Lecture Note 17 Code
Donghyung Lee
2019-11-25
Last updated: 2019-11-25
Checks: 7 0
Knit directory: STA_463_563_Fall2019/
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Prep data
bodyfat=read.table("https://raw.githubusercontent.com/dleelab/STA463_563_Fall2019/master/data/bodyfat.txt")
bodyfat1=bodyfat[-c(39,42,86,31,182),]Fit model
fit=lm(bodyfat.pct~., data=bodyfat1)
summary(fit)
Call:
lm(formula = bodyfat.pct ~ ., data = bodyfat1)
Residuals:
Min 1Q Median 3Q Max
-10.6818 -3.0914 -0.2067 2.9663 9.7495
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.713774 24.076239 0.196 0.84495
age 0.061458 0.032848 1.871 0.06260 .
weight -0.010653 0.068670 -0.155 0.87685
height -0.244939 0.193639 -1.265 0.20716
neck -0.377731 0.237760 -1.589 0.11348
chest -0.130036 0.109566 -1.187 0.23651
abdomen 0.906703 0.091554 9.903 < 2e-16 ***
hip -0.167168 0.146680 -1.140 0.25559
thigh 0.163305 0.148672 1.098 0.27315
knee -0.008593 0.250591 -0.034 0.97268
ankle -0.025386 0.361079 -0.070 0.94401
biceps 0.169556 0.172729 0.982 0.32730
forearm 0.263673 0.208082 1.267 0.20636
wrist -1.685174 0.557689 -3.022 0.00279 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.249 on 233 degrees of freedom
Multiple R-squared: 0.7474, Adjusted R-squared: 0.7333
F-statistic: 53.03 on 13 and 233 DF, p-value: < 2.2e-16Backward selection
AIC backward
AICb=step(fit,direction="backward")Start: AIC=728.3
bodyfat.pct ~ age + weight + height + neck + chest + abdomen +
hip + thigh + knee + ankle + biceps + forearm + wrist
Df Sum of Sq RSS AIC
- knee 1 0.02 4207.5 726.30
- ankle 1 0.09 4207.5 726.31
- weight 1 0.43 4207.9 726.33
- biceps 1 17.40 4224.8 727.32
- thigh 1 21.79 4229.2 727.58
- hip 1 23.45 4230.9 727.67
- chest 1 25.44 4232.9 727.79
- height 1 28.89 4236.3 727.99
- forearm 1 29.00 4236.4 728.00
<none> 4207.4 728.30
- neck 1 45.58 4253.0 728.96
- age 1 63.21 4270.7 729.98
- wrist 1 164.88 4372.3 735.79
- abdomen 1 1771.08 5978.5 813.08
Step: AIC=726.3
bodyfat.pct ~ age + weight + height + neck + chest + abdomen +
hip + thigh + ankle + biceps + forearm + wrist
Df Sum of Sq RSS AIC
- ankle 1 0.12 4207.6 724.31
- weight 1 0.46 4207.9 724.33
- biceps 1 17.42 4224.9 725.32
- thigh 1 22.90 4230.4 725.64
- hip 1 23.78 4231.2 725.69
- chest 1 25.42 4232.9 725.79
- forearm 1 28.97 4236.4 726.00
- height 1 30.22 4237.7 726.07
<none> 4207.5 726.30
- neck 1 45.65 4253.1 726.97
- age 1 67.74 4275.2 728.25
- wrist 1 164.89 4372.4 733.80
- abdomen 1 1771.25 5978.7 811.08
Step: AIC=724.31
bodyfat.pct ~ age + weight + height + neck + chest + abdomen +
hip + thigh + biceps + forearm + wrist
Df Sum of Sq RSS AIC
