Car Seats Analysis

Sameer Mathur (2018)

Regression Analysis

Read the Car Seats Data

  Sales CompPrice Income Advertising Population Price ShelveLoc Age
1  9.50       138     73          11        276   120     0-Bad  42
2  4.15       141     64           3        340   128     0-Bad  38
3 10.81       124    113          13        501    72     0-Bad  78
4  9.01       121     78           9        150   100     0-Bad  26
5 10.14       145    119          16        294   113     0-Bad  42
6  2.99       103     74           0        359    97     0-Bad  55
  Education Urban  US Revenue Profit
1        17   Yes Yes 1140.00 228.00
2        13   Yes  No  531.20 106.24
3        16    No Yes  778.32 155.66
4        10    No Yes  901.00 180.20
5        12   Yes Yes 1145.82 229.16
6        11   Yes Yes  290.03  58.01

Structure of the data frame

# data structure of the dataframe
str(carSeats.df)

'data.frame':   400 obs. of  13 variables:
 $ Sales      : num  9.5 4.15 10.81 9.01 10.14 ...
 $ CompPrice  : int  138 141 124 121 145 103 104 130 119 157 ...
 $ Income     : int  73 64 113 78 119 74 99 60 98 53 ...
 $ Advertising: int  11 3 13 9 16 0 15 0 0 0 ...
 $ Population : int  276 340 501 150 294 359 226 144 18 403 ...
 $ Price      : int  120 128 72 100 113 97 102 138 126 124 ...
 $ ShelveLoc  : Factor w/ 3 levels "0-Bad","1-Medium",..: 1 1 1 1 1 1 1 1 1 1 ...
 $ Age        : int  42 38 78 26 42 55 58 38 73 58 ...
 $ Education  : int  17 13 16 10 12 11 17 10 17 16 ...
 $ Urban      : Factor w/ 2 levels "No","Yes": 2 2 1 1 2 2 2 1 1 2 ...
 $ US         : Factor w/ 2 levels "No","Yes": 2 1 2 2 2 2 2 1 1 1 ...
 $ Revenue    : num  1140 531 778 901 1146 ...
 $ Profit     : num  228 106 156 180 229 ...

MODEL 0: Regress Profit on Advertising


Call:
lm(formula = p0, data = carSeats.df)

Residuals:
    Min      1Q  Median      3Q     Max 
-171.85  -34.38   -3.78   35.97  168.85 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 147.3290     4.0320  36.540  < 2e-16 ***
Advertising   3.0660     0.4295   7.139 4.49e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 57.05 on 398 degrees of freedom
Multiple R-squared:  0.1135,    Adjusted R-squared:  0.1113 
F-statistic: 50.97 on 1 and 398 DF,  p-value: 4.493e-12

IMPACT OF ADVERTISING ON PROFIT

An increase in advertising by $1000, increases the profit by $3066

MODEL 1: Regress Profit on ShelveLoc


Call:
lm(formula = p1, data = carSeats.df)

Residuals:
     Min       1Q   Median       3Q      Max 
-163.350  -33.330   -1.365   31.033  153.050 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)        120.041      4.808   24.97  < 2e-16 ***
ShelveLoc1-Medium   43.309      5.767    7.51 3.93e-13 ***
ShelveLoc2-Good    112.558      7.016   16.04  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 47.11 on 397 degrees of freedom
Multiple R-squared:  0.3971,    Adjusted R-squared:  0.394 
F-statistic: 130.7 on 2 and 397 DF,  p-value: < 2.2e-16

IMPACT OF SHELF LOCATION ON PROFIT

A change in Shelf Location from Bad to Medium, increases the profit by $43,309
A change in Shelf Location from Bad to Good, increases the profit by $112,558

MODEL 2: Regress Profit on both Advertising and ShelveLoc


Call:
lm(formula = p2, data = carSeats.df)

Residuals:
     Min       1Q   Median       3Q      Max 
-145.446  -25.160    0.039   24.796  104.054 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)        103.014      4.886  21.085  < 2e-16 ***
Advertising          2.738      0.328   8.347 1.18e-15 ***
ShelveLoc1-Medium   42.433      5.325   7.968 1.73e-14 ***
ShelveLoc2-Good    109.453      6.489  16.867  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 43.5 on 396 degrees of freedom
Multiple R-squared:  0.4873,    Adjusted R-squared:  0.4834 
F-statistic: 125.4 on 3 and 396 DF,  p-value: < 2.2e-16

