AS2-1 Climate Change

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【1.1】

Enter the model R2 (the “Multiple R-squared” value):

R-square = 0.7415

*****************

ClimateChange_Testing <- subset(ClimateChange, Year > 2006)
ClimateChange_Training <- subset(ClimateChange, Year <= 2006)

ClimateChange_TrainingReg <- lm(Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + TSI + Aerosols, data = ClimateChange_Training)
summary(ClimateChange_TrainingReg)

------------------------------------------------------------------

Call:
lm(formula = Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + 
    TSI + Aerosols, data = ClimateChange_Training)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.26009 -0.06126 -0.00145  0.05684  0.32530 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.231e+02  2.087e+01  -5.897 1.13e-08 ***
MEI          6.367e-02  6.685e-03   9.524  < 2e-16 ***
CO2          6.906e-03  2.395e-03   2.883 0.004262 ** 
CH4          1.645e-04  5.470e-04   0.301 0.763863    
N2O         -1.620e-02  9.461e-03  -1.712 0.088083 .  
CFC.11      -6.410e-03  1.767e-03  -3.629 0.000342 ***
CFC.12       3.625e-03  1.104e-03   3.285 0.001159 ** 
TSI          9.181e-02  1.566e-02   5.861 1.37e-08 ***
Aerosols    -1.520e+00  2.188e-01  -6.949 2.88e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.09329 on 263 degrees of freedom
Multiple R-squared:  0.7415,    Adjusted R-squared:  0.7337 
F-statistic: 94.32 on 8 and 263 DF,  p-value: < 2.2e-16

【1.2】

Which variables are significant in the model? We will consider a variable signficant only if the p-value is below 0.05. (Select all that apply.)

(1) MEI
(2) CO2 
(5) CFC.11 
(6) CFC.12 
(7) TSI 
(8) Aerosols 

【2.1】

Which of the following is the simplest correct explanation for this contradiction?

(3) All of the gas concentration variables reflect human development - N2O and CFC.11 are correlated with other variables in the data set.

【2.2】

Compute the correlations between all the variables in the training set. Which of the following independent variables is N2O highly correlated with (absolute correlation greater than 0.7)? Select all that apply.

(2) CO2
(3) CH4
(5) CFC.12

Which of the following independent variables is CFC.11 highly correlated with? Select all that apply.

(3) CH4
(5) CFC.12

------------------------------------------------------------------

> cor(ClimateChange_Training)
                Year         Month           MEI         CO2         CH4         N2O
Year      1.00000000 -0.0279419602 -0.0369876842  0.98274939  0.91565945  0.99384523
Month    -0.02794196  1.0000000000  0.0008846905 -0.10673246  0.01856866  0.01363153
MEI      -0.03698768  0.0008846905  1.0000000000 -0.04114717 -0.03341930 -0.05081978
CO2       0.98274939 -0.1067324607 -0.0411471651  1.00000000  0.87727963  0.97671982
CH4       0.91565945  0.0185686624 -0.0334193014  0.87727963  1.00000000  0.89983864
N2O       0.99384523  0.0136315303 -0.0508197755  0.97671982  0.89983864  1.00000000
CFC.11    0.56910643 -0.0131112236  0.0690004387  0.51405975  0.77990402  0.52247732
CFC.12    0.89701166  0.0006751102  0.0082855443  0.85268963  0.96361625  0.86793078
TSI       0.17030201 -0.0346061935 -0.1544919227  0.17742893  0.24552844  0.19975668
Aerosols -0.34524670  0.0148895406  0.3402377871 -0.35615480 -0.26780919 -0.33705457
Temp      0.78679714 -0.0998567411  0.1724707512  0.78852921  0.70325502  0.77863893
              CFC.11        CFC.12         TSI    Aerosols        Temp
Year      0.56910643  0.8970116635  0.17030201 -0.34524670  0.78679714
Month    -0.01311122  0.0006751102 -0.03460619  0.01488954 -0.09985674
MEI       0.06900044  0.0082855443 -0.15449192  0.34023779  0.17247075
CO2       0.51405975  0.8526896272  0.17742893 -0.35615480  0.78852921
CH4       0.77990402  0.9636162478  0.24552844 -0.26780919  0.70325502
N2O       0.52247732  0.8679307757  0.19975668 -0.33705457  0.77863893
CFC.11    1.00000000  0.8689851828  0.27204596 -0.04392120  0.40771029
CFC.12    0.86898518  1.0000000000  0.25530281 -0.22513124  0.68755755
TSI       0.27204596  0.2553028138  1.00000000  0.05211651  0.24338269
Aerosols -0.04392120 -0.2251312440  0.05211651  1.00000000 -0.38491375
Temp      0.40771029  0.6875575483  0.24338269 -0.38491375  1.00000000

【3】

Enter the coefficient of N2O in this reduced model:

2.532e-02

Enter the model R2:

0.7261

------------------------------------------------------------------

> ClimateChange_TrainingReg2 <- lm(Temp ~ MEI + N2O + TSI + Aerosols, data = ClimateChange_Training)
> summary(ClimateChange_TrainingReg2)

Call:
lm(formula = Temp ~ MEI + N2O + TSI + Aerosols, data = ClimateChange_Training)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.27916 -0.05975 -0.00595  0.05672  0.34195 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.162e+02  2.022e+01  -5.747 2.37e-08 ***
MEI          6.419e-02  6.652e-03   9.649  < 2e-16 ***
N2O          2.532e-02  1.311e-03  19.307  < 2e-16 ***
TSI          7.949e-02  1.487e-02   5.344 1.89e-07 ***
Aerosols    -1.702e+00  2.180e-01  -7.806 1.19e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.09547 on 279 degrees of freedom
Multiple R-squared:  0.7261,    Adjusted R-squared:  0.7222 
F-statistic: 184.9 on 4 and 279 DF,  p-value: < 2.2e-16

【4】

Enter the R2 value of the model produced by the step function:

0.744

Which of the following variable(s) were eliminated from the full model by the step function? Select all that apply.

(3) CH4

------------------------------------------------------------------

ClimateChangeReg <- lm(Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + TSI + Aerosols, data = ClimateChange)
> summary(ClimateChangeReg)

Call:
lm(formula = Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + 
    TSI + Aerosols, data = ClimateChange)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.26228 -0.05868  0.00051  0.05718  0.32170 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.277e+02  1.919e+01  -6.654 1.36e-10 ***
MEI          6.632e-02  6.186e-03  10.722  < 2e-16 ***
CO2          5.207e-03  2.192e-03   2.375   0.0182 *  
CH4          6.371e-05  4.977e-04   0.128   0.8982    
N2O         -1.693e-02  7.835e-03  -2.161   0.0315 *  
CFC.11      -7.278e-03  1.461e-03  -4.980 1.07e-06 ***
CFC.12       4.272e-03  8.763e-04   4.875 1.77e-06 ***
TSI          9.586e-02  1.401e-02   6.844 4.38e-11 ***
Aerosols    -1.582e+00  2.099e-01  -7.535 5.86e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.09182 on 299 degrees of freedom
Multiple R-squared:  0.744, Adjusted R-squared:  0.7371 
F-statistic: 108.6 on 8 and 299 DF,  p-value: < 2.2e-16

> summary(ClimateChangeReg)

Call:
lm(formula = Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + 
    TSI + Aerosols, data = ClimateChange)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.26228 -0.05868  0.00051  0.05718  0.32170 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.277e+02  1.919e+01  -6.654 1.36e-10 ***
MEI          6.632e-02  6.186e-03  10.722  < 2e-16 ***
CO2          5.207e-03  2.192e-03   2.375   0.0182 *  
CH4          6.371e-05  4.977e-04   0.128   0.8982    
N2O         -1.693e-02  7.835e-03  -2.161   0.0315 *  
CFC.11      -7.278e-03  1.461e-03  -4.980 1.07e-06 ***
CFC.12       4.272e-03  8.763e-04   4.875 1.77e-06 ***
TSI          9.586e-02  1.401e-02   6.844 4.38e-11 ***
Aerosols    -1.582e+00  2.099e-01  -7.535 5.86e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.09182 on 299 degrees of freedom
Multiple R-squared:  0.744, Adjusted R-squared:  0.7371 
F-statistic: 108.6 on 8 and 299 DF,  p-value: < 2.2e-16

> StepModel <- step(ClimateChangeReg)
Start:  AIC=-1462.11
Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + TSI + Aerosols

           Df Sum of Sq    RSS     AIC
- CH4       1   0.00014 2.5209 -1464.1
<none>                  2.5208 -1462.1
- N2O       1   0.03935 2.5601 -1459.3
- CO2       1   0.04756 2.5683 -1458.3
- CFC.12    1   0.20038 2.7211 -1440.5
- CFC.11    1   0.20911 2.7299 -1439.6
- TSI       1   0.39485 2.9156 -1419.3
- Aerosols  1   0.47860 2.9994 -1410.6
- MEI       1   0.96917 3.4899 -1363.9

【5】

Enter the testing set R2:

0.6547574

------------------------------------------------------------------

ClimateChangeReg <- lm(Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + TSI + Aerosols, data = ClimateChange)
summary(ClimateChangeReg)
StepModel <- step(ClimateChangeReg)

TempPredict <- predict(StepModel, newdata = ClimateChange_Testing)
summary(TempPredict)
SSE = sum((TempPredict - ClimateChange_Testing$Temp)^2)
SST = sum((mean(ClimateChange_Training$Temp)-ClimateChange_Testing$Temp)^2)
Rsquare = 1 - SSE/SST

> Rsquare
[1] 0.6547574