- weight 1 0.64 4208.2 722.35
- biceps 1 18.42 4226.0 723.39
- thigh 1 22.93 4230.5 723.65
- hip 1 23.69 4231.3 723.69
- chest 1 25.29 4232.9 723.79
- forearm 1 28.85 4236.4 724.00
- height 1 30.10 4237.7 724.07
<none> 4207.6 724.31
- neck 1 46.66 4254.3 725.03
- age 1 69.56 4277.1 726.36
- wrist 1 198.00 4405.6 733.67
- abdomen 1 1800.42 6008.0 810.29
Step: AIC=722.35
bodyfat.pct ~ age + height + neck + chest + abdomen + hip + thigh +
biceps + forearm + wrist
Df Sum of Sq RSS AIC
- biceps 1 17.78 4226.0 721.39
- thigh 1 22.50 4230.7 721.66
- forearm 1 28.22 4236.4 722.00
- hip 1 33.28 4241.5 722.29
<none> 4208.2 722.35
- chest 1 43.63 4251.9 722.89
- neck 1 51.82 4260.1 723.37
- age 1 75.45 4283.7 724.74
- height 1 82.62 4290.9 725.15
- wrist 1 217.85 4426.1 732.81
- abdomen 1 2035.08 6243.3 817.78
Step: AIC=721.39
bodyfat.pct ~ age + height + neck + chest + abdomen + hip + thigh +
forearm + wrist
Df Sum of Sq RSS AIC
- hip 1 34.08 4260.1 721.37
<none> 4226.0 721.39
- chest 1 35.29 4261.3 721.44
- thigh 1 39.74 4265.8 721.70
- neck 1 45.55 4271.6 722.04
- forearm 1 48.88 4274.9 722.23
- age 1 77.67 4303.7 723.89
- height 1 81.62 4307.6 724.11
- wrist 1 210.56 4436.6 731.40
- abdomen 1 2017.65 6243.7 815.79
Step: AIC=721.37
bodyfat.pct ~ age + height + neck + chest + abdomen + thigh +
forearm + wrist
Df Sum of Sq RSS AIC
- thigh 1 13.92 4274.0 720.18
<none> 4260.1 721.37
- neck 1 36.89 4297.0 721.50
- chest 1 44.36 4304.4 721.93
- forearm 1 48.08 4308.2 722.14
- age 1 100.32 4360.4 725.12
- height 1 125.61 4385.7 726.55
- wrist 1 242.30 4502.4 733.03
- abdomen 1 2260.98 6521.1 824.53
Step: AIC=720.18
bodyfat.pct ~ age + height + neck + chest + abdomen + forearm +
wrist
Df Sum of Sq RSS AIC
- neck 1 31.8 4305.8 720.01
<none> 4274.0 720.18
- chest 1 47.8 4321.8 720.93
- forearm 1 55.8 4329.9 721.38
- age 1 91.4 4365.4 723.40
- height 1 121.4 4395.4 725.10
- wrist 1 231.8 4505.8 731.22
- abdomen 1 3214.8 7488.8 856.71
Step: AIC=720.01
bodyfat.pct ~ age + height + chest + abdomen + forearm + wrist
Df Sum of Sq RSS AIC
<none> 4305.8 720.01
- forearm 1 38.9 4344.6 720.23
- chest 1 68.1 4373.9 721.88
- age 1 93.9 4399.6 723.33
- height 1 128.4 4434.2 725.27
- wrist 1 340.8 4646.6 736.82
- abdomen 1 3185.5 7491.2 854.79summary(AICb)
Call:
lm(formula = bodyfat.pct ~ age + height + chest + abdomen + forearm +
wrist, data = bodyfat1)
Residuals:
Min 1Q Median 3Q Max
-10.445 -3.103 -0.361 3.033 8.917
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.59694 8.21002 0.560 0.57606
age 0.05703 0.02494 2.287 0.02305 *
height -0.32321 0.12079 -2.676 0.00797 **
chest -0.16774 0.08609 -1.948 0.05253 .
