(Interpret and comments are in every step, conlusion is at the last)
extract data
#install.packages("mplot")
library(mplot)
data(bodyfat)
Fat = bodyfat$Bodyfat
Neck = bodyfat$Neck
Chest = bodyfat$Chest
Abdo = bodyfat$Abdo
Hip = bodyfat$Hip
Thigh = bodyfat$Thigh
Knee = bodyfat$Knee
Ankle = bodyfat$Ankle
Bic = bodyfat$Bic
Fore = bodyfat$Fore
Wrist = bodyfat$Wrist
Age = bodyfat$Age
Height = bodyfat$Height
Weight = bodyfat$Weight
Bodyfat<-data.frame(Fat, Age, Height, Weight, Neck, Chest, Abdo, Hip, Thigh, Knee, Ankle, Bic, Fore, Wrist)cor()
corResult<-cor(Bodyfat)
corResult[,1]## Fat Age Height Weight Neck Chest Abdo Hip
## 1.0000000 0.2543748 0.1513761 0.6615254 0.4647812 0.7379384 0.8382541 0.6273400
## Thigh Knee Ankle Bic Fore Wrist
## 0.5225635 0.4876076 0.3595300 0.4915120 0.3403827 0.4200907the correlation between the body fat and other parameters, some of them have strong correlation, the smallest is Age and Height.
build the model
Model<-lm(Fat ~ Age+Height+Weight+Neck+Chest+Abdo+Hip+Thigh+Knee+Ankle+Bic+Fore+Wrist)
summary(Model)##
## Call:
## lm(formula = Fat ~ Age + Height + Weight + Neck + Chest + Abdo +
## Hip + Thigh + Knee + Ankle + Bic + Fore + Wrist)
##
## 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
## Height 0.258388 0.320810 0.805 0.4223
## Weight -0.271016 0.243569 -1.113 0.2682
## 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-16Abdo’s coefficient was significantly not 0 at the level of P <0.05.(Neck should be considered, too)
Multiple R-squared: 0.7519 shows that predict variable explained 75.19% of the variance in bodyfat.
Residual standard error: 4.081 shows that the average estimation error of bodyfat is 4.081% in this model.
We need optimize it later.
diagnose
library(car)## Loading required package: carDataplot(Model)
qqPlot(Model,id.method='identify',simulate = TRUE,labels=row.names(Bodyfat),main='Q-Q plot')
