Last updated: 2019-10-31

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Knit directory: STA_463_563_Fall2019/

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Rmd b46eac3 dleelab 2019-10-31 size changed
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Rmd 41acc1b dleelab 2019-10-18 new 
data0120=read.table("http://www.stat.ufl.edu/~rrandles/sta4210/Rclassnotes/data/textdatasets/KutnerData/Chapter%20%201%20Data%20Sets/CH01PR20.txt")
colnames(data0120)=c("minutes","copiers_number")
attach(data0120)
fit=lm(minutes~copiers_number)

2. We Seek a Descriptive measures of How Well the Fitted Model Explains the Observed Response.

2.a. Compute SST, SSR and \(R^2\) using R.

\[SST=\sum_{i=1}^{n}(Y_i-\bar{Y})^2\]

SST=sum((data0120$minutes-mean(data0120$minutes))^2)
SST
[1] 80376.8

\[SSR=\sum_{i=1}^{n}(\hat{Y_i}-\bar{Y})^2\]

SSR=sum((fit$fitted.values-mean(data0120$minutes))^2)
SSR
[1] 76960.42

\[R^2=\frac{SSR}{SST}\]

Rsq=SSR/SST
Rsq
[1] 0.9574955
summary(fit)#compare with the summary(fit)

Call:
lm(formula = minutes ~ copiers_number)

Residuals:
     Min       1Q   Median       3Q      Max 
-22.7723  -3.7371   0.3334   6.3334  15.4039 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)     -0.5802     2.8039  -0.207    0.837    
copiers_number  15.0352     0.4831  31.123   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 8.914 on 43 degrees of freedom
Multiple R-squared:  0.9575,    Adjusted R-squared:  0.9565 
F-statistic: 968.7 on 1 and 43 DF,  p-value: < 2.2e-16

Interpretation: 95.75% of the total variation in minutes can be explained by a linear relationship with copiers_number.

2.b. Coefficient of Correlation, \(r=\pm \sqrt{R^2}\).

  • The sign is determined by the slope of the regression equation.
  • \(r\) does not have any particular interpretation in regression. It is simply a measure of the fit of the linear regression model.
  • \[-1\leq r \leq 1\]
  • Relationship between \(b_1\) and \(r\): \(b_1=r\sqrt{\frac{S_{YY}}{S_{XX}}}\).
r=sqrt(Rsq)
r
[1] 0.978517
cor(minutes,copiers_number)
[1] 0.978517

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