abdomen 0.86277 0.06475 13.325 < 2e-16 ***
forearm 0.27422 0.18631 1.472 0.14239
wrist -1.97588 0.45335 -4.358 1.94e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.236 on 240 degrees of freedom
Multiple R-squared: 0.7415, Adjusted R-squared: 0.735
F-statistic: 114.7 on 6 and 240 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(AICb,add.smooth=FALSE)
BIC backward Sometimes BIC loves a simple model (too much)
n=nrow(bodyfat1)
BICb=step(fit,direction="backward",k=log(n))Start: AIC=777.43
bodyfat.pct ~ age + weight + height + neck + chest + abdomen +
hip + thigh + knee + ankle + biceps + forearm + wrist
Df Sum of Sq RSS AIC
- knee 1 0.02 4207.5 771.92
- ankle 1 0.09 4207.5 771.93
- weight 1 0.43 4207.9 771.95
- biceps 1 17.40 4224.8 772.94
- thigh 1 21.79 4229.2 773.20
- hip 1 23.45 4230.9 773.30
- chest 1 25.44 4232.9 773.41
- height 1 28.89 4236.3 773.61
- forearm 1 29.00 4236.4 773.62
- neck 1 45.58 4253.0 774.58
- age 1 63.21 4270.7 775.61
<none> 4207.4 777.43
- wrist 1 164.88 4372.3 781.42
- abdomen 1 1771.08 5978.5 858.70
Step: AIC=771.92
bodyfat.pct ~ age + weight + height + neck + chest + abdomen +
hip + thigh + ankle + biceps + forearm + wrist
Df Sum of Sq RSS AIC
- ankle 1 0.12 4207.6 766.42
- weight 1 0.46 4207.9 766.44
- biceps 1 17.42 4224.9 767.43
- thigh 1 22.90 4230.4 767.75
- hip 1 23.78 4231.2 767.81
- chest 1 25.42 4232.9 767.90
- forearm 1 28.97 4236.4 768.11
- height 1 30.22 4237.7 768.18
- neck 1 45.65 4253.1 769.08
- age 1 67.74 4275.2 770.36
<none> 4207.5 771.92
- wrist 1 164.89 4372.4 775.91
- abdomen 1 1771.25 5978.7 853.20
Step: AIC=766.42
bodyfat.pct ~ age + weight + height + neck + chest + abdomen +
hip + thigh + biceps + forearm + wrist
Df Sum of Sq RSS AIC
- weight 1 0.64 4208.2 760.95
- biceps 1 18.42 4226.0 761.99
- thigh 1 22.93 4230.5 762.25
- hip 1 23.69 4231.3 762.30
- chest 1 25.29 4232.9 762.39
- forearm 1 28.85 4236.4 762.60
- height 1 30.10 4237.7 762.67
- neck 1 46.66 4254.3 763.64
- age 1 69.56 4277.1 764.96
<none> 4207.6 766.42
- wrist 1 198.00 4405.6 772.27
- abdomen 1 1800.42 6008.0 848.89
Step: AIC=760.95
bodyfat.pct ~ age + height + neck + chest + abdomen + hip + thigh +
biceps + forearm + wrist
Df Sum of Sq RSS AIC
- biceps 1 17.78 4226.0 756.48
- thigh 1 22.50 4230.7 756.76
- forearm 1 28.22 4236.4 757.09
- hip 1 33.28 4241.5 757.39
- chest 1 43.63 4251.9 757.99
- neck 1 51.82 4260.1 758.46
- age 1 75.45 4283.7 759.83
- height 1 82.62 4290.9 760.24
<none> 4208.2 760.95
- wrist 1 217.85 4426.1 767.91
- abdomen 1 2035.08 6243.3 852.87
Step: AIC=756.48
bodyfat.pct ~ age + height + neck + chest + abdomen + hip + thigh +
forearm + wrist
Df Sum of Sq RSS AIC
- hip 1 34.08 4260.1 752.96
- chest 1 35.29 4261.3 753.03
- thigh 1 39.74 4265.8 753.28
- neck 1 45.55 4271.6 753.62