## [1] 60 73Residuals vs Fitted: normal distribution —-> OK
Normal QQ: combine with qqplot, Within the confidence interval —-> OK
Scale-Location: The variance is basically a constant —-> OK
Residuals vs Leverage: cook’s distance is in the 0.5 —-> OK
independence
durbinWatsonTest(Model)## lag Autocorrelation D-W Statistic p-value
## 1 -0.07255836 2.121203 0.55
## Alternative hypothesis: rho != 0p=0.49>0.05. No autocorrelation.
homoscedasticity
ncvTest(Model)## Non-constant Variance Score Test
## Variance formula: ~ fitted.values
## Chisquare = 0.8365218, Df = 1, p = 0.36039p value shows the variance is constant.
VIF multicollinearity
vif(Model)## Age Height Weight Neck Chest Abdo Hip Thigh
## 2.256602 4.323426 64.038005 3.849251 13.233597 10.759846 12.318911 7.432930
## Knee Ankle Bic Fore Wrist
## 4.506660 3.382574 4.086372 2.344915 3.662724Weight is much bugger than 10 —-> delete it
Model<-lm(Fat ~ Age+Height+Neck+Chest+Abdo+Hip+Thigh+Knee+Ankle+Bic+Fore+Wrist)
summary(Model)##
## Call:
## lm(formula = Fat ~ Age + Height + Neck + Chest + Abdo + Hip +
## Thigh + Knee + Ankle + Bic + Fore + Wrist)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.3267 -2.4316 -0.1254 2.7091 9.3941
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10.31143 12.80995 -0.805 0.4225
## Age 0.01850 0.04272 0.433 0.6658
## Height -0.01202 0.20965 -0.057 0.9544
## Neck -0.67730 0.31334 -2.162 0.0327 *
## Chest -0.03454 0.12026 -0.287 0.7745
## Abdo 0.94212 0.11358 8.295 2.33e-13 ***
## Hip -0.25301 0.17872 -1.416 0.1596
## Thigh 0.06148 0.19771 0.311 0.7564
## Knee -0.14806 0.32293 -0.458 0.6475
## Ankle -0.21031 0.48680 -0.432 0.6665
## Bic 0.03261 0.24132 0.135 0.8927
## Fore 0.34778 0.26621 1.306 0.1940
## Wrist -0.39793 0.73598 -0.541 0.5898
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.085 on 115 degrees of freedom
## Multiple R-squared: 0.7492, Adjusted R-squared: 0.723
## F-statistic: 28.62 on 12 and 115 DF, p-value: < 2.2e-16vif(Model)## Age Height Neck Chest Abdo Hip Thigh Knee
## 2.174732 1.842510 3.634663 7.037885 9.144421 9.386768 7.196930 4.416175
## Ankle Bic Fore Wrist
## 3.103387 3.985286 2.239099 3.565813VIF smaller than 10—->OK
Computing best subsets regression
1.regression
2.plot
3.Model selection criteria: Adjusted R2, Cp and BIC
library(leaps)
leaps1<-regsubsets(Fat ~ Age+Height+Neck+Chest+Abdo+Hip+Thigh+Knee+Ankle+Bic+Fore+Wrist, data = Bodyfat, nvmax = 12)
leaps2<-regsubsets(Fat ~ Age+Height+Neck+Chest+Abdo+Hip+Thigh+Knee+Ankle+Bic+Fore+Wrist, data = Bodyfat, nbest = 12)
res.sum <- summary(leaps1)
res.sum## Subset selection object
## Call: regsubsets.formula(Fat ~ Age + Height + Neck + Chest + Abdo +
## Hip + Thigh + Knee + Ankle + Bic + Fore + Wrist, data = Bodyfat,
## nvmax = 12)
## 12 Variables (and intercept)
## Forced in Forced out
## Age FALSE FALSE
## Height FALSE FALSE
## Neck FALSE FALSE
## Chest FALSE FALSE
## Abdo FALSE FALSE
## Hip FALSE FALSE
## Thigh FALSE FALSE
## Knee FALSE FALSE
## Ankle FALSE FALSE
## Bic FALSE FALSE
## Fore FALSE FALSE
## Wrist FALSE FALSE
## 1 subsets of each size up to 12
## Selection Algorithm: exhaustive
## Age Height Neck Chest Abdo Hip Thigh Knee Ankle Bic Fore Wrist
## 1 ( 1 ) " " " " " " " " "*" " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " "*" " " "*" " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " "*" " " "*" "*" " " " " " " " " " " " "
## 4 ( 1 ) " " " " "*" " " "*" "*" " " " " " " " " "*" " "
## 5 ( 1 ) " " " " "*" " " "*" "*" " " " " "*" " " "*" " "
## 6 ( 1 ) " " " " "*" " " "*" "*" " " " " "*" " " "*" "*"
## 7 ( 1 ) " " " " "*" " " "*" "*" " " "*" "*" " " "*" "*"
## 8 ( 1 ) "*" " " "*" " " "*" "*" " " "*" "*" " " "*" "*"
## 9 ( 1 ) "*" " " "*" " " "*" "*" "*" "*" "*" " " "*" "*"
## 10 ( 1 ) "*" " " "*" "*" "*" "*" "*" "*" "*" " " "*" "*"
## 11 ( 1 ) "*" " " "*" "*" "*" "*" "*" "*" "*" "*" "*" "*"
## 12 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*"data.frame(
Adj.R2 = which.max(res.sum$adjr2),
CP = which.min(res.sum$cp),
BIC = which.min(res.sum$bic)
)## Adj.R2 CP BIC
## 1 4 3 3plot(leaps1,scale = 'adjr2')
leaps1 for maximum size of subsets to examine, shows choose 4 when consider R2 and choose 3 for CP and BIC.
