Multiple Linear Regression
Mikhilesh Dehane
9/15/2020
Multiple Linear Regression
Predicts mpg for a car from hp, vol, sp, and wt
Cars dataset - IV x1 hp, x2 vol, x3 sp, x4 wt input variable - DV y mpg output variable
Exploratory Data Analysis(60% of time)
1. Measures of Central Tendency
2. Measures of Dispersion
3. Third Moment Business decision
4. Fourth Moment Business decision
5. Probability distributions of variables
6. Graphical representations
> Histogram,Box plot,Dot plot,Stem & Leaf plot,
> Bar plot
7. Find the correlation b/n Output (MPG) & (HP,VOL,SP)-Scatter plot
8. Correlation Coefficient matrix - Strength & Direction of Correlation
Cars <- read.csv("E:\\mikhilesh\\Horizon 2020\\ApCoTe Yogesh Sky Analytics DA DS Learning\\Cars dataset.csv") # reading the Cars.csv data set
#View(Cars)
# Exploratory Data Analysis(60% of time)
# 1. Measures of Central Tendency
# 2. Measures of Dispersion
# 3. Third Moment Business decision
# 4. Fourth Moment Business decision
# 5. Probability distributions of variables
# 6. Graphical representations
# > Histogram,Box plot,Dot plot,Stem & Leaf plot,
# > Bar plot
dim(Cars)## [1] 81 5attach(Cars)
class(Cars)## [1] "data.frame"names(Cars)## [1] "HP" "MPG" "VOL" "SP" "WT"str(Cars)## 'data.frame': 81 obs. of 5 variables:
## $ HP : int 49 55 55 70 53 70 55 62 62 80 ...
## $ MPG: num 53.7 50 50 45.7 50.5 ...
## $ VOL: int 89 92 92 92 92 89 92 50 50 94 ...
## $ SP : num 104 105 105 113 104 ...
## $ WT : num 28.8 30.5 30.2 30.6 29.9 ...str(HP)## int [1:81] 49 55 55 70 53 70 55 62 62 80 ...summary(Cars)## HP MPG VOL SP
## Min. : 49.0 Min. :12.10 Min. : 50.00 Min. : 99.56
## 1st Qu.: 84.0 1st Qu.:27.86 1st Qu.: 89.00 1st Qu.:113.83
## Median :100.0 Median :35.15 Median :101.00 Median :118.21
## Mean :117.5 Mean :34.42 Mean : 98.77 Mean :121.54
## 3rd Qu.:140.0 3rd Qu.:39.53 3rd Qu.:113.00 3rd Qu.:126.40
## Max. :322.0 Max. :53.70 Max. :160.00 Max. :169.60
## WT
## Min. :15.71
## 1st Qu.:29.59
## Median :32.73
## Mean :32.41
## 3rd Qu.:37.39
## Max. :53.00# 7. Find the correlation b/n Output (MPG) & (HP,VOL,SP)-Scatter plot
# To check for Multicolinearity Problem
plot(Cars) # OR pairs(Cars) will give the same plots
#Plots demonstrate Multicolineariity (linear relation) between HP and SP pair and VOL and WT pair
# 8. Correlation Coefficient matrix - Strength & Direction of Correlation
#To confirm, lets check correlation r
cor(Cars) #Values greater than 0.85 are considered strong correlation## HP MPG VOL SP WT
## HP 1.00000000 -0.7250383 0.07745947 0.9738481 0.07651307
## MPG -0.72503835 1.0000000 -0.52905658 -0.6871246 -0.52675909
## VOL 0.07745947 -0.5290566 1.00000000 0.1021700 0.99920308
## SP 0.97384807 -0.6871246 0.10217001 1.0000000 0.10243919
## WT 0.07651307 -0.5267591 0.99920308 0.1024392 1.00000000# pairs with > 0.85 are HP - SP and VOL - WT
plot(HP, SP)
cor(HP, MPG) ## [1] -0.7250383cor(MPG, HP)## [1] -0.7250383# First build a basic Linear regression model with all the input variables
# The Linear Model of interest
m1 <- lm(MPG ~ VOL+ HP + SP + WT,data = Cars) #syntax m1 <- lm(y ~ x,data =dsetname)
m1##
