**4.7 Ordinary least squares regression**

For the OLS model, LDI is statistically significant (*p* < 0.0001) for and robust probability. Precipitation determines the runoff depth, however, appears to be also significant at the 5%, but not as significant according to the robust p value (*p* = 0.05, robust *p* = 0.06). The variance inflation factor (VIF) tests for the redundancy amongst the explanatory variables that are added to the model. If two or more explanatory variables tell the same story because they are linearly related, the error variances are inflated, and the resulting multicollinearity produces a higher VIF. Studies suggest that accepting VIFs fewer than 7.5 or 10 is the rule of thumb for determining if there is multicollinearity within a dataset [35].

#### **Figure 3.**

*The monthly mean runoff depth per sub-basin in the Indian River Lagoon from January 2011 to June 2011. The values increase form dark green, to warmer colors.*

**129**

**Figure 5.**

**Figure 4.**

*A GIS-Based Approach for Determining Potential Runoff Coefficient and Runoff Depth…*

Within the OLS diagnostic results, statistical values provide information describing the performance of the model along with indicators for choosing an alternative model to adequately address the overall question (**Table 1**). The Akaike's information criterion (AIC) measures the overall model performance, which can be

*Indian River Lagoon chlorophyll* a *concentrations for spring 2011. The concentrations are estimated using medium resolution imaging spectrometer normalized difference chlorophyll index (image source: [29]).*

*The monthly mean runoff depth per sub-basins in the Indian River Lagoon from July 2011 to December 2011.* 

explains how

used in comparison to other regression analyses [36]. The multiple R2

*DOI: http://dx.doi.org/10.5772/intechopen.87163*

*The values increase form dark green, to warmer colors.*

*A GIS-Based Approach for Determining Potential Runoff Coefficient and Runoff Depth… DOI: http://dx.doi.org/10.5772/intechopen.87163*

#### **Figure 4.**

*Lagoon Environments around the World - A Scientific Perspective*

**4.7 Ordinary least squares regression**

The runoff depth values for the IRL ranges from 2.51 to 141.48 cm. The monthly runoff depth was calculated for 2011, the year of the super algal bloom in the IRL, to serve as potential explanation for the contribution of high surface runoff to locations of the algal blooms (**Figures 3** and **4**). The average runoff per sub-basin (**Figures 3** and **4**) was compared to the chlorophyll *a* concentrations quantified from European Space Agency's Medium Resolution Imaging Spectrometer (MERIS) for 2011 [29] (**Figures 5** and **6**). The maps were all assigned the same symbology to aid easier depictions of changes in quantities, and for comparison between months.

For the OLS model, LDI is statistically significant (*p* < 0.0001) for and robust probability. Precipitation determines the runoff depth, however, appears to be also significant at the 5%, but not as significant according to the robust p value (*p* = 0.05, robust *p* = 0.06). The variance inflation factor (VIF) tests for the redundancy amongst the explanatory variables that are added to the model. If two or more explanatory variables tell the same story because they are linearly related, the error variances are inflated, and the resulting multicollinearity produces a higher VIF. Studies suggest that accepting VIFs fewer than 7.5 or 10 is the rule of thumb for

*The monthly mean runoff depth per sub-basin in the Indian River Lagoon from January 2011 to June 2011.* 

determining if there is multicollinearity within a dataset [35].

**4.6 Runoff depth**

**128**

**Figure 3.**

*The values increase form dark green, to warmer colors.*

*The monthly mean runoff depth per sub-basins in the Indian River Lagoon from July 2011 to December 2011. The values increase form dark green, to warmer colors.*

#### **Figure 5.**

*Indian River Lagoon chlorophyll* a *concentrations for spring 2011. The concentrations are estimated using medium resolution imaging spectrometer normalized difference chlorophyll index (image source: [29]).*

Within the OLS diagnostic results, statistical values provide information describing the performance of the model along with indicators for choosing an alternative model to adequately address the overall question (**Table 1**). The Akaike's information criterion (AIC) measures the overall model performance, which can be used in comparison to other regression analyses [36]. The multiple R2 explains how

#### **Figure 6.**

*Indian River Lagoon chlorophyll* a *concentrations for 2011 (September–December 2011). The concentrations are estimated using medium resolution imaging spectrometer normalized difference chlorophyll index (image source: [29]).*


#### **Table 1.**

*A table of the ordinary least squares model variables.*

much the independent variables explain the variation in the dependent variable. In relation to the multiple R<sup>2</sup> the Adjusted R2 accounts for the model complexity. The multiple R2 the Adjusted R2 for this tests shows a small R2 between the variables (R<sup>2</sup> = 0.15). The OLS regression also tests for the model significance with the Joint F-statistic and Joint Wald statistic to support the significance of R<sup>2</sup> values (**Table 2**).

The Koenker's BP statistic tests for nonstationary and heteroscedasticity. The null hypothesis is that the dependent and independent variables have a consistent relationship in geographic space, thus being stationary [33]. The Koenker's BP statistic shows significant existence of nonstationary trends between runoff depth and LDI (*p* = 0.004). Therefore, the model significance was interpreted based on the Joint Wald statistic (*p* < 0.0001) which also indicates that the relationship was statistically significant. However, the overall measure of how well the explanatory variables explained the variation in the runoff depth from the OLS analysis was relatively small (R2 = 0.15). The Jarque-Bera statistic tests for model bias that can arise form nonstationary data, misspecification of independent variables, and skewed residuals [30]. In this case, the Jarque-Bera statistic shows no significant model bias (*p* = 0.064). A Global Moran's Index was performed on the residuals of the output file to test for the assumption of no spatial autocorrelation or clustering in the data. The Global Moran's Index showed statistically significant clustering rejecting the null hypothesis that the data are randomly distributed spatially within a global assumption (Moran's I = 0.07, *p* < 0.0001). Therefore, the OLS results should not be used to adequately interpret relationship between the explanatory variables and runoff depth.
