in their functional forms and/or coefficients to account for the effects of landscape heterogeneity and spatial scaling on nutrient supply and transport processes. Despite advances in watershed science to address this question, uncertainties remain over the nature of these effects and how they should be represented in watershed models (e.g., McDonnell *et al.*, 2007; Kirchner, 2009). Existing national-scale SPARROW nutrient models are estimated with the assumption that the coefficients of the model that describe the marginal effect of water-quality explanatory variables on stream nutrient loads (e.g., the loading response from an incremental change in soil permeability; the change in the fraction of nutrient removed in streams from a unit change in water travel time) are the same for all locations nationally (Alexander *et al.*, 2008). This assumption insures that the widest ranges of environmental conditions are used to estimate the effects of individual model processes, and imparts a parsimonious form that facilitates the interpretation of the estimated coefficients. The assumption may be overly simplistic, however, if there exist latent processes, related to landscape heterogeneity or process interactions, for which surrogate measures of their effects are not available in geospatial datasets. Under these conditions, large-scale model development could lead to biased predictions in particular regions and under certain management or forecasting scenarios.

In recognition of potential regional variation in water-quality model processes, the USGS National Water Quality Assessment (NAWQA) Program has developed a set of SPARROW regional major river basin (MRB) nutrient models, described in this Featured Collection (for a summary of these models, see Preston *et al.*, this issue). The estimation of these regional models has benefited from a large assemblage of water-quality data, representing an approximate order of magnitude increase in the number of monitoring sites, across all regions, compared to the number of sites from which data were available to construct existing national-scale SPARROW nutrient models (Smith *et al.*, 1997; Alexander *et al.*, 2008). The inherent flexibility afforded by independent regional models allows the predictions of current water-quality conditions derived from these models to have less bias and greater precision than those of national models; thus, from the standpoint of water-quality assessment, the regional models out-perform the national models.

Regionalization of models, however, also poses a potential negative consequence: an individual regional model likely encompasses a narrower range in basin attributes than does a national model. For example, a climate variable typically exhibits less variation within a region than across the entire nation. In a model, an attribute’s effect on water quality is mediated by a coefficient, the estimation of which is made more precise the greater the variation of the attribute in the sample of data used to estimate the model. The reduced variation of an attribute within a confined region thereby reduces the precision of the attendant coefficient. Thus, users of a regional model (e.g., water resource managers) face a fundamental tradeoff in model accuracy between the bias and precision of the model predictions. On the one hand, if a model coefficient’s true value varies across regions, the loss in precision due to less within-region variability in the associated basin attribute may be an acceptable outcome for obtaining an unbiased coefficient estimate. On the other hand, if the true coefficient does not vary across regions, the reduced precision implies a loss in accuracy of the estimated coefficient, a loss that possibly could be ameliorated by expanding the spatial extent of the model to include a wider range of basin attributes. It remains an open question, therefore, whether the advantage of added flexibility afforded by individual regional models, which facilitates a less biased assessment of existing water-quality conditions, outweighs the loss of precision, which potentially diminishes a water manager’s ability to use the model for predicting changes in water quality brought about by changes in its causative factors.

This article has two major objectives: the first objective is to evaluate regional variation in the nutrient model coefficients using a “fixed-effects” approach (Judge *et al.*, 1985) to determine whether a single national model should be rejected in favor of a more regional-based approach. The “fixed-effects” approach to regionalization requires specifying multiple sets of coefficients in the national model, a separate set for each region of the analysis, with each regional set consisting of all the process coefficients included in the national model. This approach is a generalization of the fixed effects approach commonly used in longitudinal studies, which typically limit the fixed effects specification to an intercept term. Functionally, with fixed-effects specified for all process coefficients, the approach is equivalent to estimating independent regional models, the manner of regionalization implemented by the SPARROW MRB models. The fixed-effects method has been used, to varying degrees, in numerous previous SPARROW modeling studies (for examples of partial approaches, whereby fixed effects are specified only for a subset of the process coefficients, see Smith *et al.*, 2003; Hoos and McMahon, 2009). An advantage of the approach is its flexibility – it places no restrictions on the pattern of variation taken by the regional process coefficients. The chief disadvantage, as mentioned above, is that the precision of the coefficient estimates of the independently estimated models may be compromised if there is too little within-region variation in basin attributes.