## Call:
## lm(formula = MPG ~ VOL + HP + SP + WT, data = Cars)
##
## Coefficients:
## (Intercept) VOL HP SP WT
## 30.6773 -0.3361 -0.2054 0.3956 0.4006summary(m1)##
## Call:
## lm(formula = MPG ~ VOL + HP + SP + WT, data = Cars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.6320 -2.9944 -0.3705 2.2149 15.6179
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.67734 14.90030 2.059 0.0429 *
## VOL -0.33605 0.56864 -0.591 0.5563
## HP -0.20544 0.03922 -5.239 1.4e-06 ***
## SP 0.39563 0.15826 2.500 0.0146 *
## WT 0.40057 1.69346 0.237 0.8136
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.488 on 76 degrees of freedom
## Multiple R-squared: 0.7705, Adjusted R-squared: 0.7585
## F-statistic: 63.8 on 4 and 76 DF, p-value: < 2.2e-16# results show p-values > 0.05 for VOL and WT - not significant model
# Next TO check which of vol and wt are creating problems, build a separate model for vol and wt each
# Prediction based on only Volume
mv <- lm(MPG ~ VOL,data = Cars)
mv##
## Call:
## lm(formula = MPG ~ VOL, data = Cars)
##
## Coefficients:
## (Intercept) VOL
## 55.8171 -0.2166summary(mv) # Volume became significant - p-value < 0.05##
## Call:
## lm(formula = MPG ~ VOL, data = Cars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -25.3074 -5.2026 0.1902 5.4536 17.1632
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 55.81709 3.95696 14.106 < 2e-16 ***
## VOL -0.21662 0.03909 -5.541 3.82e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.798 on 79 degrees of freedom
## Multiple R-squared: 0.2799, Adjusted R-squared: 0.2708
## F-statistic: 30.71 on 1 and 79 DF, p-value: 3.823e-07# Prediction based on only Weight
mw <- lm(MPG ~ WT,data = Cars)
summary(mw) # Weight became significant - p-value < 0.05##
## Call:
## lm(formula = MPG ~ WT, data = Cars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -25.3933 -5.4377 0.2738 5.2951 16.9351
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 55.2296 3.8761 14.249 < 2e-16 ***
## WT -0.6420 0.1165 -5.508 4.38e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.811 on 79 degrees of freedom
## Multiple R-squared: 0.2775, Adjusted R-squared: 0.2683
## F-statistic: 30.34 on 1 and 79 DF, p-value: 4.383e-07# As seen from m1 model summary, HP and SP dont have any problems.
# Next we check model/Prediction based on Volume and Weight
mvw <- lm(MPG ~ VOL + WT,data = Cars)
summary(mvw) # Both became Insignificant - p-value > 0.05 for both vol and wt##
## Call:
## lm(formula = MPG ~ VOL + WT, data = Cars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -24.9939 -4.9460 0.0028 5.3905 17.6972
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 56.8847 4.5342 12.55 <2e-16 ***
## VOL -0.6983 0.9841 -0.71 0.480
## WT 1.4349 2.9291 0.49 0.626
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.835 on 78 degrees of freedom
## Multiple R-squared: 0.2821, Adjusted R-squared: 0.2637
## F-statistic: 15.33 on 2 and 78 DF, p-value: 2.434e-06library(car)## Warning: package 'car' was built under R version 3.6.2## Loading required package: carData# It is Better to delete influential observations rather than deleting entire column which is creating the multicolinearity problem