- forearm 1 48.88 4274.9 753.81
- age 1 77.67 4303.7 755.47
- height 1 81.62 4307.6 755.70
<none> 4226.0 756.48
- wrist 1 210.56 4436.6 762.98
- abdomen 1 2017.65 6243.7 847.38
Step: AIC=752.96
bodyfat.pct ~ age + height + neck + chest + abdomen + thigh +
forearm + wrist
Df Sum of Sq RSS AIC
- thigh 1 13.92 4274.0 748.25
- neck 1 36.89 4297.0 749.58
- chest 1 44.36 4304.4 750.01
- forearm 1 48.08 4308.2 750.22
<none> 4260.1 752.96
- age 1 100.32 4360.4 753.20
- height 1 125.61 4385.7 754.62
- wrist 1 242.30 4502.4 761.11
- abdomen 1 2260.98 6521.1 852.61
Step: AIC=748.25
bodyfat.pct ~ age + height + neck + chest + abdomen + forearm +
wrist
Df Sum of Sq RSS AIC
- neck 1 31.8 4305.8 744.57
- chest 1 47.8 4321.8 745.49
- forearm 1 55.8 4329.9 745.95
- age 1 91.4 4365.4 747.97
<none> 4274.0 748.25
- height 1 121.4 4395.4 749.66
- wrist 1 231.8 4505.8 755.79
- abdomen 1 3214.8 7488.8 881.27
Step: AIC=744.57
bodyfat.pct ~ age + height + chest + abdomen + forearm + wrist
Df Sum of Sq RSS AIC
- forearm 1 38.9 4344.6 741.28
- chest 1 68.1 4373.9 742.94
- age 1 93.9 4399.6 744.39
<none> 4305.8 744.57
- height 1 128.4 4434.2 746.32
- wrist 1 340.8 4646.6 757.88
- abdomen 1 3185.5 7491.2 875.85
Step: AIC=741.28
bodyfat.pct ~ age + height + chest + abdomen + wrist
Df Sum of Sq RSS AIC
- chest 1 50.1 4394.7 738.60
- age 1 68.3 4412.9 739.62
<none> 4344.6 741.28
- height 1 124.2 4468.8 742.73
- wrist 1 302.1 4646.7 752.38
- abdomen 1 3177.1 7521.8 871.34
Step: AIC=738.6
bodyfat.pct ~ age + height + abdomen + wrist
Df Sum of Sq RSS AIC
- age 1 85.4 4480.2 737.85
<none> 4394.7 738.60
- height 1 120.8 4515.5 739.79
- wrist 1 408.6 4803.3 755.05
- abdomen 1 9472.4 13867.1 1016.93
Step: AIC=737.85
bodyfat.pct ~ height + abdomen + wrist
Df Sum of Sq RSS AIC
<none> 4480.2 737.85
- height 1 233.7 4713.9 744.90
- wrist 1 346.1 4826.2 750.72
- abdomen 1 9867.5 14347.6 1019.83summary(BICb)
Call:
lm(formula = bodyfat.pct ~ height + abdomen + wrist, data = bodyfat1)
Residuals:
Min 1Q Median 3Q Max
-9.8873 -3.1289 -0.2657 3.1623 9.9508
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.45048 7.89846 0.943 0.346473
height -0.40707 0.11433 -3.561 0.000445 ***
abdomen 0.77834 0.03364 23.134 < 2e-16 ***
wrist -1.73617 0.40074 -4.332 2.16e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.294 on 243 degrees of freedom
Multiple R-squared: 0.731, Adjusted R-squared: 0.7277
F-statistic: 220.2 on 3 and 243 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(BICb,add.smooth=FALSE)
Forward selection
null=lm(bodyfat.pct~1, data=bodyfat1)
full=lm(bodyfat.pct~.,data=bodyfat1)AIC forward
AICf=step(null, direction="forward",scope=list(upper=full))Start: AIC=1042.17
bodyfat.pct ~ 1
Df Sum of Sq RSS AIC
+ abdomen 1 11272.3 5384.8 765.24
+ chest 1 8079.3 8577.8 880.24
