Data605_Discussion12
Jenny
April 26, 2018
Visualizing the Relationships in the Data
head(mtcars)## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1pairs(mtcars, gap=0.5)
##### Backward Elimination Process
car.lm <- lm(mpg ~ cyl + disp + hp + drat +
wt + qsec + vs + am +
gear + carb, data=mtcars)
summary(car.lm)##
## Call:
## lm(formula = mpg ~ cyl + disp + hp + drat + wt + qsec + vs +
## am + gear + carb, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4506 -1.6044 -0.1196 1.2193 4.6271
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.30337 18.71788 0.657 0.5181
## cyl -0.11144 1.04502 -0.107 0.9161
## disp 0.01334 0.01786 0.747 0.4635
## hp -0.02148 0.02177 -0.987 0.3350
## drat 0.78711 1.63537 0.481 0.6353
## wt -3.71530 1.89441 -1.961 0.0633 .
## qsec 0.82104 0.73084 1.123 0.2739
## vs 0.31776 2.10451 0.151 0.8814
## am 2.52023 2.05665 1.225 0.2340
## gear 0.65541 1.49326 0.439 0.6652
## carb -0.19942 0.82875 -0.241 0.8122
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.65 on 21 degrees of freedom
## Multiple R-squared: 0.869, Adjusted R-squared: 0.8066
## F-statistic: 13.93 on 10 and 21 DF, p-value: 3.793e-07# Remove the cyl
car.lm <- update(car.lm, .~. - cyl, data=mtcars)
summary(car.lm)##
## Call:
## lm(formula = mpg ~ disp + hp + drat + wt + qsec + vs + am + gear +
## carb, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4286 -1.5908 -0.0412 1.2120 4.5961
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.96007 13.53030 0.810 0.4266
## disp 0.01283 0.01682 0.763 0.4538
## hp -0.02191 0.02091 -1.048 0.3062
## drat 0.83520 1.53625 0.544 0.5921
## wt -3.69251 1.83954 -2.007 0.0572 .
## qsec 0.84244 0.68678 1.227 0.2329
## vs 0.38975 1.94800 0.200 0.8433
## am 2.57743 1.94035 1.328 0.1977
## gear 0.71155 1.36562 0.521 0.6075
## carb -0.21958 0.78856 -0.278 0.7833
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.59 on 22 degrees of freedom
## Multiple R-squared: 0.8689, Adjusted R-squared: 0.8153
## F-statistic: 16.21 on 9 and 22 DF, p-value: 9.031e-08# Remove vs
car.lm <- update(car.lm, .~. - vs, data=mtcars)
summary(car.lm)##
## Call:
## lm(formula = mpg ~ disp + hp + drat + wt + qsec + am + gear +
## carb, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.356 -1.576 -0.149 1.218 4.604
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.76828 11.89230 0.821 0.4199