4 is Neck+Abdo+Hip+Fore
3 is -Fore
res.sum <- summary(leaps2)
res.sum## Subset selection object
## Call: regsubsets.formula(Fat ~ Age + Height + Neck + Chest + Abdo +
## Hip + Thigh + Knee + Ankle + Bic + Fore + Wrist, data = Bodyfat,
## nbest = 12)
## 12 Variables (and intercept)
## Forced in Forced out
## Age FALSE FALSE
## Height FALSE FALSE
## Neck FALSE FALSE
## Chest FALSE FALSE
## Abdo FALSE FALSE
## Hip FALSE FALSE
## Thigh FALSE FALSE
## Knee FALSE FALSE
## Ankle FALSE FALSE
## Bic FALSE FALSE
## Fore FALSE FALSE
## Wrist FALSE FALSE
## 12 subsets of each size up to 8
## Selection Algorithm: exhaustive
## Age Height Neck Chest Abdo Hip Thigh Knee Ankle Bic Fore Wrist
## 1 ( 1 ) " " " " " " " " "*" " " " " " " " " " " " " " "
## 1 ( 2 ) " " " " " " "*" " " " " " " " " " " " " " " " "
## 1 ( 3 ) " " " " " " " " " " "*" " " " " " " " " " " " "
## 1 ( 4 ) " " " " " " " " " " " " "*" " " " " " " " " " "
## 1 ( 5 ) " " " " " " " " " " " " " " " " " " "*" " " " "
## 1 ( 6 ) " " " " " " " " " " " " " " "*" " " " " " " " "
## 1 ( 7 ) " " " " "*" " " " " " " " " " " " " " " " " " "
## 1 ( 8 ) " " " " " " " " " " " " " " " " " " " " " " "*"
## 1 ( 9 ) " " " " " " " " " " " " " " " " "*" " " " " " "
## 1 ( 10 ) " " " " " " " " " " " " " " " " " " " " "*" " "
## 1 ( 11 ) "*" " " " " " " " " " " " " " " " " " " " " " "
## 1 ( 12 ) " " "*" " " " " " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " "*" " " "*" " " " " " " " " " " " " " "
## 2 ( 2 ) " " " " " " " " "*" "*" " " " " " " " " " " " "
## 2 ( 3 ) " " " " " " " " "*" " " " " " " " " " " " " "*"
## 2 ( 4 ) " " " " " " " " "*" " " " " " " "*" " " " " " "
## 2 ( 5 ) " " " " " " " " "*" " " " " "*" " " " " " " " "
## 2 ( 6 ) " " " " " " " " "*" " " "*" " " " " " " " " " "
## 2 ( 7 ) " " "*" " " " " "*" " " " " " " " " " " " " " "
## 2 ( 8 ) " " " " " " " " "*" " " " " " " " " "*" " " " "
## 2 ( 9 ) " " " " " " "*" "*" " " " " " " " " " " " " " "
## 2 ( 10 ) "*" " " " " " " "*" " " " " " " " " " " " " " "
## 2 ( 11 ) " " " " " " " " "*" " " " " " " " " " " "*" " "
## 2 ( 12 ) " " " " "*" "*" " " " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " "*" " " "*" "*" " " " " " " " " " " " "
## 3 ( 2 ) " " " " " " " " "*" "*" " " " " " " " " " " "*"
## 3 ( 3 ) " " " " "*" " " "*" " " " " " " "*" " " " " " "
## 3 ( 4 ) " " " " "*" " " "*" " " " " "*" " " " " " " " "
## 3 ( 5 ) " " " " "*" " " "*" " " "*" " " " " " " " " " "
## 3 ( 6 ) " " "*" "*" " " "*" " " " " " " " " " " " " " "
## 3 ( 7 ) "*" " " "*" " " "*" " " " " " " " " " " " " " "
## 3 ( 8 ) " " " " "*" " " "*" " " " " " " " " " " " " "*"
## 3 ( 9 ) " " " " "*" " " "*" " " " " " " " " " " "*" " "
## 3 ( 10 ) " " " " " " " " "*" "*" " " " " "*" " " " " " "
## 3 ( 11 ) " " " " " " " " "*" "*" " " "*" " " " " " " " "
## 3 ( 12 ) " " " " " " " " "*" " " "*" " " " " " " " " "*"
## 4 ( 1 ) " " " " "*" " " "*" "*" " " " " " " " " "*" " "
## 4 ( 2 ) " " " " "*" " " "*" "*" " " " " " " "*" " " " "
## 4 ( 3 ) " " " " "*" " " "*" "*" " " "*" " " " " " " " "
## 4 ( 4 ) " " " " "*" " " "*" "*" " " " " "*" " " " " " "
## 4 ( 5 ) " " " " "*" " " "*" "*" " " " " " " " " " " "*"
## 4 ( 6 ) " " " " "*" " " "*" "*" "*" " " " " " " " " " "
## 4 ( 7 ) " " "*" "*" " " "*" "*" " " " " " " " " " " " "
## 4 ( 8 ) "*" " " "*" " " "*" "*" " " " " " " " " " " " "
## 4 ( 9 ) " " " " "*" "*" "*" "*" " " " " " " " " " " " "
## 4 ( 10 ) " " " " "*" " " "*" " " " " " " "*" " " "*" " "
## 4 ( 11 ) " " " " "*" " " "*" " " " " "*" " " " " "*" " "
## 4 ( 12 ) " " " " "*" " " "*" " " " " "*" "*" " " " " " "
## 5 ( 1 ) " " " " "*" " " "*" "*" " " " " "*" " " "*" " "
## 5 ( 2 ) " " " " "*" " " "*" "*" " " "*" " " " " "*" " "
## 5 ( 3 ) " " " " "*" " " "*" "*" " " " " " " " " "*" "*"
## 5 ( 4 ) " " "*" "*" " " "*" "*" " " " " " " " " "*" " "
## 5 ( 5 ) " " " " "*" "*" "*" "*" " " " " " " " " "*" " "
## 5 ( 6 ) " " " " "*" " " "*" "*" " " " " " " "*" "*" " "
## 5 ( 7 ) " " " " "*" " " "*" "*" "*" " " " " " " "*" " "
## 5 ( 8 ) "*" " " "*" " " "*" "*" " " " " " " " " "*" " "
## 5 ( 9 ) " " " " "*" " " "*" "*" " " "*" " " "*" " " " "
## 5 ( 10 ) " " " " "*" " " "*" "*" " " " " "*" "*" " " " "
## 5 ( 11 ) " " " " "*" " " "*" "*" " " " " " " "*" " " "*"
## 5 ( 12 ) " " " " "*" " " "*" "*" " " "*" " " " " " " "*"
## 6 ( 1 ) " " " " "*" " " "*" "*" " " " " "*" " " "*" "*"
## 6 ( 2 ) " " " " "*" " " "*" "*" " " "*" "*" " " "*" " "
## 6 ( 3 ) " " " " "*" " " "*" "*" " " "*" " " " " "*" "*"
## 6 ( 4 ) " " " " "*" " " "*" "*" "*" " " "*" " " "*" " "
## 6 ( 5 ) " " " " "*" "*" "*" "*" " " " " "*" " " "*" " "
## 6 ( 6 ) " " "*" "*" " " "*" "*" " " " " "*" " " "*" " "
## 6 ( 7 ) " " " " "*" " " "*" "*" " " " " "*" "*" "*" " "
## 6 ( 8 ) "*" " " "*" " " "*" "*" " " " " "*" " " "*" " "
## 6 ( 9 ) " " "*" "*" " " "*" "*" " " " " " " " " "*" "*"
## 6 ( 10 ) " " " " "*" " " "*" "*" "*" "*" " " " " "*" " "
## 6 ( 11 ) "*" " " "*" " " "*" "*" " " " " " " " " "*" "*"
## 6 ( 12 ) " " " " "*" "*" "*" "*" " " "*" " " " " "*" " "
## 7 ( 1 ) " " " " "*" " " "*" "*" " " "*" "*" " " "*" "*"
## 7 ( 2 ) " " "*" "*" " " "*" "*" " " " " "*" " " "*" "*"
## 7 ( 3 ) " " " " "*" " " "*" "*" "*" "*" "*" " " "*" " "
## 7 ( 4 ) " " " " "*" "*" "*" "*" " " " " "*" " " "*" "*"
## 7 ( 5 ) "*" " " "*" " " "*" "*" " " "*" " " " " "*" "*"
## 7 ( 6 ) " " " " "*" " " "*" "*" "*" " " "*" " " "*" "*"
## 7 ( 7 ) "*" " " "*" " " "*" "*" " " " " "*" " " "*" "*"
## 7 ( 8 ) " " " " "*" "*" "*" "*" " " "*" "*" " " "*" " "
## 7 ( 9 ) " " " " "*" " " "*" "*" " " " " "*" "*" "*" "*"