+ hip 1 6595.8 10061.3 919.64
+ weight 1 6271.6 10385.5 927.48
+ thigh 1 4914.0 11743.0 957.82
+ neck 1 3928.2 12728.8 977.73
+ knee 1 3918.2 12738.9 977.93
+ biceps 1 3777.8 12879.3 980.64
+ forearm 1 2091.3 14565.8 1011.03
+ wrist 1 1806.0 14851.1 1015.82
+ age 1 1381.7 15275.4 1022.78
+ ankle 1 1377.0 15280.1 1022.85
<none> 16657.1 1042.17
+ height 1 14.7 16642.4 1043.95
Step: AIC=765.24
bodyfat.pct ~ abdomen
Df Sum of Sq RSS AIC
+ weight 1 741.54 4643.2 730.64
+ wrist 1 670.86 4713.9 734.37
+ height 1 558.54 4826.2 740.19
+ neck 1 431.71 4953.1 746.60
+ ankle 1 408.95 4975.8 747.73
+ hip 1 368.94 5015.8 749.71
+ knee 1 297.48 5087.3 753.20
+ chest 1 245.61 5139.2 755.71
+ age 1 134.15 5250.6 761.01
+ forearm 1 126.12 5258.6 761.38
+ thigh 1 122.60 5262.2 761.55
+ biceps 1 106.56 5278.2 762.30
<none> 5384.8 765.24
Step: AIC=730.64
bodyfat.pct ~ abdomen + weight
Df Sum of Sq RSS AIC
+ wrist 1 194.342 4448.9 722.08
+ thigh 1 64.555 4578.7 729.18
+ neck 1 60.599 4582.6 729.40
+ biceps 1 43.814 4599.4 730.30
<none> 4643.2 730.64
+ height 1 29.879 4613.3 731.05
+ forearm 1 15.642 4627.6 731.81
+ ankle 1 11.512 4631.7 732.03
+ chest 1 9.686 4633.5 732.13
+ age 1 3.135 4640.1 732.48
+ hip 1 0.864 4642.4 732.60
+ knee 1 0.264 4643.0 732.63
Step: AIC=722.08
bodyfat.pct ~ abdomen + weight + wrist
Df Sum of Sq RSS AIC
+ biceps 1 63.711 4385.2 720.52
+ forearm 1 45.452 4403.4 721.55
<none> 4448.9 722.08
+ height 1 32.899 4416.0 722.25
+ thigh 1 27.087 4421.8 722.57
+ age 1 25.900 4423.0 722.64
+ neck 1 8.228 4440.7 723.62
+ chest 1 5.234 4443.6 723.79
+ knee 1 4.308 4444.6 723.84
+ hip 1 2.773 4446.1 723.93
+ ankle 1 0.302 4448.6 724.07
Step: AIC=720.52
bodyfat.pct ~ abdomen + weight + wrist + biceps
Df Sum of Sq RSS AIC
<none> 4385.2 720.52
+ age 1 30.4493 4354.7 720.80
+ neck 1 19.9062 4365.3 721.40
+ forearm 1 18.0195 4367.2 721.50
+ height 1 13.2257 4371.9 721.77
+ thigh 1 10.4040 4374.8 721.93
+ chest 1 9.6830 4375.5 721.97
+ knee 1 5.8835 4379.3 722.19
+ hip 1 2.6655 4382.5 722.37
+ ankle 1 2.2862 4382.9 722.39summary(AICf)
Call:
lm(formula = bodyfat.pct ~ abdomen + weight + wrist + biceps,
data = bodyfat1)
Residuals:
Min 1Q Median 3Q Max
-9.9094 -3.0949 -0.2513 3.2278 9.2880
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -30.98611 7.01286 -4.418 1.50e-05 ***
abdomen 0.96241 0.05496 17.511 < 2e-16 ***
weight -0.11673 0.02745 -4.253 3.02e-05 ***
wrist -1.48815 0.43280 -3.438 0.000689 ***
biceps 0.28245 0.15063 1.875 0.061985 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.257 on 242 degrees of freedom
Multiple R-squared: 0.7367, Adjusted R-squared: 0.7324
F-statistic: 169.3 on 4 and 242 DF, p-value: < 2.2e-16BIC forward
BICf=step(null, direction="forward", scope=list(upper=full),k=log(n))Start: AIC=1045.68
bodyfat.pct ~ 1
