Homework6_2
Kunal Dolas
10/16/2020
Using the fat data, fit the model described in section 4.2 of the Faraway text using the following methods: Least squares, Least absolute deviations, and Huber method.
require(faraway)## Loading required package: farawaydata(fat, package="faraway")
head(fat)## brozek siri density age weight height adipos free neck chest abdom hip
## 1 12.6 12.3 1.0708 23 154.25 67.75 23.7 134.9 36.2 93.1 85.2 94.5
## 2 6.9 6.1 1.0853 22 173.25 72.25 23.4 161.3 38.5 93.6 83.0 98.7
## 3 24.6 25.3 1.0414 22 154.00 66.25 24.7 116.0 34.0 95.8 87.9 99.2
## 4 10.9 10.4 1.0751 26 184.75 72.25 24.9 164.7 37.4 101.8 86.4 101.2
## 5 27.8 28.7 1.0340 24 184.25 71.25 25.6 133.1 34.4 97.3 100.0 101.9
## 6 20.6 20.9 1.0502 24 210.25 74.75 26.5 167.0 39.0 104.5 94.4 107.8
## thigh knee ankle biceps forearm wrist
## 1 59.0 37.3 21.9 32.0 27.4 17.1
## 2 58.7 37.3 23.4 30.5 28.9 18.2
## 3 59.6 38.9 24.0 28.8 25.2 16.6
## 4 60.1 37.3 22.8 32.4 29.4 18.2
## 5 63.2 42.2 24.0 32.2 27.7 17.7
## 6 66.0 42.0 25.6 35.7 30.6 18.8lmod<-lm(brozek ~ age + weight + height + neck + chest + abdom + hip + thigh + knee + ankle + biceps + forearm + wrist, data=fat)
summary(lmod)##
## Call:
## lm(formula = brozek ~ age + weight + height + neck + chest +
## abdom + hip + thigh + knee + ankle + biceps + forearm + wrist,
## data = fat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.264 -2.572 -0.097 2.898 9.327
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -15.29255 16.06992 -0.952 0.34225
## age 0.05679 0.02996 1.895 0.05929 .
## weight -0.08031 0.04958 -1.620 0.10660
## height -0.06460 0.08893 -0.726 0.46830
## neck -0.43754 0.21533 -2.032 0.04327 *
## chest -0.02360 0.09184 -0.257 0.79740
## abdom 0.88543 0.08008 11.057 < 2e-16 ***
## hip -0.19842 0.13516 -1.468 0.14341
## thigh 0.23190 0.13372 1.734 0.08418 .
## knee -0.01168 0.22414 -0.052 0.95850
## ankle 0.16354 0.20514 0.797 0.42614
## biceps 0.15280 0.15851 0.964 0.33605
## forearm 0.43049 0.18445 2.334 0.02044 *
## wrist -1.47654 0.49552 -2.980 0.00318 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.988 on 238 degrees of freedom
## Multiple R-squared: 0.749, Adjusted R-squared: 0.7353
## F-statistic: 54.63 on 13 and 238 DF, p-value: < 2.2e-16cor(residuals(lmod)[-1],residuals(lmod)[-length(residuals(lmod))])## [1] 0.1030356##least squares
require(nlme)## Loading required package: nlmeglmod <- gls(brozek ~ age + weight + height + neck + chest + abdom + hip + thigh + knee + ankle + biceps + forearm + wrist, data=na.omit(fat))
summary(glmod)## Generalized least squares fit by REML
## Model: brozek ~ age + weight + height + neck + chest + abdom + hip + thigh + knee + ankle + biceps + forearm + wrist
## Data: na.omit(fat)
## AIC BIC logLik
## 1464.835 1516.919 -717.4177
##
## Coefficients:
## Value Std.Error t-value p-value
## (Intercept) -15.292549 16.069921 -0.951626 0.3423
## age 0.056786 0.029965 1.895105 0.0593
## weight -0.080310 0.049581 -1.619787 0.1066
## height -0.064600 0.088930 -0.726414 0.4683
## neck -0.437541 0.215334 -2.031920 