MLR assignment
Marcin Kyć
summary(full.mod)##
## Call:
## lm(formula = Bodyfat ~ ., data = subset(bodyfat, select = -Id))
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.3767 -2.5514 -0.1723 2.6391 9.1393
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -52.553646 40.062856 -1.312 0.1922
## Age 0.009288 0.043470 0.214 0.8312
## Weight -0.271016 0.243569 -1.113 0.2682
## Height 0.258388 0.320810 0.805 0.4223
## Neck -0.592669 0.322125 -1.840 0.0684 .
## Chest 0.090883 0.164738 0.552 0.5822
## Abdo 0.995184 0.123072 8.086 7.29e-13 ***
## Hip -0.141981 0.204533 -0.694 0.4890
## Thigh 0.101272 0.200714 0.505 0.6148
## Knee -0.096682 0.325889 -0.297 0.7673
## Ankle -0.048017 0.507695 -0.095 0.9248
## Bic 0.075332 0.244105 0.309 0.7582
## Fore 0.412107 0.272144 1.514 0.1327
## Wrist -0.263067 0.745145 -0.353 0.7247
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.081 on 114 degrees of freedom
## Multiple R-squared: 0.7519, Adjusted R-squared: 0.7236
## F-statistic: 26.57 on 13 and 114 DF, p-value: < 2.2e-16cor(bodyfat)## Id Bodyfat Age Weight Height Neck
## Id 1.00000000 0.03939478 0.31247917 0.02991081 0.1005728 0.01905489
## Bodyfat 0.03939478 1.00000000 0.25437478 0.66152537 0.1513761 0.46478119
## Age 0.31247917 0.25437478 1.00000000 0.03625955 -0.1456736 0.16143675
## Weight 0.02991081 0.66152537 0.03625955 1.00000000 0.5566971 0.77653328
## Height 0.10057276 0.15137609 -0.14567365 0.55669710 1.0000000 0.32875108
## Neck 0.01905489 0.46478119 0.16143675 0.77653328 0.3287511 1.00000000
## Chest 0.08919333 0.73793844 0.21663744 0.89169209 0.2738272 0.72333247
## Abdo 0.05306424 0.83825415 0.24349337 0.88518199 0.2855097 0.69104708
## Hip -0.05670558 0.62734000 -0.03808131 0.93299025 0.4593269 0.67828816
## Thigh -0.11581914 0.52256353 -0.20937442 0.83565003 0.3633394 0.63693632
## Knee 0.08259872 0.48760762 0.09193971 0.82244800 0.5652882 0.61183461
## Ankle -0.05909733 0.35953002 -0.06344775 0.73209696 0.4387430 0.54684519
## Bic -0.07388071 0.49151203 -0.07291758 0.78242651 0.2999832 0.68510986
## Fore -0.01516617 0.34038274 -0.05128106 0.61700438 0.2959517 0.62291587
## Wrist 0.07399466 0.42009072 0.30484297 0.70467729 0.2724456 0.75856061
## Chest Abdo Hip Thigh Knee Ankle
## Id 0.08919333 0.05306424 -0.05670558 -0.1158191 0.08259872 -0.05909733
## Bodyfat 0.73793844 0.83825415 0.62734000 0.5225635 0.48760762 0.35953002
## Age 0.21663744 0.24349337 -0.03808131 -0.2093744 0.09193971 -0.06344775
## Weight 0.89169209 0.88518199 0.93299025 0.8356500 0.82244800 0.73209696
## Height 0.27382720 0.28550967 0.45932694 0.3633394 0.56528815 0.43874296
## Neck 0.72333247 0.69104708 0.67828816 0.6369363 0.61183461 0.54684519
## Chest 1.00000000 0.90988148 0.80172538 0.6793998 0.65754967 0.55066510
## Abdo 0.90988148 1.00000000 0.84601360 0.7159499 0.68721784 0.55320958
## Hip 0.80172538 0.84601360 1.00000000 0.8717518 0.77475311 0.69969995
## Thigh 0.67939979 0.71594992 0.87175176 1.0000000 0.74233690 0.68393641
## Knee 0.65754967 0.68721784 0.77475311 0.7423369 1.00000000 0.73889645
## Ankle 0.55066510 