Chapter 7 Building Models for Interpretation

7.1 Plots for Model Selection

7.1.1 Correlation Matrix and Plot

Cars_Cat <- select_if(Cars2015, is.factor)
summary(Cars_Cat)
##         Make            Model            Type    Drive          Size   
##  Chevrolet: 8   CTS        :  2   7Pass    :15   AWD:25   Large   :29  
##  Ford     : 7   2 Touring  :  1   Hatchback:11   FWD:63   Midsized:34  
##  Hyundai  : 7   200        :  1   Sedan    :46   RWD:22   Small   :47  
##  Toyoto   : 7   3 i Touring:  1   Sporty   :11                         
##  Audi     : 6   3 Series GT:  1   SUV      :18                         
##  Nissan   : 6   300        :  1   Wagon    : 9                         
##  (Other)  :69   (Other)    :103
Cars_Num <- select_if(Cars2015, is.numeric)
C <- cor(Cars_Num, use = "pairwise.complete.obs")
round(C,2)
##           LowPrice HighPrice CityMPG HwyMPG FuelCap Length Width Wheelbase
## LowPrice      1.00      0.91   -0.65  -0.59    0.57   0.47  0.48      0.46
## HighPrice     0.91      1.00   -0.56  -0.49    0.47   0.39  0.37      0.39
## CityMPG      -0.65     -0.56    1.00   0.93   -0.77  -0.72 -0.78     -0.69
## HwyMPG       -0.59     -0.49    0.93   1.00   -0.75  -0.64 -0.75     -0.64
## FuelCap       0.57      0.47   -0.77  -0.75    1.00   0.82  0.85      0.79
## Length        0.47      0.39   -0.72  -0.64    0.82   1.00  0.81      0.92
## Width         0.48      0.37   -0.78  -0.75    0.85   0.81  1.00      0.76
## Wheelbase     0.46      0.39   -0.69  -0.64    0.79   0.92  0.76      1.00
## Height        0.02     -0.10   -0.39  -0.54    0.58   0.46  0.62      0.49
## UTurn         0.40      0.31   -0.73  -0.68    0.76   0.84  0.77      0.81
## Weight        0.55      0.43   -0.83  -0.84    0.91   0.82  0.91      0.81
## Acc030       -0.76     -0.74    0.64   0.51   -0.47  -0.38 -0.41     -0.31
## Acc060       -0.74     -0.72    0.68   0.52   -0.49  -0.47 -0.46     -0.38
## QtrMile      -0.76     -0.76    0.65   0.49   -0.45  -0.42 -0.41     -0.35
## PageNum      -0.23     -0.20    0.28   0.15   -0.15  -0.23 -0.20     -0.24
##           Height UTurn Weight Acc030 Acc060 QtrMile PageNum
## LowPrice    0.02  0.40   0.55  -0.76  -0.74   -0.76   -0.23
## HighPrice  -0.10  0.31   0.43  -0.74  -0.72   -0.76   -0.20
## CityMPG    -0.39 -0.73  -0.83   0.64   0.68    0.65    0.28
## HwyMPG     -0.54 -0.68  -0.84   0.51   0.52    0.49    0.15
## FuelCap     0.58  0.76   0.91  -0.47  -0.49   -0.45   -0.15
## Length      0.46  0.84   0.82  -0.38  -0.47   -0.42   -0.23
## Width       0.62  0.77   0.91  -0.41  -0.46   -0.41   -0.20
## Wheelbase   0.49  0.81   0.81  -0.31  -0.38   -0.35   -0.24
## Height      1.00  0.55   0.71   0.21   0.21    0.25    0.06
## UTurn       0.55  1.00   0.80  -0.36  -0.41   -0.37   -0.22
## Weight      0.71  0.80   1.00  -0.41  -0.43   -0.39   -0.20
## Acc030      0.21 -0.36  -0.41   1.00   0.95    0.95    0.25
## Acc060      0.21 -0.41  -0.43   0.95   1.00    0.99    0.26
## QtrMile     0.25 -0.37  -0.39   0.95   0.99    1.00    0.26
## PageNum     0.06 -0.22  -0.20   0.25   0.26    0.26    1.00
library(corrplot)
C <- corrplot(C)