Df Sum of Sq RSS AIC
+ abdomen 1 11272.3 5384.8 772.26
+ chest 1 8079.3 8577.8 887.26
+ hip 1 6595.8 10061.3 926.66
+ weight 1 6271.6 10385.5 934.50
+ thigh 1 4914.0 11743.0 964.84
+ neck 1 3928.2 12728.8 984.75
+ knee 1 3918.2 12738.9 984.95
+ biceps 1 3777.8 12879.3 987.65
+ forearm 1 2091.3 14565.8 1018.05
+ wrist 1 1806.0 14851.1 1022.84
+ age 1 1381.7 15275.4 1029.80
+ ankle 1 1377.0 15280.1 1029.87
<none> 16657.1 1045.68
+ height 1 14.7 16642.4 1050.97
Step: AIC=772.26
bodyfat.pct ~ abdomen
Df Sum of Sq RSS AIC
+ weight 1 741.54 4643.2 741.17
+ wrist 1 670.86 4713.9 744.90
+ height 1 558.54 4826.2 750.72
+ neck 1 431.71 4953.1 757.13
+ ankle 1 408.95 4975.8 758.26
+ hip 1 368.94 5015.8 760.24
+ knee 1 297.48 5087.3 763.73
+ chest 1 245.61 5139.2 766.24
+ age 1 134.15 5250.6 771.54
+ forearm 1 126.12 5258.6 771.91
+ thigh 1 122.60 5262.2 772.08
<none> 5384.8 772.26
+ biceps 1 106.56 5278.2 772.83
Step: AIC=741.17
bodyfat.pct ~ abdomen + weight
Df Sum of Sq RSS AIC
+ wrist 1 194.342 4448.9 736.12
<none> 4643.2 741.17
+ thigh 1 64.555 4578.7 743.22
+ neck 1 60.599 4582.6 743.44
+ biceps 1 43.814 4599.4 744.34
+ height 1 29.879 4613.3 745.09
+ forearm 1 15.642 4627.6 745.85
+ ankle 1 11.512 4631.7 746.07
+ chest 1 9.686 4633.5 746.16
+ age 1 3.135 4640.1 746.51
+ hip 1 0.864 4642.4 746.63
+ knee 1 0.264 4643.0 746.67
Step: AIC=736.12
bodyfat.pct ~ abdomen + weight + wrist
Df Sum of Sq RSS AIC
<none> 4448.9 736.12
+ biceps 1 63.711 4385.2 738.07
+ forearm 1 45.452 4403.4 739.09
+ height 1 32.899 4416.0 739.80
+ thigh 1 27.087 4421.8 740.12
+ age 1 25.900 4423.0 740.19
+ neck 1 8.228 4440.7 741.17
+ chest 1 5.234 4443.6 741.34
+ knee 1 4.308 4444.6 741.39
+ hip 1 2.773 4446.1 741.47
+ ankle 1 0.302 4448.6 741.61summary(BICf)
Call:
lm(formula = bodyfat.pct ~ abdomen + weight + wrist, data = bodyfat1)
Residuals:
Min 1Q Median 3Q Max
-9.7874 -3.1975 -0.4396 3.2762 9.9296
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -26.92797 6.70500 -4.016 7.89e-05 ***
abdomen 0.95312 0.05502 17.323 < 2e-16 ***
weight -0.09151 0.02405 -3.805 0.00018 ***
wrist -1.41093 0.43306 -3.258 0.00128 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.279 on 243 degrees of freedom
Multiple R-squared: 0.7329, Adjusted R-squared: 0.7296
F-statistic: 222.3 on 3 and 243 DF, p-value: < 2.2e-16Stepwise
AIC both
AICs=step(null, direction="both",scope=list(upper=full))Start: AIC=1042.17
bodyfat.pct ~ 1
Df Sum of Sq RSS AIC
+ abdomen 1 11272.3 5384.8 765.24
+ chest 1 8079.3 8577.8 880.24
+ hip 1 6595.8 10061.3 919.64
+ weight 1 6271.6 10385.5 927.48
+ thigh 1 4914.0 11743.0 957.82
+ neck 1 3928.2 12728.8 977.73
+ knee 1 3918.2 12738.9 977.93
+ biceps 1 3777.8 12879.3 980.64
+ forearm 1 2091.3 14565.8 1011.03
+ wrist 1 1806.0 14851.1 1015.82