## 7 ( 10 ) " " " " "*" " " "*" "*" " " "*" "*" "*" "*" " "
## 7 ( 11 ) " " " " "*" "*" "*" "*" " " "*" " " " " "*" "*"
## 7 ( 12 ) " " "*" "*" " " "*" "*" " " "*" "*" " " "*" " "
## 8 ( 1 ) "*" " " "*" " " "*" "*" " " "*" "*" " " "*" "*"
## 8 ( 2 ) " " " " "*" " " "*" "*" "*" "*" "*" " " "*" "*"
## 8 ( 3 ) " " " " "*" "*" "*" "*" " " "*" "*" " " "*" "*"
## 8 ( 4 ) "*" " " "*" " " "*" "*" "*" "*" " " " " "*" "*"
## 8 ( 5 ) " " "*" "*" "*" "*" "*" " " " " "*" " " "*" "*"
## 8 ( 6 ) " " "*" "*" " " "*" "*" " " "*" "*" " " "*" "*"
## 8 ( 7 ) " " " " "*" " " "*" "*" " " "*" "*" "*" "*" "*"
## 8 ( 8 ) "*" "*" "*" " " "*" "*" " " " " "*" " " "*" "*"
## 8 ( 9 ) " " " " "*" "*" "*" "*" "*" "*" "*" " " "*" " "
## 8 ( 10 ) "*" " " "*" " " "*" "*" "*" "*" "*" " " "*" " "
## 8 ( 11 ) "*" " " "*" " " "*" "*" "*" " " "*" " " "*" "*"
## 8 ( 12 ) "*" " " "*" " " "*" "*" " " "*" " " "*" "*" "*"data.frame(
Adj.R2 = which.max(res.sum$adjr2),
CP = which.min(res.sum$cp),
BIC = which.min(res.sum$bic)
)## Adj.R2 CP BIC
## 1 37 25 25plot(leaps2,scale = 'adjr2')
leaps2 for number of subsets of each size to record, shows choose 31 when consider R2 and choose 21 for CP and BIC.
37 is Neck+Abdo+Hip+Fore
25 is -Fore
So all information said that the best choice is Neck+Abdo+Hip (+Fore(if consider R2))
stepwise method
AIC | backward
library(MASS)
stepAIC(Model,direction = "backward")## Start: AIC=372.58
## Fat ~ Age + Height + Neck + Chest + Abdo + Hip + Thigh + Knee +
## Ankle + Bic + Fore + Wrist
##
## Df Sum of Sq RSS AIC
## - Height 1 0.05 1919.3 370.58
## - Bic 1 0.30 1919.5 370.60
## - Chest 1 1.38 1920.6 370.67
## - Thigh 1 1.61 1920.8 370.69
## - Ankle 1 3.11 1922.3 370.79
## - Age 1 3.13 1922.3 370.79
## - Knee 1 3.51 1922.7 370.81
## - Wrist 1 4.88 1924.1 370.90
## - Fore 1 28.48 1947.7 372.46
## <none> 1919.2 372.58
## - Hip 1 33.45 1952.7 372.79
## - Neck 1 77.97 1997.2 375.68
## - Abdo 1 1148.35 3067.6 430.61
##
## Step: AIC=370.58
## Fat ~ Age + Neck + Chest + Abdo + Hip + Thigh + Knee + Ankle +
## Bic + Fore + Wrist
##
## Df Sum of Sq RSS AIC
## - Bic 1 0.32 1919.6 368.60
## - Chest 1 1.34 1920.6 368.67
## - Thigh 1 1.95 1921.2 368.71
## - Ankle 1 3.12 1922.4 368.79
## - Age 1 3.44 1922.7 368.81
## - Wrist 1 4.82 1924.1 368.90
## - Knee 1 4.93 1924.2 368.91
## - Fore 1 28.48 1947.7 370.47
## <none> 1919.3 370.58
## - Hip 1 37.35 1956.6 371.05
## - Neck 1 80.13 1999.4 373.82
## - Abdo 1 1158.13 3077.4 429.02
##
## Step: AIC=368.6
## Fat ~ Age + Neck + Chest + Abdo + Hip + Thigh + Knee + Ankle +
## Fore + Wrist
##
## Df Sum of Sq RSS AIC
## - Chest 1 1.16 1920.8 366.68
## - Thigh 1 2.98 1922.6 366.80
## - Age 1 3.38 1923.0 366.83
## - Ankle 1 3.62 1923.2 366.84
## - Wrist 1 4.64 1924.2 366.91
## - Knee 