Multiple Liner Regression Model
Cars <- read.csv(file.choose()) # choose the Cars.csv data set
View(Cars)
dim(Cars) # Dimension of DataSet
## [1] 81 5
attach(Cars)
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
Cars[15:20,c(1,4)]
## HP SP
## 15 66 108.1854
## 16 73 111.1854
## 17 78 114.3693
## 18 92 117.5985
## 19 78 114.3693
## 20 90 118.4729
# 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
head(Cars)
## HP MPG VOL SP WT
## 1 49 53.70068 89 104.1854 28.76206
## 2 55 50.01340 92 105.4613 30.46683
## 3 55 50.01340 92 105.4613 30.19360
## 4 70 45.69632 92 113.4613 30.63211
## 5 53 50.50423 92 104.4613 29.88915
## 6 70 45.69632 89 113.1854 29.59177
colnames(Cars)
## [1] "HP" "MPG" "VOL" "SP" "WT"
# 7. Find the correlation b/n Output (MPG) & (HP,VOL,SP)-Scatter plot
pairs(Cars)
plot(Cars)
windows()
plot(Cars)
# 8. Correlation Coefficient matrix - Strength & Direction of Correlation
cor(Cars)
## 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
library(caret)
## Warning: package 'caret' was built under R version 3.5.2
## Loading required package: lattice
## Warning: package 'lattice' was built under R version 3.5.2
## Loading required package: ggplot2
## Warning: package 'ggplot2' was built under R version 3.5.2
inTrain <- createDataPartition(y = MPG, p=0.80,list = FALSE)
inTrain
## Resample1
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train <- Cars[inTrain,]
test <- Cars[-inTrain,]
# The Linear Model of interest
m1 <- lm(MPG ~ . ,data = train)
coef(m1)
## (Intercept) HP VOL SP WT
## 29.6430209 -0.2073439 -0.1788465 0.4151304 -0.1094680
summary(m1)
##
## Call:
## lm(formula = MPG ~ ., data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5636 -3.0325 -0.3562 2.1273 14.3246
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.64302 16.70098 1.775 0.0810 .
## HP -0.20734 0.04324 -4.795 1.11e-05 ***
## VOL -0.17885 0.65743 -0.272 0.7865
## SP 0.41513 0.17645 2.353 0.0219 *
## WT -0.10947 1.95909 -0.056 0.9556
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.611 on 60 degrees of freedom
## Multiple R-squared: 0.7567, Adjusted R-squared: 0.7404
## F-statistic: 46.64 on 4 and 60 DF, p-value: < 2.2e-16
# Prediction based on only Volume
model.carV <- lm(MPG ~ VOL,data = train)
summary(model.carV) # Volume became significant
##
## Call:
## lm(formula = MPG ~ VOL, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -24.4732 -4.7976 0.1821 5.4507 17.3269
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 54.1234 4.6333 11.681 < 2e-16 ***
## VOL -0.1994 0.0459 -4.345 5.17e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.001 on 63 degrees of freedom
## Multiple R-squared: 0.2306, Adjusted R-squared: 0.2184
## F-statistic: 18.88 on 1 and 63 DF, p-value: 5.169e-05
# Prediction based on only Weight
model.carW <- lm(MPG ~ WT,data = train)