+ age 1 1381.7 15275.4 1022.78
+ ankle 1 1377.0 15280.1 1022.85
<none> 16657.1 1042.17
+ height 1 14.7 16642.4 1043.95
Step: AIC=765.24
bodyfat.pct ~ abdomen
Df Sum of Sq RSS AIC
+ weight 1 741.5 4643.2 730.64
+ wrist 1 670.9 4713.9 734.37
+ height 1 558.5 4826.2 740.19
+ neck 1 431.7 4953.1 746.60
+ ankle 1 408.9 4975.8 747.73
+ hip 1 368.9 5015.8 749.71
+ knee 1 297.5 5087.3 753.20
+ chest 1 245.6 5139.2 755.71
+ age 1 134.1 5250.6 761.01
+ forearm 1 126.1 5258.6 761.38
+ thigh 1 122.6 5262.2 761.55
+ biceps 1 106.6 5278.2 762.30
<none> 5384.8 765.24
- abdomen 1 11272.3 16657.1 1042.17
Step: AIC=730.64
bodyfat.pct ~ abdomen + weight
Df Sum of Sq RSS AIC
+ wrist 1 194.3 4448.9 722.08
+ thigh 1 64.6 4578.7 729.18
+ neck 1 60.6 4582.6 729.40
+ biceps 1 43.8 4599.4 730.30
<none> 4643.2 730.64
+ height 1 29.9 4613.3 731.05
+ forearm 1 15.6 4627.6 731.81
+ ankle 1 11.5 4631.7 732.03
+ chest 1 9.7 4633.5 732.13
+ age 1 3.1 4640.1 732.48
+ hip 1 0.9 4642.4 732.60
+ knee 1 0.3 4643.0 732.63
- weight 1 741.5 5384.8 765.24
- abdomen 1 5742.3 10385.5 927.48
Step: AIC=722.08
bodyfat.pct ~ abdomen + weight + wrist
Df Sum of Sq RSS AIC
+ biceps 1 63.7 4385.2 720.52
+ forearm 1 45.5 4403.4 721.55
<none> 4448.9 722.08
+ height 1 32.9 4416.0 722.25
+ thigh 1 27.1 4421.8 722.57
+ age 1 25.9 4423.0 722.64
+ neck 1 8.2 4440.7 723.62
+ chest 1 5.2 4443.6 723.79
+ knee 1 4.3 4444.6 723.84
+ hip 1 2.8 4446.1 723.93
+ ankle 1 0.3 4448.6 724.07
- wrist 1 194.3 4643.2 730.64
- weight 1 265.0 4713.9 734.37
- abdomen 1 5494.1 9943.0 918.72
Step: AIC=720.52
bodyfat.pct ~ abdomen + weight + wrist + biceps
Df Sum of Sq RSS AIC
<none> 4385.2 720.52
+ age 1 30.4 4354.7 720.80
+ neck 1 19.9 4365.3 721.40
+ forearm 1 18.0 4367.2 721.50
+ height 1 13.2 4371.9 721.77
+ thigh 1 10.4 4374.8 721.93
+ chest 1 9.7 4375.5 721.97
- biceps 1 63.7 4448.9 722.08
+ knee 1 5.9 4379.3 722.19
+ hip 1 2.7 4382.5 722.37
+ ankle 1 2.3 4382.9 722.39
- wrist 1 214.2 4599.4 730.30
- weight 1 327.7 4712.9 736.32
- abdomen 1 5556.2 9941.3 920.68summary(AICs)
Call:
lm(formula = bodyfat.pct ~ abdomen + weight + wrist + biceps,
data = bodyfat1)
Residuals:
Min 1Q Median 3Q Max
-9.9094 -3.0949 -0.2513 3.2278 9.2880
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -30.98611 7.01286 -4.418 1.50e-05 ***
abdomen 0.96241 0.05496 17.511 < 2e-16 ***
weight -0.11673 0.02745 -4.253 3.02e-05 ***
wrist -1.48815 0.43280 -3.438 0.000689 ***
biceps 0.28245 0.15063 1.875 0.061985 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.257 on 242 degrees of freedom
Multiple R-squared: 0.7367, Adjusted R-squared: 0.7324
F-statistic: 169.3 on 4 and 242 DF, p-value: < 2.2e-16BIC both
BICs=step(null, direction="both", scope=list(upper=full),k=log(n))Start: AIC=1045.68
bodyfat.pct ~ 1
Df Sum of Sq RSS AIC