1 4.86 1924.5 366.93
## <none> 1919.6 368.60
## - Fore 1 36.09 1955.7 368.99
## - Hip 1 37.04 1956.6 369.05
## - Neck 1 79.81 1999.4 371.82
## - Abdo 1 1164.23 3083.8 427.28
##
## Step: AIC=366.68
## Fat ~ Age + Neck + Abdo + Hip + Thigh + Knee + Ankle + Fore +
## Wrist
##
## Df Sum of Sq RSS AIC
## - Thigh 1 3.26 1924.0 364.90
## - Age 1 3.33 1924.1 364.90
## - Ankle 1 3.60 1924.3 364.92
## - Wrist 1 4.73 1925.5 364.99
## - Knee 1 4.82 1925.6 365.00
## <none> 1920.7 366.68
## - Fore 1 34.94 1955.7 366.99
## - Hip 1 38.40 1959.1 367.21
## - Neck 1 85.07 2005.8 370.23
## - Abdo 1 1924.58 3845.3 453.53
##
## Step: AIC=364.9
## Fat ~ Age + Neck + Abdo + Hip + Knee + Ankle + Fore + Wrist
##
## Df Sum of Sq RSS AIC
## - Age 1 1.52 1925.5 363.00
## - Ankle 1 3.17 1927.2 363.11
## - Knee 1 3.32 1927.3 363.12
## - Wrist 1 5.13 1929.1 363.24
## <none> 1924.0 364.90
## - Hip 1 36.09 1960.1 365.28
## - Fore 1 37.24 1961.2 365.35
## - Neck 1 82.15 2006.2 368.25
## - Abdo 1 1961.69 3885.7 452.87
##
## Step: AIC=363
## Fat ~ Neck + Abdo + Hip + Knee + Ankle + Fore + Wrist
##
## Df Sum of Sq RSS AIC
## - Knee 1 2.62 1928.1 361.17
## - Wrist 1 3.70 1929.2 361.24
## - Ankle 1 4.36 1929.9 361.29
## <none> 1925.5 363.00
## - Fore 1 35.85 1961.4 363.36
## - Hip 1 56.44 1982.0 364.70
## - Neck 1 84.61 2010.1 366.50
## - Abdo 1 2556.02 4481.5 469.13
##
## Step: AIC=361.17
## Fat ~ Neck + Abdo + Hip + Ankle + Fore + Wrist
##
## Df Sum of Sq RSS AIC
## - Wrist 1 3.79 1931.9 359.42
## - Ankle 1 9.03 1937.2 359.77
## <none> 1928.1 361.17
## - Fore 1 36.30 1964.4 361.56
## - Hip 1 67.91 1996.1 363.60
## - Neck 1 87.51 2015.7 364.85
## - Abdo 1 2570.36 4498.5 467.61
##
## Step: AIC=359.42
## Fat ~ Neck + Abdo + Hip + Ankle + Fore
##
## Df Sum of Sq RSS AIC
## - Ankle 1 14.86 1946.8 358.40
## <none> 1931.9 359.42
## - Fore 1 36.06 1968.0 359.79
## - Hip 1 67.31 1999.2 361.81
## - Neck 1 140.04 2072.0 366.38
## - Abdo 1 2574.16 4506.1 465.83
##
## Step: AIC=358.4
## Fat ~ Neck + Abdo + Hip + Fore
##
## Df Sum of Sq RSS AIC
## - Fore 1 28.63 1975.4 358.27
## <none> 1946.8 358.40
## - Hip 1 126.23 2073.0 364.45
## - Neck 1 147.87 2094.7 365.78
## - Abdo 1 2670.93 4617.7 466.96
##
## Step: AIC=358.27
## Fat ~ Neck + Abdo + Hip
##
## Df Sum of Sq RSS AIC
## <none> 1975.4 358.27
## - Hip 1 107.53 2083.0 363.06
## - Neck 1 119.24 2094.7 363.78
## - Abdo 1 2642.74 4618.2 464.97##
## Call:
## lm(formula = Fat ~ Neck + Abdo + Hip)
##
## Coefficients:
## (Intercept) Neck Abdo Hip
## -14.2955 -0.6266 0.9290 -0.2863least AIC: Neck+Abdo+Hip
anova and AIC test AIC for fore (and chest for interest)
Model1<-lm(Fat ~ Neck+Abdo+Hip)
Model2<-lm(Fat ~ Neck+Abdo+Hip+Fore)
#test for interest
#Because in cor() the chest has a high correlation.