## plotting Influential measures
influencePlot(m1) # A user friendly representation of the above
## StudRes Hat CookD
## 1 2.421762 0.05200781 0.06047977
## 71 -2.100131 0.22253511 0.24164401
## 77 4.503603 0.25138750 1.08651940#COOK'S DISTANCE is a measure to identify which is the most influential record.
# Regression after deleting the most influential - 77th observation, which is influential observation
m2 <- lm(MPG ~ VOL+SP+HP+WT,data=Cars[-77,])
summary(m2) # Summary demonstrate even after removing the most influential record 77, vol and wt are still insignificant p-vale > 0.05##
## Call:
## lm(formula = MPG ~ VOL + SP + HP + WT, data = Cars[-77, ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.3943 -2.3555 -0.5913 1.8978 12.0184
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.82675 13.32251 2.089 0.04013 *
## VOL -0.18546 0.50895 -0.364 0.71659
## SP 0.41189 0.14139 2.913 0.00471 **
## HP -0.22664 0.03534 -6.413 1.14e-08 ***
## WT 0.03754 1.51458 0.025 0.98029
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.008 on 75 degrees of freedom
## Multiple R-squared: 0.8192, Adjusted R-squared: 0.8096
## F-statistic: 84.96 on 4 and 75 DF, p-value: < 2.2e-16#SO NEXT WE WILL REMOVE INFULENTIAL REMOVES and check the significance of model
# Regression after deleting the 77th, 71st, and 1st Observations
m3 <- lm(MPG ~ VOL+ SP+ HP+ WT,data=Cars[-c(71,77,1),])
summary(m3) # Summary demonstrate even after removing all the influential record 77, 71 & 1st - vol and wt are still insignificant p-vale > 0.05##
## Call:
## lm(formula = MPG ~ VOL + SP + HP + WT, data = Cars[-c(71, 77,
## 1), ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.7300 -2.5391 -0.3696 2.1482 10.7151
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 24.82062 13.01740 1.907 0.06049 .
## VOL -0.31823 0.49668 -0.641 0.52372
## SP 0.44618 0.13881 3.214 0.00195 **
## HP -0.22688 0.03413 -6.647 4.67e-09 ***
## WT 0.40617 1.48045 0.274 0.78459
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.859 on 73 degrees of freedom
## Multiple R-squared: 0.821, Adjusted R-squared: 0.8112
## F-statistic: 83.72 on 4 and 73 DF, p-value: < 2.2e-16# Variance Inflation factor to check collinearity b/n variables
vif(m1)## VOL HP SP WT
## 638.80608 19.92659 20.00764 639.53382#(USusally if vif > 10, you should remove that variable)
# In this model m1, all vif > 10 - then there exists collinearity among all the variables
#But we cannot remove all the variables,
#SO we will remove the highest vif variable first - in this case - WT
# check the new model again after removing highest VIF variable - wt
finalmodel <- lm(MPG ~ VOL + SP + HP,data=Cars)
finalmodel##
## Call:
## lm(formula = MPG ~ VOL + SP + HP, data = Cars)
##
## Coefficients:
## (Intercept) VOL SP HP
## 29.9234 -0.2017 0.4007 -0.2067summary(finalmodel) # Summary demonstrate after removing highest VIF wt, model is significant, all other variables p-value < 0.05##
## Call:
## lm(formula = MPG ~ VOL + SP + HP, data = Cars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.5869 -2.8942 -0.3157 2.1291 15.6669
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.92339 14.46589 2.069 0.0419 *
## VOL -0.20165 0.02259 -8.928 1.65e-13 ***
## SP 0.40066 0.15586 2.571 0.0121 *
## HP -0.20670 0.03861 -5.353 8.64e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.46 on 77 degrees of freedom
## Multiple R-squared: 0.7704, Adjusted R-squared: 0.7614
## F-statistic: 86.11 on 3 and 77 DF, p-value: < 2.2e-16# JUST for PRACTICE and UNDERSTANDING- check the new model again after removing second highest VIF variable - vol
finalmodel1 <- lm(MPG ~ WT + SP + HP,data=Cars)
finalmodel1##
## Call:
## lm(formula = MPG ~ WT + SP + HP, data = Cars)
##
## Coefficients:
## (Intercept) WT SP HP
## 28.7848 -0.5994 0.4078 -0.2085summary(finalmodel1) # Summary demonstrate after removing swcond highest VIF vol, model is significant, intercept p-value > 0.05 - not significant##
## Call:
## lm(formula = MPG ~ WT + SP + HP, data = Cars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.7567 -2.7652 -0.3683 1.8589 15.7690
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 28.78481 14.49047 1.986 0.0505 .