0.55320958 0.69969995 0.6839364 0.73889645 1.00000000
## Bic 0.69793237 0.66113863 0.75209342 0.7880858 0.63122086 0.55134883
## Fore 0.52877383 0.45341062 0.52056277 0.5413877 0.47546587 0.51538622
## Wrist 0.64290732 0.62450167 0.64123969 0.5427466 0.61037364 0.63668233
## Bic Fore Wrist
## Id -0.07388071 -0.01516617 0.07399466
## Bodyfat 0.49151203 0.34038274 0.42009072
## Age -0.07291758 -0.05128106 0.30484297
## Weight 0.78242651 0.61700438 0.70467729
## Height 0.29998318 0.29595168 0.27244564
## Neck 0.68510986 0.62291587 0.75856061
## Chest 0.69793237 0.52877383 0.64290732
## Abdo 0.66113863 0.45341062 0.62450167
## Hip 0.75209342 0.52056277 0.64123969
## Thigh 0.78808577 0.54138768 0.54274663
## Knee 0.63122086 0.47546587 0.61037364
## Ankle 0.55134883 0.51538622 0.63668233
## Bic 1.00000000 0.67831620 0.58626764
## Fore 0.67831620 1.00000000 0.52990372
## Wrist 0.58626764 0.52990372 1.00000000corrplot(cor(bodyfat), method = "number", type = "upper", diag =FALSE)
##I assumed that variables are “highly correlated” if correlation is >0.6.
##Therefore, Weight is higly correlated with: Chest, Abdo, Hip, Thigh, Knee, Ankle, Bic, Fore, Wrist, Neck.
##Thus, I remove them from model and create the new one with only Weight, Height, Age.
ggplot(bodyfat, aes(Bodyfat, Weight+Age+Height)) +
geom_point(na.rm = TRUE) +
geom_smooth(method = "lm", na.rm = TRUE)## `geom_smooth()` using formula 'y ~ x'
full_mod<-lm(Bodyfat~Weight+Age+Height, data=bodyfat)
par(mfrow=c(2,2))
with(bodyfat, plot(Age,resid(full_mod),
xlab="Age",
ylab="Residual"))
abline(h=0, lwd=2, lty=2, col="blue")
with(bodyfat, plot(Height,resid(full_mod),
xlab="Height",
ylab="Residual"))
abline(h=0, lwd=2, lty=2, col="blue")
with(bodyfat, plot(Weight,resid(full_mod),
xlab="Weight",
ylab="Residual"))
abline(h=0, lwd=2, lty=2, col="blue")
##Explanatory variables are linearly related to percent body fat
qqnorm(resid(full_mod))
qqline(resid(full_mod))
##Residuals are normally distributed
summary(full_mod)##
## Call:
## lm(formula = Bodyfat ~ Weight + Age + Height, data = bodyfat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.2015 -3.8572 0.3377 3.6076 11.5864
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.11780 15.95284 2.139 0.03442 *
## Weight 0.52509 0.04838 10.853 < 2e-16 ***
## Age 0.11531 0.03863 2.985 0.00341 **
## Height -0.89161 0.24774 -3.599 0.00046 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.334 on 124 degrees of freedom
## Multiple R-squared: 0.5389, Adjusted R-squared: 0.5278
## F-statistic: 48.31 on 3 and 124 DF, p-value: < 2.2e-16##Equation:
Body fat = 34.11780 ± 15.95284 + (0.52509 ± 0.04838) * Weight + (0.11531 ± 0.03863) * Age + (-0.89161 ± 0.24774) * Height ± 5.334
##Body fat changes approx.:
increases 0.52509 ± 0.04838 every additional Weight unit
increases 0.11531 ± 0.03863 every additional Age unit
decreases 0.89161 ± 0.24774 every additional Height unit
predict.lm(full_mod, newdata=data.frame(Weight=90, Age=30, Height=74),
interval="prediction", level=0.95)## fit lwr upr
## 1 18.85597 8.119608 29.59234