IMPACT OF ADVERTISING ON PROFIT

An increase in advertising by $1, increases the profit by $2.74

IMPACT OF SHELF LOCATION ON PROFIT

A change in Shelf Location from Bad to Medium, increases the profit by $42,433
A change in Shelf Location from Bad to Good, increases the profit by $109,453
Notice that when we jointly consider the role of Advertising and Shelf Location, the 'answers' are different.
We were 'over-predicting' the impact of Advertising when we ignored the impact of Shelf Location (See Model 0)
We were 'over-predicting' the impact of Shelf Location when we ignored the impact of Advertising (See Model 1)

MODEL 3: Is there an INTERACTION between Advertising and Shelf Location?

Regress Profit on both Advertising and ShelveLoc


Call:
lm(formula = p3, data = carSeats.df)

Residuals:
     Min       1Q   Median       3Q      Max 
-143.299  -24.554    1.032   24.480  106.201 

Coefficients:
                              Estimate Std. Error t value Pr(>|t|)    
(Intercept)                   108.6051     6.1743  17.590  < 2e-16 ***
Advertising                     1.8390     0.6903   2.664  0.00803 ** 
ShelveLoc1-Medium              34.6937     7.4206   4.675 4.04e-06 ***
ShelveLoc2-Good               103.1563     9.3093  11.081  < 2e-16 ***
Advertising:ShelveLoc1-Medium   1.2276     0.8190   1.499  0.13472    
Advertising:ShelveLoc2-Good     0.9950     0.9812   1.014  0.31118    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 43.48 on 394 degrees of freedom
Multiple R-squared:  0.4902,    Adjusted R-squared:  0.4837 
F-statistic: 75.77 on 5 and 394 DF,  p-value: < 2.2e-16

MODEL 4:

p4 <- Profit ~ Advertising + ShelveLoc + CompPrice + Population + Income + Age + Education + Urban + US
fitp4 = lm(p4 ,data=carSeats.df )
summary (fitp4)

MODEL 4:


Call:
lm(formula = p4, data = carSeats.df)

Residuals:
     Min       1Q   Median       3Q      Max 
-159.654  -17.856    2.075   20.401   68.740 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)       -60.136334  18.171281  -3.309  0.00102 ** 
Advertising         2.959638   0.334864   8.838  < 2e-16 ***
ShelveLoc1-Medium  44.625673   3.797403  11.752  < 2e-16 ***
ShelveLoc2-Good   109.212841   4.608355  23.699  < 2e-16 ***
CompPrice           1.563453   0.101946  15.336  < 2e-16 ***
Population          0.005803   0.011149   0.521  0.60299    
Income              0.370657   0.055560   6.671 8.73e-11 ***
Age                -0.952238   0.095685  -9.952  < 2e-16 ***
Education          -0.679343   0.593883  -1.144  0.25337    
UrbanYes            2.981043   3.402003   0.876  0.38143    
USYes              -5.783463   4.511614  -1.282  0.20064    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 30.69 on 389 degrees of freedom
Multiple R-squared:  0.7493,    Adjusted R-squared:  0.7429 
F-statistic: 116.3 on 10 and 389 DF,  p-value: < 2.2e-16

NOTES

The Adjusted R-squared is higher (74% compared to 48%)
The impact of Advertising and Shelf Location after controling for the competition and the consumer demographics is higher than the predicted impact when we ignored these control variables
The control factors that statistically significant are {CompPrice, Income, Age}

Comparing MODEL 2, MODEL 4, : How Is Model 4 'better' than Model 2?

# Print the summary of the two regression models
summary(fitp2)


Call:
lm(formula = p2, data = carSeats.df)

Residuals:
     Min       1Q   Median       3Q      Max 
-145.446  -25.160    0.039   24.796  104.054 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)        103.014      4.886  21.085  < 2e-16 ***
Advertising          2.738      0.328   8.347 1.18e-15 ***
ShelveLoc1-Medium   42.433      5.325   7.968 1.73e-14 ***
ShelveLoc2-Good    109.453      6.489  16.867  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 43.5 on 396 degrees of freedom
Multiple R-squared:  0.4873,    Adjusted R-squared:  0.4834 
F-statistic: 125.4 on 3 and 396 DF,  p-value: < 2.2e-16

#summary (fitp4)

Comparing MODEL 2, MODEL 4, : How Is Model 4 'better' than Model 2?