0.0433
## chest -0.023603 0.091839 -0.257007 0.7974
## abdom 0.885429 0.080077 11.057242 0.0000
## hip -0.198419 0.135156 -1.468068 0.1434
## thigh 0.231895 0.133718 1.734211 0.0842
## knee -0.011677 0.224143 -0.052095 0.9585
## ankle 0.163536 0.205143 0.797178 0.4261
## biceps 0.152799 0.158513 0.963954 0.3360
## forearm 0.430489 0.184452 2.333874 0.0204
## wrist -1.476537 0.495519 -2.979779 0.0032
##
## Correlation:
## (Intr) age weight height neck chest abdom hip thigh knee
## age 0.041
## weight 0.898 0.207
## height -0.587 0.133 -0.454
## neck -0.299 -0.091 -0.292 0.025
## chest -0.429 -0.035 -0.408 0.214 -0.032
## abdom 0.001 -0.451 -0.202 0.110 -0.026 -0.469
## hip -0.604 0.080 -0.541 0.252 0.174 0.181 -0.247
## thigh -0.075 0.366 -0.056 0.203 -0.106 0.157 -0.137 -0.324
## knee -0.232 -0.251 -0.255 -0.086 0.150 0.088 0.069 -0.043 -0.278
## ankle -0.179 0.104 -0.183 0.062 0.104 0.025 0.093 0.053 0.010 -0.168
## biceps -0.067 -0.072 -0.177 0.049 -0.082 -0.080 0.119 0.079 -0.268 0.076
## forearm -0.082 0.163 -0.011 0.021 -0.114 -0.118 0.053 0.095 -0.010 -0.090
## wrist -0.181 -0.339 -0.143 -0.099 -0.290 0.017 0.149 -0.020 0.064 -0.075
## ankle biceps forerm
## age
## weight
## height
## neck
## chest
## abdom
## hip
## thigh
## knee
## ankle
## biceps 0.036
## forearm 0.021 -0.295
## wrist -0.240 -0.039 -0.177
##
## Standardized residuals:
## Min Q1 Med Q3 Max
## -2.57362104 -0.64497638 -0.02432769 0.72670538 2.33871677
##
## Residual standard error: 3.987973
## Degrees of freedom: 252 total; 238 residualintervals(glmod,which="var-cov")## Approximate 95% confidence intervals
##
## Residual standard error:
## lower est. upper
## 3.659628 3.987973 4.381550##Huber
require(MASS)## Loading required package: MASSrlmod <- rlm(brozek ~ age + weight + height + neck + chest + abdom + hip + thigh + knee + ankle + biceps + forearm + wrist, data=fat)
summary(rlmod)##
## Call: rlm(formula = brozek ~ age + weight + height + neck + chest +
## abdom + hip + thigh + knee + ankle + biceps + forearm + wrist,
## data = fat)
## Residuals:
## Min 1Q Median 3Q Max
## -10.3964 -2.7352 -0.1171 2.8008 9.4446
##
## Coefficients:
## Value Std. Error t value
## (Intercept) -11.3460 17.1216 -0.6627
## age 0.0650 0.0319 2.0368
## weight -0.0643 0.0528 -1.2163
## height -0.0625 0.0948 -0.6595
## neck -0.4553 0.2294 -1.9846
## chest -0.0256 0.0978 -0.2614
## abdom 0.8778 0.0853 10.2891
## hip -0.2142 0.1440 -1.4872
## thigh 0.2632 0.1425 1.8473
## knee -0.1076 0.2388 -0.4505
## ankle 0.1815 0.2186 0.8306
## biceps 0.1367 0.1689 0.8091
## forearm 0.4152 0.1965 2.1126
## wrist -1.5739 0.5279 -2.9812
##
## Residual standard error: 4.073 on 238 degrees of freedom(a)Comment on any substantial differences between the least squares fit and the robust methods. The values of the intercepts are different and most of the cases its lower. The significanct variables forund in the both models are same. In the huber model, age, appears to be significant, while it doesn’t in the basic lm model. The standard error is not greatly affected.