## WT -0.59941 0.06739 -8.895 1.91e-13 ***
## SP 0.40775 0.15626 2.609 0.0109 *
## HP -0.20850 0.03871 -5.386 7.56e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.469 on 77 degrees of freedom
## Multiple R-squared: 0.7695, Adjusted R-squared: 0.7605
## F-statistic: 85.68 on 3 and 77 DF, p-value: < 2.2e-16# SO removing wt is better model than removing vol#OTHER OPTION TO VIF METHOD IS AVPLOT
## Added Variable plot to check correlation b/n variables and o/p variable
avPlots(m1)
# In this model m1, plots show correlation b/w hp-mpg, sp-mpg. If observed minutely, there is a slight negative coorelation b/w vol-mpg but wt-mpg line is almost flat - showing no relation
avPlots(finalmodel)
# In this finalmodel, plots show all the variables have correlation after removing wt
# VIF and AV plot has given us an indication to delete "wt" variablelibrary(MASS)
stepAIC(m1) #Akaike Information Criterion## Start: AIC=248.06
## MPG ~ VOL + HP + SP + WT
##
## Df Sum of Sq RSS AIC
## - WT 1 1.13 1531.8 246.12
## - VOL 1 7.03 1537.7 246.43
## <none> 1530.7 248.06
## - SP 1 125.87 1656.5 252.46
## - HP 1 552.74 2083.4 271.03
##
## Step: AIC=246.12
## MPG ~ VOL + HP + SP
##
## Df Sum of Sq RSS AIC
## <none> 1531.8 246.12
## - SP 1 131.46 1663.3 250.79
## - HP 1 570.08 2101.9 269.75
## - VOL 1 1585.81 3117.6 301.68##
## Call:
## lm(formula = MPG ~ VOL + HP + SP, data = Cars)
##
## Coefficients:
## (Intercept) VOL HP SP
## 29.9234 -0.2017 -0.2067 0.4007#StepAIC take all input variables from the basic model m1 - removes all influential variables, and runs all the models and stops at the best model
#Notice stepAIC automatically removed wt variable and given final model as lm(formula = MPG ~ VOL + HP + SP, data = Cars)
#Lesser the AIC better the model#Predicting values for MPG using the final model
pv <- predict(finalmodel,Cars)
pv## 1 2 3 4 5 6 7 8
## 43.59077 42.25679 42.25679 42.36150 42.26954 42.85590 42.25679 48.13221
## 9 10 11 12 13 14 15 16
## 48.13221 40.76616 41.43447 47.94095 39.86565 41.43447 41.67943 41.43447
## 17 18 19 20 21 22 23 24
## 41.27332 47.94095 41.27332 38.01722 38.66367 37.46001 38.11462 39.42854
## 25 26 27 28 29 30 31 32
## 40.09224 46.73898 35.69724 38.66367 38.09467 35.87282 35.04067 37.18309
## 33 34 35 36 37 38 39 40
## 37.32689 34.69006 37.40255 37.63925 39.28727 38.33839 38.33839 35.96910
## 41 42 43 44 45 46 47 48
## 34.13369 35.28731 37.34958 38.25375 35.95927 36.20872 34.23109 35.56226
## 49 50 51 52 53 54 55 56
## 36.95791 33.17920 33.17920 33.17920 29.38875 27.38159 28.31041 28.69214
## 57 58 59 60 61 62 63 64
## 35.78519 33.17920 35.43009 32.36991 29.73729 28.87233 25.07082 26.38923
## 65 66 67 68 69 70 71 72
## 25.85377 36.45744 25.91011 23.76768 24.42689 20.12047 27.91145 22.66872
## 73 74 75 76 77 78 79 80
## 23.16313 18.68892 23.79778 20.97037 21.23314 17.86773 26.21686 12.23755
## 81
## 15.59296# converting numeric pv into dataframe
pv <- as.data.frame(pv)
