[R Studio][13.2.4][DS13.2.4] 문제풀이

 

 

 

 < 문제 출처 >

이공학도를 위한 확률과 통계 3판 한글판

 

 

 

 

ds13.2.4-vo2max-aerobic-fitness.txt

0.00MB

 

 

 

 

 

 

 

 

1. 데이터 불러오고 심장박동, 체지방률, 몸무게, 나이 모두 포함한 모델로 lm3 생성 및 확인

 

> raw_datas <- read.table("ds13.2.4-vo2max-aerobic-fitness.txt", header=T)
> raw_datas
   VO2_max Heart_Rate_at_Rest Age Body_Fat Weight
1       23                 62  59       26  182.0
2       45                 59  47       18  175.0
3       29                 82  44       22  200.0
4       55                 61  32       10  168.5
5       48                 60  45       19  193.0
6       42                 58  61       22  170.0
7       32                 76  71       28  193.0
8       33                 70  32       23  218.0
9       34                 68  28       27  228.0
10      52                 76  36       10  128.0
11      40                 67  36       18  167.0
12      35                 66  51       29  194.0
13      45                 50  31       29  219.0
14      47                 57  44       13  215.0
15      26                 61  73       28  246.0
16      42                 51  47       19  171.0
17      35                 60  40       25  212.0
18      41                 63  43       16  167.0
19      29                 66  68       22  162.0
20      38                 57  40       28  239.0
> lm3 <- lm(VO2_max~., data=raw_datas)
> summary(lm3)

Call:
lm(formula = VO2_max ~ ., data = raw_datas)

Residuals:
     Min       1Q   Median       3Q      Max 
-10.3426  -3.3262  -0.5895   4.1191   8.2448 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)        95.05546   14.64884   6.489 1.02e-05 ***
Heart_Rate_at_Rest -0.36228    0.14997  -2.416   0.0289 *  
Age                -0.21471    0.09798  -2.191   0.0446 *  
Body_Fat           -0.77537    0.28823  -2.690   0.0168 *  
Weight             -0.03530    0.05697  -0.620   0.5449    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.167 on 15 degrees of freedom
Multiple R-squared:  0.7197,	Adjusted R-squared:  0.645 
F-statistic:  9.63 on 4 and 15 DF,  p-value: 0.0004602

 

 

 

 

 

2. 나이만 포함한 모델 lm3_age 확인

 

> lm3_age <- lm(VO2_max~Age, data=raw_datas)
> summary(lm3_age)

Call:
lm(formula = VO2_max ~ Age, data = raw_datas)

Residuals:
     Min       1Q   Median       3Q      Max 
-11.2953  -4.1037  -0.3734   6.8993  11.5875 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  54.2181     6.1842   8.767 6.49e-08 ***
Age          -0.3377     0.1282  -2.634   0.0168 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.57 on 18 degrees of freedom
Multiple R-squared:  0.2782,	Adjusted R-squared:  0.2381 
F-statistic: 6.939 on 1 and 18 DF,  p-value: 0.01684

lm3의 residual standard error < lm3_age의 residual standard error

이기때문에 모두 포함된 모델을 선택하는 것이 현명합니다.

 

 

 

 

 

3. 최종 모델 도출

 

 lm3의 결과 summary에서 Weight의 경우 p-value가 0.5449로 상당히 크기 때문에 유의하지 않으므로 해당 데이터는 모델에서 제외시킵니다.

> lm3_final <- lm(VO2_max ~ Heart_Rate_at_Rest + Age + Body_Fat, data=raw_datas)
> summary(lm3_final)

Call:
lm(formula = VO2_max ~ Heart_Rate_at_Rest + Age + Body_Fat, data = raw_datas)

Residuals:
   Min     1Q Median     3Q    Max 
-9.645 -3.124 -1.861  4.796  8.296 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)        88.82036   10.43735   8.510 2.47e-07 ***
Heart_Rate_at_Rest -0.34282    0.14379  -2.384 0.029841 *  
Age                -0.19496    0.09084  -2.146 0.047553 *  
Body_Fat           -0.90070    0.20132  -4.474 0.000384 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.067 on 16 degrees of freedom
Multiple R-squared:  0.7126,	Adjusted R-squared:  0.6587 
F-statistic: 13.22 on 3 and 16 DF,  p-value: 0.0001342

 

 - 추정값

βˆ 0 = 88.82036

βˆ 1 = -0.34282

βˆ 2 = -0.19496

βˆ 3 = -0.90070

 

이므로

 

Y = 88.82036 - 0.34282 X1 - 0.19496 X2 - 0.90070 X3

 

입니다.