#But all result shows we can delete them
Model3<-lm(Fat ~ Chest+Neck+Abdo+Hip)
anova(Model1,Model2)## Analysis of Variance Table
##
## Model 1: Fat ~ Neck + Abdo + Hip
## Model 2: Fat ~ Neck + Abdo + Hip + Fore
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 124 1975.4
## 2 123 1946.8 1 28.626 1.8086 0.1812AIC(Model1,Model2,Model3)## df AIC
## Model1 5 723.5212
## Model2 6 723.6528
## Model3 6 725.5200p = 0.1812 means we can delete fore. And AIC has same result
Result
summary(Model1)##
## Call:
## lm(formula = Fat ~ Neck + Abdo + Hip)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.0590 -2.5669 -0.0023 2.6107 8.7289
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -14.29546 7.49107 -1.908 0.05866 .
## Neck -0.62659 0.22903 -2.736 0.00713 **
## Abdo 0.92901 0.07213 12.880 < 2e-16 ***
## Hip -0.28631 0.11020 -2.598 0.01051 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.991 on 124 degrees of freedom
## Multiple R-squared: 0.7418, Adjusted R-squared: 0.7356
## F-statistic: 118.8 on 3 and 124 DF, p-value: < 2.2e-16So the regression equation:
bodyfat = -14.29546 - 0.62659 * Neck + 0.92901 * Abdomen - 0.28631 * Hip
The result is the same with when we only consider the circumference without age, height, weight. (https://rpubs.com/YifeiLiu/RegressionBodyfat)
Consider weight(same result)
because weight is an important parameter in common sense
newModel1<-lm(Fat ~ Weight+Neck+Abdo+Hip)
vif(newModel1)## Weight Neck Abdo Hip
## 13.310543 2.631873 4.694637 8.183684newModel2<-lm(Fat ~ Weight+Neck+Abdo+Fore)
vif(newModel2)## Weight Neck Abdo Fore
## 7.058795 2.774132 4.979725 1.899964summary(newModel2)##
## Call:
## lm(formula = Fat ~ Weight + Neck + Abdo + Fore)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.1637 -2.8476 0.0877 2.7522 8.3342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -41.88325 8.30967 -5.040 1.62e-06 ***
## Weight -0.22932 0.07868 -2.915 0.00423 **
## Neck -0.61856 0.26606 -2.325 0.02172 *
## Abdo 0.98046 0.08146 12.036 < 2e-16 ***
## Fore 0.43409 0.23834 1.821 0.07099 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.971 on 123 degrees of freedom
## Multiple R-squared: 0.7466, Adjusted R-squared: 0.7383
## F-statistic: 90.58 on 4 and 123 DF, p-value: < 2.2e-16plot(newModel2)
Consider weight and hip will let VIF > 10.
Consider weight and fore, the summary is not better than not consider weight above.
So the regression equation is still:
bodyfat = -14.29546 - 0.62659 * Neck + 0.92901 * Abdomen - 0.28631 * Hip
The result is the same with when we only consider the circumference without age, height, weight. (https://rpubs.com/YifeiLiu/RegressionBodyfat)