# ADDING the predicted values of MPG to original datset
final <- cbind(Cars,pv)
final## HP MPG VOL SP WT pv
## 1 49 53.70068 89 104.18535 28.76206 43.59077
## 2 55 50.01340 92 105.46126 30.46683 42.25679
## 3 55 50.01340 92 105.46126 30.19360 42.25679
## 4 70 45.69632 92 113.46126 30.63211 42.36150
## 5 53 50.50423 92 104.46126 29.88915 42.26954
## 6 70 45.69632 89 113.18535 29.59177 42.85590
## 7 55 50.01340 92 105.46126 30.30848 42.25679
## 8 62 46.71655 50 102.59851 15.84776 48.13221
## 9 62 46.71655 50 102.59851 16.35948 48.13221
## 10 80 42.29908 94 115.64520 30.92015 40.76616
## 11 73 44.65283 89 111.18535 29.36334 41.43447
## 12 92 39.35409 50 117.59851 15.75353 47.94095
## 13 92 39.35409 99 122.10506 32.81359 39.86565
## 14 73 44.65283 89 111.18535 29.37844 41.43447
## 15 66 45.73489 89 108.18535 29.34728 41.67943
## 16 73 44.65283 89 111.18535 29.60453 41.43447
## 17 78 42.78991 91 114.36929 29.53578 41.27332
## 18 92 39.35409 50 117.59851 16.19412 47.94095
## 19 78 42.78991 91 114.36929 29.92939 41.27332
## 20 90 38.90183 103 118.47294 33.51697 38.01722
## 21 92 38.41100 99 119.10506 32.32465 38.66367
## 22 74 42.82848 107 110.84082 34.90821 37.46001
## 23 95 38.31061 101 120.28900 32.67583 38.11462
## 24 81 40.47472 96 113.82914 31.83712 39.42854
## 25 95 38.31061 89 119.18535 28.78173 40.09224
## 26 92 38.41100 50 114.59851 16.04317 46.73898
## 27 92 38.41100 117 120.76052 38.06282 35.69724
## 28 92 38.41100 99 119.10506 32.83507 38.66367
## 29 52 43.46943 104 99.56491 34.48321 38.09467
## 30 103 35.40419 107 121.84082 35.54936 35.87282
## 31 84 39.43124 114 113.48461 37.04235 35.04067
## 32 84 39.43124 101 112.28900 33.23436 37.18309
## 33 102 36.28546 97 119.92111 31.38004 37.32689
## 34 102 36.28546 113 121.39264 37.57329 34.69006
## 35 81 39.53163 101 111.28900 32.70164 37.40255
## 36 90 37.95874 98 115.01309 31.91122 37.63925
## 37 90 37.95874 88 114.09338 28.75400 39.28727
## 38 102 34.07067 86 116.90944 27.87992 38.33839
## 39 102 34.07067 86 116.90944 28.63050 38.33839
## 40 130 31.01413 92 128.46126 30.11543 35.96910
## 41 95 35.15273 113 116.39264 37.39252 34.13369
## 42 95 35.15273 106 115.74885 35.02718 35.28731
## 43 102 34.07067 92 117.46126 30.52743 37.34958
## 44 95 35.15273 88 114.09338 28.34398 38.25375
## 45 93 35.64356 102 114.38097 33.07863 35.95927
## 46 100 34.56150 99 117.10506 32.62192 36.20872
## 47 100 34.56150 111 118.20870 36.49862 34.23109
## 48 98 35.05233 103 116.47294 33.91006 35.56226
## 49 130 31.01413 86 127.90944 28.07060 36.95791
## 50 115 29.62994 101 118.28900 33.45847 33.17920
## 51 115 29.62994 101 118.28900 33.21395 33.17920
## 52 115 29.62994 101 118.28900 33.43671 33.17920
## 53 115 29.62994 