Lecture Note 16 Code

Prep data
Fit models
ANOVA
Y~X1+X2 vs. Y~X1+X2+X3
Y~X1 vs. Y~X1+X2+X3
- Option 1:calculate directly use formula for F star
- Option 2
Compute VIF
- Drop X1

Last updated: 2019-11-25

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Knit directory: STA_463_563_Fall2019/

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Prep data

bodyfat=read.table("http://users.stat.ufl.edu/~rrandles/sta4210/Rclassnotes/data/textdatasets/KutnerData/Chapter%20%207%20Data%20Sets/CH07TA01.txt")
colnames(bodyfat)=c("X1","X2","X3","Y")
head(bodyfat)

    X1   X2   X3    Y
1 19.5 43.1 29.1 11.9
2 24.7 49.8 28.2 22.8
3 30.7 51.9 37.0 18.7
4 29.8 54.3 31.1 20.1
5 19.1 42.2 30.9 12.9
6 25.6 53.9 23.7 21.7

Fit models

fit1=lm(Y~X1,data=bodyfat)
fit2=lm(Y~X1+X2,data=bodyfat)
fit3=lm(Y~X1+X2+X3,data=bodyfat)

ANOVA

anova(fit1)

Analysis of Variance Table

Response: Y
          Df Sum Sq Mean Sq F value    Pr(>F)    
X1         1 352.27  352.27  44.305 3.024e-06 ***
Residuals 18 143.12    7.95                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

anova(fit2)

Analysis of Variance Table

Response: Y
          Df Sum Sq Mean Sq F value    Pr(>F)    
X1         1 352.27  352.27 54.4661 1.075e-06 ***
X2         1  33.17   33.17  5.1284    0.0369 *  
Residuals 17 109.95    6.47                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

anova(fit3)

Analysis of Variance Table

Response: Y
          Df Sum Sq Mean Sq F value    Pr(>F)    
X1         1 352.27  352.27 57.2768 1.131e-06 ***
X2         1  33.17   33.17  5.3931   0.03373 *  
X3         1  11.55   11.55  1.8773   0.18956    
Residuals 16  98.40    6.15                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Y~X1+X2 vs. Y~X1+X2+X3

Option 1, directly use SSR(X3|X1,X2)=SSE(X1,X2)-SSE(X1,X2,X3)

n=nrow(bodyfat)
p=4
q=3
#SSE(X1,X2)
SSE_X12=anova(fit2)$"Sum Sq"[3]
#SSE(X1,X2,X3)
SSE_X123=anova(fit3)$"Sum Sq"[4]
Fstar=(SSE_X12-SSE_X123)/(p-q)/((SSE_X123)/(n-p))
Fstar

[1] 1.877289

1-pf(Fstar,(p-q),(n-p))

[1] 0.1895628

Option 2,read anova table directly.

anova(fit3)

Analysis of Variance Table

Response: Y
          Df Sum Sq Mean Sq F value    Pr(>F)    
X1         1 352.27  352.27 57.2768 1.131e-06 ***
X2         1  33.17   33.17  5.3931   0.03373 *  
X3         1  11.55   11.55  1.8773   0.18956    
Residuals 16  98.40    6.15                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Option 3, use anova(fit1, fit2) to conduct the hypothesis test.

#fit1: reduced model, fit2: full model
anova(fit2,fit3)

Analysis of Variance Table

Model 1: Y ~ X1 + X2
Model 2: Y ~ X1 + X2 + X3
  Res.Df     RSS Df Sum of Sq      F Pr(>F)
1     17 109.951                           
2     16  98.405  1    11.546 1.8773 0.1896

Y~X1 vs. Y~X1+X2+X3

Option 1:calculate directly use formula for F star

SSE_X1=anova(fit1)$"Sum Sq"[2]
Fstar2=(SSE_X1-SSE_X123)/(4-2)/(SSE_X123/(n-4))
Fstar2

[1] 3.63517

1-pf(Fstar2,2,(n-4))

[1] 0.04995028

Option 2

anova(fit1,fit3)

Analysis of Variance Table

Model 1: Y ~ X1
Model 2: Y ~ X1 + X2 + X3
  Res.Df     RSS Df Sum of Sq      F  Pr(>F)  
1     18 143.120                              
2     16  98.405  2    44.715 3.6352 0.04995 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Compute VIF

library(fmsb)
library(car)

Loading required package: carData

vif(fit3)

      X1       X2       X3 
708.8429 564.3434 104.6060

# Verify vif function result
fitvif1=lm(X1~X2+X3,data=bodyfat)
1/(1-summary(fitvif1)$r.squared)

[1] 708.8429

# cor
cor(bodyfat)

          X1        X2        X3         Y
X1 1.0000000 0.9238425 0.4577772 0.8432654
X2 0.9238425 1.0000000 0.0846675 0.8780896
X3 0.4577772 0.0846675 1.0000000 0.1424440
Y  0.8432654 0.8780896 0.1424440 1.0000000

Drop X1

summary(fit3)


Call:
lm(formula = Y ~ X1 + X2 + X3, data = bodyfat)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.7263 -1.6111  0.3923  1.4656  4.1277 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  117.085     99.782   1.173    0.258
X1             4.334      3.016   1.437    0.170
X2            -2.857      2.582  -1.106    0.285
X3            -2.186      1.595  -1.370    0.190

Residual standard error: 2.48 on 16 degrees of freedom
Multiple R-squared:  0.8014,    Adjusted R-squared:  0.7641 
F-statistic: 21.52 on 3 and 16 DF,  p-value: 7.343e-06

fit4=lm(Y~X2+X3,data=bodyfat)
summary(fit4)


Call:
lm(formula = Y ~ X2 + X3, data = bodyfat)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.0777 -1.8296  0.1893  1.3545  4.1275 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -25.99695    6.99732  -3.715  0.00172 ** 
X2            0.85088    0.11245   7.567 7.72e-07 ***
X3            0.09603    0.16139   0.595  0.55968    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.557 on 17 degrees of freedom
Multiple R-squared:  0.7757,    Adjusted R-squared:  0.7493 
F-statistic:  29.4 on 2 and 17 DF,  p-value: 3.033e-06

vif(fit4)

     X2      X3 
1.00722 1.00722

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     

other attached packages:
[1] car_3.0-4     carData_3.0-2 fmsb_0.6.3   

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.2        knitr_1.24        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.4.0   hms_0.5.1        
 [7] rlang_0.4.0       stringr_1.4.0     tools_3.6.1      
[10] data.table_1.12.2 xfun_0.9          rio_0.5.16       
[13] git2r_0.26.1      htmltools_0.3.6   abind_1.4-5      
[16] readxl_1.3.1      yaml_2.2.0        rprojroot_1.3-2  
[19] digest_0.6.20     tibble_2.1.3      crayon_1.3.4     
[22] zip_2.0.4         vctrs_0.2.0       fs_1.3.1         
[25] zeallot_0.1.0     curl_4.0          glue_1.3.1       
[28] evaluate_0.14     haven_2.1.1       rmarkdown_1.15   
[31] openxlsx_4.1.3    stringi_1.4.3     cellranger_1.1.0 
[34] pillar_1.4.2      compiler_3.6.1    forcats_0.4.0    
[37] backports_1.1.4   foreign_0.8-71    pkgconfig_2.0.2