## disp 0.01214 0.01612 0.753 0.4590
## hp -0.02095 0.01993 -1.051 0.3040
## drat 0.87510 1.49113 0.587 0.5630
## wt -3.71151 1.79834 -2.064 0.0505 .
## qsec 0.91083 0.58312 1.562 0.1319
## am 2.52390 1.88128 1.342 0.1928
## gear 0.75984 1.31577 0.577 0.5692
## carb -0.24796 0.75933 -0.327 0.7470
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.535 on 23 degrees of freedom
## Multiple R-squared: 0.8687, Adjusted R-squared: 0.823
## F-statistic: 19.02 on 8 and 23 DF, p-value: 2.008e-08# Remove carb
car.lm <- update(car.lm, .~. - carb, data=mtcars)
summary(car.lm)##
## Call:
## lm(formula = mpg ~ disp + hp + drat + wt + qsec + am + gear,
## data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.1200 -1.7753 -0.1446 1.0903 4.7172
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.19763 11.54220 0.797 0.43334
## disp 0.01552 0.01214 1.278 0.21342
## hp -0.02471 0.01596 -1.548 0.13476
## drat 0.81023 1.45007 0.559 0.58151
## wt -4.13065 1.23593 -3.342 0.00272 **
## qsec 1.00979 0.48883 2.066 0.04981 *
## am 2.58980 1.83528 1.411 0.17104
## gear 0.60644 1.20596 0.503 0.61964
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.488 on 24 degrees of freedom
## Multiple R-squared: 0.8681, Adjusted R-squared: 0.8296
## F-statistic: 22.56 on 7 and 24 DF, p-value: 4.218e-09# Remove gear
car.lm <- update(car.lm, .~. - gear, data=mtcars)
summary(car.lm)##
## Call:
## lm(formula = mpg ~ disp + hp + drat + wt + qsec + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.2669 -1.6148 -0.2585 1.1220 4.5564
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.71062 10.97539 0.976 0.33848
## disp 0.01310 0.01098 1.193 0.24405
## hp -0.02180 0.01465 -1.488 0.14938
## drat 1.02065 1.36748 0.746 0.46240
## wt -4.04454 1.20558 -3.355 0.00254 **
## qsec 0.99073 0.48002 2.064 0.04955 *
## am 2.98469 1.63382 1.827 0.07969 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.45 on 25 degrees of freedom
## Multiple R-squared: 0.8667, Adjusted R-squared: 0.8347
## F-statistic: 27.09 on 6 and 25 DF, p-value: 8.637e-10# Remove drat
car.lm <- update(car.lm, .~. - drat, data=mtcars)
summary(car.lm)##
## Call:
## lm(formula = mpg ~ disp + hp + wt + qsec + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.5399 -1.7398 -0.3196 1.1676 4.5534
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.36190 9.74079 1.474 0.15238
## disp 0.01124 0.01060 1.060 0.29897
## hp -0.02117 0.01450 -1.460 0.15639
## wt -4.08433 1.19410 -3.420 0.00208 **
## qsec 1.00690 0.47543 2.118 0.04391 *
## am 3.47045 1.48578 2.336 0.02749 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.429 on 26 degrees of freedom
## Multiple R-squared: 0.8637, Adjusted R-squared: 0.8375
## F-statistic: 32.96 on 5 and 26 DF, p-value: 1.844e-10# Remove disp
car.lm <- update(car.lm, .~. - disp, data=mtcars)
summary(car.lm)##
## Call:
## lm(formula = mpg ~ hp + wt + qsec + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4975 -1.5902 -0.1122 1.1795 4.5404
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.44019 9.31887 1.871 0.07215 .
## hp -0.01765 0.01415 -1.247 0.22309
## wt -3.23810 0.88990 -3.639 0.00114 **
## qsec 0.81060 0.43887 1.847 0.07573 .
## am 2.92550 1.39715 2.094 0.04579 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.435 on 27 degrees of freedom
## Multiple R-squared: 0.8579, Adjusted R-squared: 0.8368
## F-statistic: 40.74 on 4 and 27 DF, p-value: 4.589e-11# Remove hp
car.lm <- update(car.lm, .~. - hp, data=mtcars)
summary(car.lm)##
## Call:
## lm(formula = mpg ~ wt + qsec + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4811 -1.5555 -0.7257 1.4110 4.6610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.6178 6.9596 1.382 0.177915
## wt -3.9165 0.7112 -5.507 6.95e-06 ***
## qsec 1.2259 0.2887 4.247 0.000216 ***
## am 2.9358 1.4109 2.081 0.046716 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.459 on 28 degrees of freedom
## Multiple R-squared: 0.8497, Adjusted R-squared: 0.8336
## F-statistic: 52.75 on 3 and 28 DF, p-value: 1.21e-11The final model is mpg = 9.6178 - 3.9165wt + 1.2259qsec + 2.9358*am
Residual Analysis
plot(fitted(car.lm), resid(car.lm))
qqnorm(resid(car.lm))
qqline(resid(car.lm))