Arboriculture & Urban Forestry 42(5): September 2016 ample, in one area, a small reduction in peak flow may significantly reduce the number of sewer back- ups. However, in an area that has pipes that are of sufficient diameter to accommodate peak flow, that same reduction in peak flow may have no effect on fixed costs. Therefore, the effects of veg- etation on treatments costs are highly situational. To investigate the influence of trees on fixed and variable treatment costs, researchers estimated a total flow model and a change-in-flow model. These two measures of flow are shown in Figure 4, which graphs sewer flow as a function of time. Total flow is the integral of this function between two points in time (in this case, these two points are 15 minutes apart). The change-in-flow model is the first differential of this function. The first differential was approximated using the slope of the ray connecting the two points f(t0 ) and f(t15 ). 323 Statistical Analysis The data are structured as repeated measure- ments on the same observational unit (sewer shed). Data of this type can be analyzed using regression models of the following general form: ized change in flow for the ith sewer shed at time t, Xi,t [3] where Yi,t ing tree cover), εi,t 𝑌𝑌▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 = 𝛼𝛼▮䯧 +𝜷𝜷▮䯧 𝑿𝑿▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 +𝜐𝜐▮䯧 𝑖𝑖▮䯧 +𝜀𝜀▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 coefficients to be estimated in the regression step. Typically, linear models of this form are estimated using either fixed-effects or random-effects estima- tors. The researchers chose between the two based on a Hausman specification test (Hausman 1978). Variables were selected for inclusion in the is either normalized flow or normal- is a vector of independent variables (includ- with the unit-specific residual υi, and α and β are is an i.i.d. error term uncorrelated final model using iterative backward selec- tion. Variables were dropped from the model using progressively smaller p-value thresh- olds, with a final threshold of 0.1. The only exception to this selection criterion was rain- fall. Table 2 shows a complete list of candidate variables. A variance-covariance matrix was used to avoid including highly collinear com- binations of variables in the same model. It takes time for rainfall to pass through a sewer Figure 4. Total flow and change in flow. Not all the sewer sheds are the same size, so total flow was normalized by sewer-shed area (units: cubic meters per hectare per 15 minute increment). To make the interpretation of model coefficients more intuitive, rainfall was measured in the same units. In the change-in-flow model, different sized sewer sheds were accommodated by using absolute percentage change in flow from one 15-minute increment to the next. Formally, normalized total flow (NF) and normalized change in flow (NCF) are defined as: [1] 𝑁𝑁▮䯧 𝑁𝑁▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 = 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 𝐴𝐴▮䯧 𝐴𝐴▮䯧 𝐴𝐴▮䯧 𝐴𝐴▮䯧 𝑖𝑖▮䯧 [2] where i indexes sewer shed and t indexes time. 𝑁𝑁▮䯧 𝑁𝑁▮䯧 𝑁𝑁▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 = ���𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 +15−𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 � 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝐹𝐹▮䯧 𝑖𝑖▮䯧 ,𝑡𝑡▮䯧 � ∗ 100� system and reach a flow meter. Therefore, lagged rainfall were included in both models. For exam- ple, the variable rain denotes rainfall in the current 15-minute period, rain (15-minute lag) denotes rain- fall in the previous 15-minute period, and so forth. Several statistical issues can complicate the estimation of regression models using repeated- measurements data. Data are typically not inde- pendent. A random or fixed effect model addresses some of this dependence, but temporal autocorrela- tion can also be an issue. Autocorrelation was tested by using a Wooldridge test (Wooldridge 2002). As in simple linear regression, heteroskedasticity can also be an issue in repeated-measurements data. Heteroskedasticity was tested by compar- ing a model that assumes panel-level homoske- dasticity (error-term variance is the same across sewer sheds) to one that assumes panel-level het- eroskedasticity (error-term variance varies across sewer sheds) using a log-likelihood ratio test. ©2016 International Society of Arboriculture
September 2016
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