+ abdomen 1 11272.3 5384.8 772.26
+ chest 1 8079.3 8577.8 887.26
+ hip 1 6595.8 10061.3 926.66
+ weight 1 6271.6 10385.5 934.50
+ thigh 1 4914.0 11743.0 964.84
+ neck 1 3928.2 12728.8 984.75
+ knee 1 3918.2 12738.9 984.95
+ biceps 1 3777.8 12879.3 987.65
+ forearm 1 2091.3 14565.8 1018.05
+ wrist 1 1806.0 14851.1 1022.84
+ age 1 1381.7 15275.4 1029.80
+ ankle 1 1377.0 15280.1 1029.87
<none> 16657.1 1045.68
+ height 1 14.7 16642.4 1050.97
Step: AIC=772.26
bodyfat.pct ~ abdomen
Df Sum of Sq RSS AIC
+ weight 1 741.5 4643.2 741.17
+ wrist 1 670.9 4713.9 744.90
+ height 1 558.5 4826.2 750.72
+ neck 1 431.7 4953.1 757.13
+ ankle 1 408.9 4975.8 758.26
+ hip 1 368.9 5015.8 760.24
+ knee 1 297.5 5087.3 763.73
+ chest 1 245.6 5139.2 766.24
+ age 1 134.1 5250.6 771.54
+ forearm 1 126.1 5258.6 771.91
+ thigh 1 122.6 5262.2 772.08
<none> 5384.8 772.26
+ biceps 1 106.6 5278.2 772.83
- abdomen 1 11272.3 16657.1 1045.68
Step: AIC=741.17
bodyfat.pct ~ abdomen + weight
Df Sum of Sq RSS AIC
+ wrist 1 194.3 4448.9 736.12
<none> 4643.2 741.17
+ thigh 1 64.6 4578.7 743.22
+ neck 1 60.6 4582.6 743.44
+ biceps 1 43.8 4599.4 744.34
+ height 1 29.9 4613.3 745.09
+ forearm 1 15.6 4627.6 745.85
+ ankle 1 11.5 4631.7 746.07
+ chest 1 9.7 4633.5 746.16
+ age 1 3.1 4640.1 746.51
+ hip 1 0.9 4642.4 746.63
+ knee 1 0.3 4643.0 746.67
- weight 1 741.5 5384.8 772.26
- abdomen 1 5742.3 10385.5 934.50
Step: AIC=736.12
bodyfat.pct ~ abdomen + weight + wrist
Df Sum of Sq RSS AIC
<none> 4448.9 736.12
+ biceps 1 63.7 4385.2 738.07
+ forearm 1 45.5 4403.4 739.09
+ height 1 32.9 4416.0 739.80
+ thigh 1 27.1 4421.8 740.12
+ age 1 25.9 4423.0 740.19
- wrist 1 194.3 4643.2 741.17
+ neck 1 8.2 4440.7 741.17
+ chest 1 5.2 4443.6 741.34
+ knee 1 4.3 4444.6 741.39
+ hip 1 2.8 4446.1 741.47
+ ankle 1 0.3 4448.6 741.61
- weight 1 265.0 4713.9 744.90
- abdomen 1 5494.1 9943.0 929.25summary(BICs)
Call:
lm(formula = bodyfat.pct ~ abdomen + weight + wrist, data = bodyfat1)
Residuals:
Min 1Q Median 3Q Max
-9.7874 -3.1975 -0.4396 3.2762 9.9296
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -26.92797 6.70500 -4.016 7.89e-05 ***
abdomen 0.95312 0.05502 17.323 < 2e-16 ***
weight -0.09151 0.02405 -3.805 0.00018 ***
wrist -1.41093 0.43306 -3.258 0.00128 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.279 on 243 degrees of freedom
Multiple R-squared: 0.7329, Adjusted R-squared: 0.7296
F-statistic: 222.3 on 3 and 243 DF, p-value: < 2.2e-16
sessionInfo()R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] workflowr_1.4.0 Rcpp_1.0.2 digest_0.6.20 rprojroot_1.3-2
[5] backports_1.1.4 git2r_0.26.1 magrittr_1.5 evaluate_0.14
[9] stringi_1.4.3 fs_1.3.1 whisker_0.3-2 rmarkdown_1.15
[13] tools_3.6.1 stringr_1.4.0 glue_1.3.1 xfun_0.9
[17] yaml_2.2.0 compiler_3.6.1 htmltools_0.3.6 knitr_1.24