summary(model.carW) # Weight became significant
##
## Call:
## lm(formula = MPG ~ WT, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -24.5381 -4.7742 0.2513 5.2685 17.1143
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 53.5477 4.5431 11.787 < 2e-16 ***
## WT -0.5897 0.1369 -4.306 5.92e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.018 on 63 degrees of freedom
## Multiple R-squared: 0.2274, Adjusted R-squared: 0.2151
## F-statistic: 18.54 on 1 and 63 DF, p-value: 5.922e-05
# Prediction based on Volume and Weight
model.carVW <- lm(MPG ~ VOL + WT,data = train)
summary(model.carVW) # Both became Insignificant
##
## Call:
## lm(formula = MPG ~ VOL + WT, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -23.9579 -4.9849 -0.0751 5.3210 18.1646
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 55.6837 5.2465 10.614 1.42e-15 ***
## VOL -0.9283 1.1314 -0.820 0.415
## WT 2.1718 3.3687 0.645 0.521
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.038 on 62 degrees of freedom
## Multiple R-squared: 0.2357, Adjusted R-squared: 0.211
## F-statistic: 9.56 on 2 and 62 DF, p-value: 0.0002406
# It is Better to delete influential observations rather than deleting entire column which is
# costliest process
# Deletion Diagnostics for identifying influential observations
influence.measures(m1)
## Influence measures of
## lm(formula = MPG ~ ., data = train) :
##
## dfb.1_ dfb.HP dfb.VOL dfb.SP dfb.WT dffit cov.r
## 1 0.043641 -0.069513 0.407022 -1.91e-02 -0.412769 0.64405 0.751
## 2 0.159944 0.054096 -0.235953 -1.08e-01 0.233456 0.41894 0.888
## 4 -0.049196 -0.089031 -0.136315 7.57e-02 0.134702 0.20868 1.123
## 5 0.102858 0.017165 0.216214 -8.35e-02 -0.219065 0.44172 0.862
## 6 -0.035886 -0.070263 -0.105375 5.88e-02 0.103614 0.16773 1.133
## 7 0.130729 0.033950 -0.112154 -8.89e-02 0.109597 0.35942 0.880
## 9 -0.132037 -0.085103 0.077234 1.01e-01 -0.072082 -0.20146 1.279
## 10 -0.030324 -0.041236 -0.005804 3.65e-02 0.005255 0.06148 1.120
## 11 0.016423 -0.016442 -0.058861 2.13e-03 0.057229 0.13044 1.083
## 13 0.030335 0.034133 0.013041 -3.35e-02 -0.012882 -0.04273 1.193
## 15 0.065626 0.019406 -0.075291 -4.21e-02 0.073392 0.17757 1.060
## 17 -0.030481 -0.039225 0.042246 3.35e-02 -0.042983 0.07022 1.139
## 18 0.114515 0.277069 0.109953 -2.52e-01 -0.082232 -0.78312 0.822
## 19 -0.019326 -0.031359 -0.009096 2.62e-02 0.008393 0.05680 1.117
## 21 0.011718 0.012895 -0.008037 -1.21e-02 0.008095 -0.01751 1.141
## 22 -0.006748 -0.027583 0.155318 -9.41e-04 -0.153052 0.25419 1.003
## 23 -0.003249 -0.003289 0.003981 3.13e-03 -0.003988 0.00525 1.206
## 24 0.001156 -0.007731 -0.030797 4.25e-03 0.030619 0.04670 1.129
## 25 0.064985 