(b)Identify which two cases have the lowest weights in the Huber fit. What is surprising about these two points?
wts <- rlmod$w
names(wts) <- row.names(fat)
head(sort(wts),2)## 224 207
## 0.5269652 0.5800712colMeans(fat)## brozek siri density age weight height adipos
## 18.938492 19.150794 1.055574 44.884921 178.924405 70.148810 25.436905
## free neck chest abdom hip thigh knee
## 143.713889 37.992063 100.824206 92.555952 99.904762 59.405952 38.590476
## ankle biceps forearm wrist
## 23.102381 32.273413 28.663889 18.229762Cases 224 and 207 have lowest weights. The values of body fat are closer to the lowest and highest values of the data. Values of other variables for this row is close to the men values of the other variables of the data.
(c)Plot weight (of the man) against height. Identify the two outlying cases. Are these the same as those identified in the previous question?
plot(weight ~ height, fat)
These outliers are not the same as found in question b. Points 39, 42 were identified as outliers. One has the lowest height and the other the highest weight and are substantially separate from the rest of the points.
test <- fat[-c(42),]
test1<- test[-c(39),]
glmod1 <- gls(brozek ~ age + weight + height + neck + chest + abdom + hip + thigh + knee + ankle + biceps + forearm + wrist, data=na.omit(test1))
summary(glmod)## Generalized least squares fit by REML
## Model: brozek ~ age + weight + height + neck + chest + abdom + hip + thigh + knee + ankle + biceps + forearm + wrist
## Data: na.omit(fat)
## AIC BIC logLik
## 1464.835 1516.919 -717.4177
##
## Coefficients:
## Value Std.Error t-value p-value
## (Intercept) -15.292549 16.069921 -0.951626 0.3423
## age 0.056786 0.029965 1.895105 0.0593
## weight -0.080310 0.049581 -1.619787 0.1066
## height -0.064600 0.088930 -0.726414 0.4683
## neck -0.437541 0.215334 -2.031920 0.0433
## chest -0.023603 0.091839 -0.257007 0.7974
## abdom 0.885429 0.080077 11.057242 0.0000
## hip -0.198419 0.135156 -1.468068 0.1434
## thigh 0.231895 0.133718 1.734211 0.0842
## knee -0.011677 0.224143 -0.052095 0.9585
## ankle 0.163536 0.205143 0.797178 0.4261
## biceps 0.152799 0.158513 0.963954 0.3360
## forearm 0.430489 0.184452 2.333874 0.0204
## wrist -1.476537 0.495519 -2.979779 0.0032
##
## Correlation:
## (Intr) age weight height neck chest abdom hip thigh knee
## age 0.041
## weight 0.898 0.207
## height -0.587 0.133 -0.454
## neck -0.299 -0.091 -0.292 0.025
## chest -0.429 -0.035 -0.408 0.214 -0.032
## abdom 0.001 -0.451 -0.202 0.110 -0.026 -0.469
## hip -0.604 0.080 -0.541 0.252 0.174 0.181 -0.247
## thigh -0.075 0.366 -0.056 0.203 -0.106 0.157 -0.137 -0.324
## knee -0.232 -0.251 -0.255 -0.086 0.150 0.088 0.069 -0.043 -0.278
## ankle -0.179 0.104 -0.183 0.062 0.104 0.025 0.093 0.053 0.010 -0.168
## biceps -0.067 -0.072 -0.177 0.049 -0.082 -0.080 0.119 0.079 -0.268 0.076
## forearm -0.082 0.163 -0.011 0.021 -0.114 -0.118 0.053 0.095 -0.010 -0.090
## wrist -0.181 -0.339 -0.143 -0.099 -0.290 0.017 0.149 -0.020 0.064 -0.075
## ankle biceps forerm
## age
## weight
## height
## neck
## chest
## abdom
## hip
## thigh
## knee
## ankle
## biceps 0.036
## forearm 0.021 -0.295
## wrist -0.240 -0.039 -0.177
##
## Standardized residuals:
## Min Q1 Med Q3 Max
## -2.57362104 -0.64497638 -0.02432769 0.72670538 2.33871677
##
## Residual standard error: 3.987973
## Degrees of freedom: 252 total; 238 residualintervals(glmod1,which="var-cov")## Approximate 95% confidence intervals
##
## Residual standard error:
## lower est. upper
## 3.616618 3.942473 4.333366due to removal of potential outliers, there is not much effect on the residual error of the model.