Question

Question 3: Consider the data of exercise 12-6 page 508. Drop column 4 (the data related to x4) and assume that y is function of x1, x2, and x3 only. a) What proportion of total variability is explained by this model? b) Plot the residuals vs. y-hat and versus each regress or. Provide detailed discussion on each plot? c) Calculate Cook's distance for the observations in the data set, are any observations influential? d) Test the hypothesis that Bl=B3=0 vs. at least one of them is not zero

Answer 1

> data=read.csv(abc.csv")
> data
y x1 x2 x3 x4
1 240 30 24 91 100
2 236 31 25 90 95
3 270 45 24 88 110
4 274 60 25 87 88
5 301 65 25 91 94
6 316 72 26 94 105
7 300 80 25 87 97
8 296 84 25 86 100
9 267 75 24 88 110
10 276 60 25 95 105
11 288 50 25 90 100
12 261 38 23 89 98
> Lin=lm(y~x1+x2+x3,data=data)
> summary(Lin)

Call:
lm(formula = y ~ x1 + x2 + x3, data = data)

Residuals:
Min 1Q Median 3Q Max
-20.4714 -9.5878 -0.2082 11.7837 15.6250

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -59.1253 182.0786 -0.325 0.7537
x1 0.9649 0.3113 3.099 0.0147 *
x2 6.1437 7.7197 0.796 0.4491
x3 1.4407 1.9510 0.738 0.4813
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 15.27 on 8 degrees of freedom
Multiple R-squared: 0.7162, Adjusted R-squared: 0.6097
F-statistic: 6.728 on 3 and 8 DF, p-value: 0.01404




a)

> #71.62% of the variation is explained by the model



b)
> p=predict(Lin,data)
> p
1 2 3 4 5 6 7 8
248.3747 254.0426 258.5255 277.7015 288.2887 305.5086 296.9988 299.4175
9 10 11 12
287.4714 289.2272 272.3750 247.0685
> res=data$y-p
> par(mfrow=(c(2,2)))
> plot(res,p,xlab="Residual",ylab="Fit", main="Residual vs Fit")
> plot(res,data$x1,xlab="Residual",ylab="X1", main="Residual vs Predictor")
> plot(res,data$x2,xlab="Residual",ylab="X2", main="Residual vs Predictor")
> plot(res,data$x3,xlab="Residual",ylab="X3", main="Residual vs Predictor")

> #We observe that except for the Residual vs X2, each of the scatter plots does not follow any specific pattern. Hence, except for X2, other predictors as well as response values are homoskedastic.



c)
> library(olsrr)
> ols_plot_resid_stand(Lin)

> #There are no influential observation in the data



d)
> library(car)
> linearHypothesis(Lin, c("x1=0", "x3=0"))
Linear hypothesis test

Hypothesis:
x1 = 0
x3 = 0

Model 1: restricted model
Model 2: y ~ x1 + x2 + x3

Res.Df RSS Df Sum of Sq F Pr(>F)
1 10 4216.7
2 8 1865.6 2 2351 5.0407 0.03832 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> #Since, Pr(>F) is less than 0.05, the Null is rejected, and B1=B3=0 does not hold good.
>






I hope this answer will help you to a great extent. Thank You :)