0.073228 -0.082254 -6.83e-02 0.083670 -0.12201 1.145
## 26 -0.110803 0.036874 -0.001870 -4.38e-03 0.027102 -0.69656 0.834
## 27 -0.165332 -0.158029 0.164687 1.51e-01 -0.162564 0.23850 1.194
## 28 0.006640 0.008542 0.007946 -8.05e-03 -0.007909 -0.01533 1.145
## 29 0.391567 0.321425 -0.212956 -3.72e-01 0.216315 0.48521 1.070
## 30 0.010096 0.011513 0.012788 -1.14e-02 -0.012921 -0.02300 1.149
## 32 0.031262 0.018457 -0.013510 -2.72e-02 0.013883 0.07857 1.091
## 33 0.031577 0.032579 -0.057235 -3.03e-02 0.057503 -0.07130 1.157
## 34 -0.037171 -0.040968 -0.052758 4.02e-02 0.053859 0.09412 1.140
## 35 0.011083 0.004547 0.097224 -1.53e-02 -0.096933 0.12458 1.141
## 36 -0.000938 -0.001820 0.006337 9.82e-04 -0.006354 0.00999 1.123
## 38 -0.020422 -0.000347 -0.108480 1.10e-02 0.111437 -0.18941 1.036
## 39 -0.093759 -0.053169 0.218665 5.97e-02 -0.215933 -0.26609 1.072
## 40 0.175513 0.176923 -0.042317 -1.88e-01 0.045024 -0.24548 1.013
## 41 0.003812 0.001112 -0.019743 -3.67e-03 0.020628 0.04622 1.115
## 42 -0.000438 -0.000250 0.000817 3.70e-04 -0.000838 -0.00181 1.116
## 43 -0.044258 -0.020911 0.121476 2.58e-02 -0.120416 -0.15864 1.084
## 44 -0.033764 -0.020657 -0.145150 3.29e-02 0.146922 -0.19325 1.109
## 45 -0.001225 -0.000770 -0.013775 1.97e-03 0.013711 -0.01769 1.149
## 46 -0.010360 -0.003520 0.018069 6.64e-03 -0.018121 -0.05154 1.097
## 47 -0.002851 -0.003133 0.002108 2.39e-03 -0.001854 0.01420 1.112
## 48 -0.003016 -0.001449 0.002781 2.50e-03 -0.002897 -0.01476 1.108
## 51 -0.085770 -0.081095 0.024176 8.43e-02 -0.025025 -0.13085 1.062
## 52 -0.103342 -0.093969 0.100159 9.62e-02 -0.101077 -0.16432 1.078
## 54 0.171132 0.142275 0.110879 -1.71e-01 -0.112661 -0.26680 1.195
## 55 0.048097 0.036056 0.002394 -4.44e-02 -0.003366 -0.07562 1.131
## 56 0.015978 0.010452 -0.022018 -1.14e-02 0.021152 -0.03792 1.153
## 57 -0.255657 -0.227176 0.029699 2.46e-01 -0.029162 -0.28930 1.062
## 59 -0.204693 -0.185122 -0.081266 2.03e-01 0.081650 -0.28472 1.018
## 60 -0.038185 -0.041704 -0.063019 5.03e-02 0.059973 -0.14674 1.081
## 61 0.025578 0.017373 0.077517 -2.66e-02 -0.078983 -0.12136 1.118
## 62 -0.130160 -0.150254 -0.120059 1.55e-01 0.115449 -0.25902 1.092
## 64 -0.043459 -0.059998 0.048115 5.39e-02 -0.051828 -0.13634 1.113
## 65 -0.030981 -0.044813 0.059554 3.72e-02 -0.062251 -0.11440 1.133
## 66 0.147464 0.137955 0.069769 -1.29e-01 -0.076883 0.27397 1.243
## 67 -0.033113 -0.050918 0.126663 3.50e-02 -0.129226 -0.17595 1.136
## 68 0.008927 0.001167 -0.017926 -3.99e-03 0.017085 -0.03422 1.168
## 69 0.008896 -0.005099 -0.009598 -1.26e-03 0.007993 -0.06027 1.133
## 71 0.552529 0.384804 -0.271075 -6.15e-01 0.305532 -1.46402 0.880
## 72 -0.019803 -0.006860 0.044213 1.04e-02 -0.042743 0.06360 1.242
## 74 0.020759 0.039497 -0.005235 -3.68e-02 0.009481 0.11399 1.286
## 75 -0.198629 -0.235529 0.050154 2.29e-01 -0.058720 -0.32646 1.107
## 77 0.103902 0.462358 -0.347098 6.61e-06 0.295485 2.52470 0.428
## 78 -0.005679 0.016498 -0.009584 -1.22e-03 0.010666 0.07945 1.190
## 79 0.262522 0.347413 0.128376 -2.34e-01 -0.147226 0.83737 1.099
## 80 -0.313594 -0.073315 -0.181897 2.53e-01 0.190966 0.89338 1.045
## 81 -0.350321 -0.433345 -0.031047 3.87e-01 0.026850 -0.49233 1.311
## cook.d hat inf
## 1 7.72e-02 0.0704
## 2 3.39e-02 0.0533
## 4 8.78e-03 0.0746
## 5 3.75e-02 0.0530
## 6 5.69e-03 0.0698
## 7 2.50e-02 0.0405
## 9 8.23e-03 0.1641 *
## 10 7.68e-04 0.0369
## 11 3.43e-03 0.0349
## 13 3.71e-04 0.0894
## 15 6.33e-03 0.0389
## 17 1.00e-03 0.0524
## 18 1.15e-01 0.1114
## 19 6.55e-04 0.0335
## 21 6.23e-05 0.0474
## 22 1.28e-02 0.0418
## 23 5.60e-06 0.0984
## 24 4.43e-04 0.0411
## 25 3.02e-03 0.0668
## 26 9.17e-02 0.0974
## 27 1.15e-02 0.1202
## 28 4.78e-05 0.0508
## 29 4.65e-02 0.1199
## 30 1.08e-04 0.0543
## 32 1.25e-03 0.0241
## 33 1.03e-03 0.0657
## 34 1.80e-03 0.0574
## 35 3.14e-03 0.0646
## 36 2.03e-05 0.0320
## 38 7.18e-03 0.0347
## 39 1.42e-02 0.0665
## 40 1.20e-02 0.0423
## 41 4.34e-04 0.0305
## 42 6.66e-07 0.0255
## 43 5.07e-03 0.0430
## 44 7.53e-03 0.0638
## 45 6.36e-05 0.0536
## 46 5.39e-04 0.0194
## 47 4.10e-05 0.0227
## 48 4.43e-05 0.0189
## 51 3.45e-03 0.0271
## 52 5.43e-03 0.0421
## 54 1.44e-02 0.1266
## 55 1.16e-03 0.0476
## 56 2.92e-04 0.0585
## 57 1.67e-02 0.0689
## 59 1.61e-02 0.0531
## 60 4.34e-03 0.0383
## 61 2.98e-03 0.0502
## 62 1.35e-02 0.0728
## 64 3.76e-03 0.0507
## 65 2.65e-03 0.0574
## 66 1.52e-02 0.1549
## 67 6.26e-03 0.0738
## 68 2.38e-04 0.0702
## 69 7.38e-04 0.0463
## 71 3.94e-01 0.2545 *
## 72 8.22e-04 0.1261
## 74 2.64e-03 0.1594 *
## 75 2.13e-02 0.0964
## 77 1.01e+00 0.2748 *
## 78 1.28e-03 0.0910
## 79 1.36e-01 0.2075
## 80 1.54e-01 0.2000 *
## 81 4.86e-02 0.2262 *
library(car)
## Warning: package 'car' was built under R version 3.5.2
## Loading required package: carData
## Warning: package 'carData' was built under R version 3.5.2
## plotting Influential measures
windows()
influenceIndexPlot(m1) # index plots for infuence measures
influencePlot(m1) # A user friendly representation of the above
## StudRes Hat CookD
## 71 -2.505789 0.2544835 0.3940027
## 77 4.101218 0.2748165 1.0088307
# Regression after deleting the 77th observation, which is influential observation
model.car1 <- lm(MPG ~ VOL + SP + HP + WT,data =train[-77,])
summary(model.car1)
##
## Call:
## lm(formula = MPG ~ VOL + SP + HP + WT, data = train[-77, ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5636 -3.0325 -0.3562 2.1273 14.3246
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.64302 16.70098 1.775 0.0810 .
