324 Donovan et al.: Analysis of Vegetation and Stormwater Runoff in an Urban Watershed Table 2. Candidate variables for possible inclusion in total flow and change-in-flow models. Variable Definition Total flow Flow (15 minute lag) Change in flow Sewer shed area Rain Rain (15-minute lag) Percent tree cover Percent grass and shrubs Percent impervious Slope Sewer flow in cubic meters per hectare per 15-minute increment Total flow in previous 15-minute increment Absolute percentage change in flow previous 15-minute increment Area of sewer shed in hectares Rainfall in cubic meters per hectare per 15-minute increment Rainfall in previous 15-minute increment Percent of sewer shed covered by tree canopy Percent of sewer shed covered by grass and shrubs Percent of sewer shed covered by impervious surface Mean slope of a sewer shed RESULTS Hausman specification tests of all models found no statistically significant difference between co- efficients estimated using random-effects estima- tors and those using fixed-effects estimators (P < 0.01). Under these conditions, both estimators are consistent, but only the random-effects estimators are efficient (Baum 2001). Therefore, all models were estimated using random-effects estimators. In the flow model, researchers found evidence of autocorrelation (P < 0.001), so a 15-minute lag of sewer flow was included as an independent variable, which removed the autocorrelation (P = 0.732). In the change-in-flow model, researchers found no evidence of autocorrelation (P = 0.149). There was, however, evidence of heteroskedas- ticity in both models (P < 0.001). Therefore, standard errors were estimated with the usual model-based techniques and with sandwich esti- mators (Greene 2000), which are robust to some forms of misspecification including heteroskedas- ticity and non-normally distributed error terms. Table 3 shows the results of the model of June sewer-flow rate. As expected, the coefficients on lagged sewer flow and rain are positive. Per- centage tree cover is negatively associated with sewer flow, but the coefficient on trees is only significant with model-based standard errors. Standard errors were included using both estima- tors, as both estimators have shortcomings. Het- eroskedasticity can lead to inefficient coefficient estimates when using model-based estimators. In contrast, sandwich estimators are robust to heteroskedasticity, but they are sensitive to mis- specification of the likelihood function. In addi- tion, the data have only 34 observational units, and sandwich estimators are only asymptoti- cally efficient and consistent (Rabe-Hesketh and Skrondal 2012). Therefore, it is not clear which ©2016 International Society of Arboriculture is the appropriate estimator, and readers should interpret results from this model cautiously. To provide some context for coefficient on tree cover, if tree cover had been one percentage point higher (mean tree-canopy cover in the sample was 28%), then sewer flow would have been reduced by 4,550 cubic meters over the two-day storm. Tree cover was not significant in the change- in-flow model (Table 4), but grass and shrubs were, which suggests that groundcover is more effective at slowing runoff than tree cover. The coefficient on grass and shrubs is -0.5, which means that a one percentage point increase in grass and shrubs would result in a 0.5 percent- age point decrease in absolute percentage flow. To provide some context for this number, the mean absolute percentage change in flow for the sample (including only those observations where absolute percentage change in flow is non-zero) is 19.5. A one percentage point increase in grass and shrubs would reduce this to 19. This is a modest change, but under the right circumstances, such a change might prevent a pipe from backing up. In addi- tion, these results apply to all grass and shrubs. Groundcover specifically designed to slow run- off (e.g., bioswales) is likely to be more effective at moderating peak flow. The coefficient on trees and shrubs is significant for both estimators, so results are less ambiguous than the flow model. When comparing the benefits of trees with the benefits of shrubs and grass, it is impor- tant to recall that increasing tree canopy does not require reducing the amount of impervi- ous surface or grass and shrubs. Therefore, the coefficient on trees should be interpreted as the marginal effect of additional tree canopy. In contrast, increasing grass and shrubs neces- sarily requires reducing impervious surface, so the coefficient on grass and shrubs should be
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