7.2 Residual by Explanatory Variable Plots

First, we fit a model:

Cars_M6 <- lm(data=Cars2015, log(LowPrice) ~ QtrMile + Weight + HwyMPG)
summary(Cars_M6)
## 
## Call:
## lm(formula = log(LowPrice) ~ QtrMile + Weight + HwyMPG, data = Cars2015)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.82308 -0.14513 -0.01922  0.16732  0.41390 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  6.550e+00  4.220e-01  15.522  < 2e-16 ***
## QtrMile     -2.170e-01  1.844e-02 -11.770  < 2e-16 ***
## Weight       1.592e-04  4.456e-05   3.573 0.000532 ***
## HwyMPG      -9.611e-03  7.377e-03  -1.303 0.195410    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2198 on 106 degrees of freedom
## Multiple R-squared:  0.7732, Adjusted R-squared:  0.7668 
## F-statistic: 120.5 on 3 and 106 DF,  p-value: < 2.2e-16

7.2.1 Creating Residual by Explanatory Variable Plot

The code for residual by explanatory plot is shown. More plots can be added if there are additional explanatory variables.

P1 <- ggplot(data=data.frame(Cars_M6$residuals), 
             aes(y=Cars_M6$residuals, x=Cars_M6$model$QtrMile)) + 
  geom_point() + ggtitle("Cars Model Residual Plot") + 
  xlab("Quarter Mile Time") + ylab("Residuals") 
P2 <- ggplot(data=data.frame(Cars_M6$residuals), 
             aes(y=Cars_M6$residuals, x=Cars_M6$model$Weight)) + 
  geom_point() + ggtitle("Cars Model Residual Plot") + 
  xlab("Weight") + ylab("Residuals")
grid.arrange(P1, P2, ncol=2)

7.3 Section 7.3 Polynomial Regression

7.3.1 Plotting Polynomial Curves

ggplot(data=Cars2015, aes(y=LowPrice, x=FuelCap)) +
  geom_point()+ stat_smooth(method="lm", se=FALSE) + 
  stat_smooth(method="lm", se=TRUE, fill=NA,formula=y ~ poly(x, 2, raw=TRUE),colour="red") + 
  stat_smooth(method="lm", se=TRUE, fill=NA,formula=y ~ poly(x, 3, raw=TRUE),colour="green") + 
  stat_smooth(method="lm", se=TRUE, fill=NA,formula=y ~ poly(x, 4, raw=TRUE),colour="orange") + 
  stat_smooth(method="lm", se=TRUE, fill=NA,formula=y ~ poly(x, 5, raw=TRUE),colour="purple") + 
  stat_smooth(method="lm", se=TRUE, fill=NA,formula=y ~ poly(x, 6, raw=TRUE),colour="darkgreen")

7.3.2 Fitting Polynomial Models

To fit a model with higher powers, use I(Variable^k).

CarLength_M1 <- lm(data=Cars2015, LowPrice~FuelCap)
CarLength_M2 <- lm(data=Cars2015, LowPrice~FuelCap + I(FuelCap^2))
CarLength_M3 <- lm(data=Cars2015, LowPrice~FuelCap + I(FuelCap^2) + I(FuelCap^3))
CarLength_M4 <- lm(data=Cars2015, LowPrice~FuelCap + I(FuelCap^2) + I(FuelCap^3) + I(FuelCap^4))
CarLength_M5 <- lm(data=Cars2015, LowPrice~FuelCap + I(FuelCap^2) + I(FuelCap^3) + I(FuelCap^4) + 
                     I(FuelCap^5))
CarLength_M6 <- lm(data=Cars2015, LowPrice~FuelCap + I(FuelCap^2) + I(FuelCap^3) + I(FuelCap^4) + 
                     I(FuelCap^5)+ I(FuelCap^6))

We can view the model summary, using the summary() command, as usual.

7.4 Adjusted \(R^2\), AIC, and BIC

We’ll calculate ddjusted \(R^2\), AIC, and BIC, for Model M6, above.

7.4.1 Example: Calculating adjusted \(R^2\), AIC, and BIC

summary(CarLength_M6)$adj.r.squared
## [1] 0.3573088
AIC(CarLength_M6)
## [1] 881.2588
BIC(CarLength_M6)
## [1] 902.8627