## VOL -0.17885 0.65743 -0.272 0.7865
## SP 0.41513 0.17645 2.353 0.0219 *
## HP -0.20734 0.04324 -4.795 1.11e-05 ***
## WT -0.10947 1.95909 -0.056 0.9556
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.611 on 60 degrees of freedom
## Multiple R-squared: 0.7567, Adjusted R-squared: 0.7404
## F-statistic: 46.64 on 4 and 60 DF, p-value: < 2.2e-16
# Regression after deleting the 77th & 71st Observations
model.car3 <-lm(MPG ~ VOL + SP + HP + WT,data=train[-c(71,77),])
summary(model.car3)
##
## Call:
## lm(formula = MPG ~ VOL + SP + HP + WT, data = train[-c(71, 77),
## ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5636 -3.0325 -0.3562 2.1273 14.3246
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.64302 16.70098 1.775 0.0810 .
## VOL -0.17885 0.65743 -0.272 0.7865
## SP 0.41513 0.17645 2.353 0.0219 *
## HP -0.20734 0.04324 -4.795 1.11e-05 ***
## WT -0.10947 1.95909 -0.056 0.9556
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.611 on 60 degrees of freedom
## Multiple R-squared: 0.7567, Adjusted R-squared: 0.7404
## F-statistic: 46.64 on 4 and 60 DF, p-value: < 2.2e-16
## Variance Inflation factor to check collinearity b/n variables
vif(m1)
## HP VOL SP WT
## 19.73908 617.79515 19.64267 618.80888
## vif>10 then there exists collinearity among all the variables
## Added Variable plot to check correlation b/n variables and o/p variable
windows()
avPlots(m1)
## VIF and AV plot has given us an indication to delete "wt" variable
finalmodel <- lm(MPG ~ WT + SP + HP,data = train)
summary(finalmodel)
##
## Call:
## lm(formula = MPG ~ WT + SP + HP, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.6808 -2.8470 -0.4567 1.8888 14.4150
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 28.62774 16.15461 1.772 0.0814 .
## WT -0.64198 0.07881 -8.145 2.47e-11 ***
## SP 0.42198 0.17332 2.435 0.0178 *
## HP -0.20917 0.04239 -4.935 6.52e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.576 on 61 degrees of freedom
## Multiple R-squared: 0.7564, Adjusted R-squared: 0.7444
## F-statistic: 63.12 on 3 and 61 DF, p-value: < 2.2e-16
## Final model
finalmodel <- lm(MPG ~ VOL + SP + HP,data = train)
summary(finalmodel)
##
## Call:
## lm(formula = MPG ~ VOL + SP + HP, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5365 -3.0712 -0.3303 2.1783 14.3088
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.85011 16.15094 1.848 0.0694 .
## VOL -0.21555 0.02643 -8.155 2.38e-11 ***
## SP 0.41367 0.17309 2.390 0.0200 *
## HP -0.20696 0.04233 -4.889 7.70e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.573 on 61 degrees of freedom
## Multiple R-squared: 0.7566, Adjusted R-squared: 0.7447
## F-statistic: 63.22 on 3 and 61 DF, p-value: < 2.2e-16
windows()
avPlots(finalmodel)
# Evaluate model LINE assumptions
windows()
plot(finalmodel)
m2 <- lm(MPG ~ . ,data = Cars)
vif(m2)
## HP VOL SP WT
## 19.92659 638.80608 20.00764 639.53382
windows()
avPlots(m2)
summary(m2)
##
## Call:
## lm(formula = MPG ~ ., 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 *
## HP -0.20544 0.03922 -5.239 1.4e-06 ***
## VOL -0.33605 0.56864 -0.591 0.5563
## 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
library(MASS)
## Warning: package 'MASS' was built under R version 3.5.2
stepAIC(m2)
## Start: AIC=248.06
## MPG ~ HP + VOL + 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 ~ HP + VOL + 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 ~ HP + VOL + SP, data = Cars)
##
## Coefficients:
## (Intercept) HP VOL SP
## 29.9234 -0.2067 -0.2017 0.4007
#Residual plots,QQplot,std-Residuals Vs Fitted,Cook's Distance
windows()
qqPlot(finalmodel)
## 71 77
## 57 61
# QQ plot of studentized residuals helps in identifying outlier