124 120.40431 40.39816 29.38875
## 54 180 24.48737 113 143.39264 37.62069 27.38159
## 55 160 26.85228 113 135.39264 37.25439 28.31041
## 56 130 27.85625 124 126.40431 40.58907 28.69214
## 57 96 31.11358 92 110.46126 30.14754 35.78519
## 58 115 29.62994 101 118.28900 32.73452 33.17920
## 59 100 30.13192 94 112.64520 30.61528 35.43009
## 60 100 28.86023 115 115.57658 37.66287 32.36991
## 61 145 27.35427 111 130.20870 36.88815 29.73729
## 62 120 24.60913 116 117.66855 37.86041 28.87233
## 63 140 23.51592 131 126.04810 43.39099 25.07082
## 64 140 23.51592 123 125.31234 40.72283 26.38923
## 65 150 23.60516 121 128.12840 40.15948 25.85377
## 66 165 40.05000 50 126.59851 15.71286 36.45744
## 67 165 23.10317 114 132.48461 37.97996 25.91011
## 68 165 23.10317 127 133.68022 41.57397 23.76768
## 69 165 23.10317 123 133.31234 40.47204 24.42689
## 70 245 21.27371 112 158.30067 37.14173 20.12047
## 71 280 19.67851 50 164.59851 15.82306 27.91145
## 72 162 23.20357 135 133.41598 44.01314 22.66872
## 73 162 23.20357 132 133.14007 43.35312 23.16313
## 74 140 19.08634 160 124.71524 52.99775 18.68892
## 75 140 19.08634 129 121.86416 42.61870 23.79778
## 76 175 18.76284 129 132.86416 42.77822 20.97037
## 77 322 36.90000 50 169.59851 16.13295 21.23314
## 78 238 19.19789 115 150.57658 37.92311 17.86773
## 79 263 34.00000 50 151.59851 15.76963 26.21686
## 80 295 19.83373 119 167.94446 39.42310 12.23755
## 81 236 12.10126 107 139.84082 34.94861 15.59296summary(Cars)## HP MPG VOL SP
## Min. : 49.0 Min. :12.10 Min. : 50.00 Min. : 99.56
## 1st Qu.: 84.0 1st Qu.:27.86 1st Qu.: 89.00 1st Qu.:113.83
## Median :100.0 Median :35.15 Median :101.00 Median :118.21
## Mean :117.5 Mean :34.42 Mean : 98.77 Mean :121.54
## 3rd Qu.:140.0 3rd Qu.:39.53 3rd Qu.:113.00 3rd Qu.:126.40
## Max. :322.0 Max. :53.70 Max. :160.00 Max. :169.60
## WT
## Min. :15.71
## 1st Qu.:29.59
## Median :32.73
## Mean :32.41
## 3rd Qu.:37.39
## Max. :53.00Testcar <- read.csv("E:\\mikhilesh\\Horizon 2020\\ApCoTe Yogesh Sky Analytics DA DS Learning\\cars testset.csv") # reading the test.csv data set
Testcar## HP VOL SP
## 1 70 92 113.4613
## 2 53 92 104.4613
## 3 70 89 113.1854#predicting mpg values for testcars
pv1 <- predict(finalmodel,newdata = Testcar)
pv1## 1 2 3
## 42.36150 42.26954 42.85590# converting numeric pv into dataframe
pv1 <- as.data.frame(pv1)
# ADDING the predicted values of MPG to original datset
finaltest <- cbind(Testcar,pv1)
finaltest## HP VOL SP pv1
## 1 70 92 113.4613 42.36150
## 2 53 92 104.4613 42.26954
## 3 70 89 113.1854 42.85590summary(Testcar)## HP VOL SP
## Min. :53.00 Min. :89.0 Min. :104.5
## 1st Qu.:61.50 1st Qu.:90.5 1st Qu.:108.8
## Median :70.00 Median :92.0 Median :113.2
## Mean :64.33 Mean :91.0 Mean :110.4
## 3rd Qu.:70.00 3rd Qu.:92.0 3rd Qu.:113.3
## Max. :70.00 Max. :92.0 Max. :113.5