The second major objective of this article is to evaluate the potential for improving the precision of the regional coefficient estimates. As suggested above, if it is known that regional coefficients are similar, it may be possible to improve precision by pooling information across regions. In the context of the fixed-effects approach, one way to do this is to estimate the model with cross-region constraints placed on the coefficients. The resulting model, which we call a hybrid model, effectively pools all the region-specific information, allowing the estimation of process coefficients that are not statistically different across regions to benefit from the greater variation in basin attributes afforded by a national-scale analysis. The constrained estimates of the fixed-effects coefficients retain some degree of regional variation, with the extent of variation depending on the particular cross-region constraints imposed as part of the specification of the hybrid model.

A distinct advantage of the hybrid model, from the standpoint of the analysis described here, is that it encompasses both the regional fixed-effects model and the national model. At one extreme, if there are no constraints, the hybrid model is identical to the regional fixed-effects model, the model that is functionally equivalent to specifying independent regional models. At the other extreme, if all possible independent constraints are applied, the hybrid model becomes identical to the national model. Thus, by partially constraining the regional fixed-effects model, the hybrid model provides a framework for evaluating the tradeoff between achieving prediction accuracy and improving coefficient precision in a regional context.

The national models serving as the subject of the fixed-effects analysis are the previously published SPARROW models for total nitrogen (TN) and total phosphorus (TP), estimated for the base year 1992 (Alexander *et al.*, 2008). As our interest is to isolate the effects of regionalization, yet retain as much as possible the specifications of the existing national models, this analysis uses data for the same set of monitoring stations that were used by Alexander *et al.* (2008). This collection of stations is too small, however, to obtain reasonable regional estimates for the number of regions used in the NAWQA MRB regional studies, which are based on many more stations. To compensate, this study adopts a rather coarse partitioning of the nation into only three regions: the East, Northwest, and Southwest regions of the conterminous U.S. (see Figure 1). This coarse regionalization makes it unlikely that the full range of regional effects present in the U.S. are properly accounted for, and supporting evidence of this shortcoming is presented. The limited size of the dataset used in this analysis also makes it likely that some of the regional variation exhibited by the hybrid model coefficient estimates is spurious. For these reasons, it is prudent to view the hybrid model coefficient estimates presented here as preliminary. The subsequent production of “management-ready” coefficient estimates should be possible by applying the hybrid method to the expansive set of data incorporated in the NAWQA MRB studies, the analysis of which will likely also benefit from altering the specification of the models beyond the existing national SPARROW models. Despite the shortcomings, we believe the data are sufficient to demonstrate the potential utility of moving beyond modeling frameworks that are either exclusively national or strictly regionally independent.

The following analysis begins with a brief description of the recently published national nutrient SPARROW models (Alexander *et al.*, 2008), developed for the 1:500,000-scale Reach File 1 network (Nolan *et al.*, 2002). An initial evaluation of the national nutrient models is undertaken to determine if regional fixed effects are present in three major regions of the conterminous U.S., with attention given to whether the regional effects can be associated with specific groups of transport processes that are an intrinsic feature of a SPARROW model specification. A subsequent section describes a practical approach for identifying constraints to be applied to the regional fixed-effects model to estimate the hybrid model. A section describing the results of the estimation of the hybrid model follows, with emphasis placed on demonstrating the effects of agglomerating information across regions on the precision of coefficient estimates and model predictions. The concluding sections contain a discussion of regionalization methodology in the context of other approaches found in the literature and a summary of the findings of this study. All mathematical derivations used in the analysis are described in an Appendix and in the Supporting Information for this article. Additionally, the Supporting Information includes a brief description of the SPARROW model and a detailed presentation of results from the regional fixed-effects and hybrid models.