# Print the summary of the two regression models
#summary(fitp2)
summary (fitp4)

Comparing MODEL 2, MODEL 4, : How Is Model 4 'better' than Model 2?


Call:
lm(formula = p4, data = carSeats.df)

Residuals:
     Min       1Q   Median       3Q      Max 
-159.654  -17.856    2.075   20.401   68.740 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)       -60.136334  18.171281  -3.309  0.00102 ** 
Advertising         2.959638   0.334864   8.838  < 2e-16 ***
ShelveLoc1-Medium  44.625673   3.797403  11.752  < 2e-16 ***
ShelveLoc2-Good   109.212841   4.608355  23.699  < 2e-16 ***
CompPrice           1.563453   0.101946  15.336  < 2e-16 ***
Population          0.005803   0.011149   0.521  0.60299    
Income              0.370657   0.055560   6.671 8.73e-11 ***
Age                -0.952238   0.095685  -9.952  < 2e-16 ***
Education          -0.679343   0.593883  -1.144  0.25337    
UrbanYes            2.981043   3.402003   0.876  0.38143    
USYes              -5.783463   4.511614  -1.282  0.20064    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 30.69 on 389 degrees of freedom
Multiple R-squared:  0.7493,    Adjusted R-squared:  0.7429 
F-statistic: 116.3 on 10 and 389 DF,  p-value: < 2.2e-16

Conceptually: Model 4 studies the joint effect of many more factors than Model 2

Method 1: Compare the Adjusted R Squared

(summary(fitp2))$adj.r.squared

[1] 0.483383

(summary(fitp4))$adj.r.squared

[1] 0.7428985

Method 2: Compare the AIC

AIC(fitp2)

[1] 4159.3

AIC(fitp4)

[1] 3887.034

Method 3: ANOVA (since Model 2 is nested within Model 4)

anova(fitp2, fitp4)

Analysis of Variance Table

Model 1: Profit ~ Advertising + ShelveLoc
Model 2: Profit ~ Advertising + ShelveLoc + CompPrice + Population + Income + 
    Age + Education + Urban + US
  Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
1    396 749257                                  
2    389 366287  7    382970 58.103 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

MODEL 5:

p5 <- Profit ~ Advertising*Income + ShelveLoc + CompPrice + Population + Age + Education + Urban + US
fitp5 = lm(p5 ,data=carSeats.df )
summary (fitp5)


Call:
lm(formula = p5, data = carSeats.df)

Residuals:
     Min       1Q   Median       3Q      Max 
-159.421  -17.994    1.991   19.878   73.950 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)        -53.675367  18.315500  -2.931  0.00358 ** 
Advertising          1.650310   0.679306   2.429  0.01558 *  
Income               0.248046   0.078290   3.168  0.00165 ** 
ShelveLoc1-Medium   44.793177   3.779306  11.852  < 2e-16 ***
ShelveLoc2-Good    109.503315   4.587353  23.871  < 2e-16 ***
CompPrice            1.572174   0.101517  15.487  < 2e-16 ***
Population           0.004853   0.011102   0.437  0.66225    
Age                 -0.937659   0.095438  -9.825  < 2e-16 ***
Education           -0.700463   0.591011  -1.185  0.23667    
UrbanYes             3.243896   3.387197   0.958  0.33881    
USYes               -5.243940   4.495835  -1.166  0.24417    
Advertising:Income   0.018512   0.008370   2.212  0.02756 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 30.53 on 388 degrees of freedom
Multiple R-squared:  0.7525,    Adjusted R-squared:  0.7454 
F-statistic: 107.2 on 11 and 388 DF,  p-value: < 2.2e-16

Conceptually: Model 5 is 'better than Model 4

Method 1: Compare the Adjusted R Squared

(summary(fitp4))$adj.r.squared

[1] 0.7428985

(summary(fitp5))$adj.r.squared

[1] 0.7454454

Method 2: Compare the AIC

AIC(fitp4)

[1] 3887.034

AIC(fitp5)

[1] 3884.022

Method 3: ANOVA (since Model 4 is nested within Model 5)

anova(fitp4, fitp5)

Analysis of Variance Table

Model 1: Profit ~ Advertising + ShelveLoc + CompPrice + Population + Income + 
    Age + Education + Urban + US
Model 2: Profit ~ Advertising * Income + ShelveLoc + CompPrice + Population + 
    Age + Education + Urban + US
  Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
1    389 366287                              
2    388 361726  1    4560.8 4.8921 0.02756 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1