Compare raw and area-normalized random data
## Randomly generated CO2 (kg) & Area (km2) are uncorrelated: 0.0163
## `geom_smooth()` using formula 'y ~ x'
## Correlation Area-normalized CO2 (kg km-2) & area (km2): -0.2626
## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = CO2.km2 ~ area.ran)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9734973 -1409980 -540263 654075 120309598
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4468517 558315 8.004 8.57e-15 ***
## area.ran -19533 3216 -6.074 2.47e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6272000 on 498 degrees of freedom
## Multiple R-squared: 0.06898, Adjusted R-squared: 0.06711
## F-statistic: 36.9 on 1 and 498 DF, p-value: 2.473e-09
## ..as are area-normalized CO2 (kg/km2) & ln(Area): -0.4855
## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = CO2.km2 ~ log(area.ran))
##
## Residuals:
## Min 1Q Median 3Q Max
## -19962710 -1398123 72259 1241339 108063412
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17381189 1303399 13.34 <2e-16 ***
## log(area.ran) -3355781 270745 -12.39 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5682000 on 498 degrees of freedom
## Multiple R-squared: 0.2358, Adjusted R-squared: 0.2342
## F-statistic: 153.6 on 1 and 498 DF, p-value: < 2.2e-16
Compare raw and energy-normalized random data
## Correlation of CO2 (kg) & electricity generation (kWh): -0.0502
## `geom_smooth()` using formula 'y ~ x'
## Energy-normalized CO2 (kg kWh-1) & generation (kWh) are correlated: -0.4212
## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = CO2.GWh ~ elec.ran)
##
## Residuals:
## Min 1Q Median 3Q Max
## -325.82 -78.63 -27.15 37.43 2411.73
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.527e+02 1.879e+01 13.45 <2e-16 ***
## elec.ran -6.056e-05 5.844e-06 -10.36 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 206.1 on 498 degrees of freedom
## Multiple R-squared: 0.1774, Adjusted R-squared: 0.1757
## F-statistic: 107.4 on 1 and 498 DF, p-value: < 2.2e-16
## Scherer & Pfister (2016) model involves area on the right hand side
## Energy-normalized CO2 (kg kWh-1) & area/generation are correlated: 0.0718
## `geom_smooth()` using formula 'y ~ x'
##
## Call:
## lm(formula = CO2.GWh ~ ATE.km2.GWh)
##
## Residuals:
## Min 1Q Median 3Q Max
## -516.51 -71.32 -52.77 -25.52 2563.89
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 81.37 10.19 7.984 9.89e-15 ***
## ATE.km2.GWh 7321.29 4555.62 1.607 0.109
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 226.7 on 498 degrees of freedom
## Multiple R-squared: 0.005159, Adjusted R-squared: 0.003162
## F-statistic: 2.583 on 1 and 498 DF, p-value: 0.1087