(a) Eight experiments were carried out a various condition of Saturation (X1) and Transisomers (X2). The values of the response Y are listed below together with the corresponding levels of X1 and X2.
|
yi
|
Saturation xi1
|
Transisomers xi2
|
|
66.0
|
38
|
47.5
|
|
43.0
|
41
|
21.3
|
|
36.0
|
34
|
36.5
|
|
23.0
|
35
|
18.0
|
|
22.0
|
31
|
29.5
|
|
14.0
|
34
|
14.2
|
|
12.0
|
29
|
21.0
|
|
7.6
|
32
|
10.0
|
Some results from fitting a linear model to this data set are given below.
|
Estimates:
|
(Intercept)
-94.552
|
X1
2.802
|
X2
1.073
|
|
Var-cov:
|
(Intercept)
|
X1
|
X2
|
|
(Intercept)
|
99.270
|
-2.910
|
0.056
|
|
X1
|
-2.910
|
0.091
|
-0.008
|
|
X2
|
0.56
|
-0.008
|
0.009
|
(i) Using a significance level of 5%, test whether the two regressors Saturation and Transisomers have the same effect on the response.
(ii) Find the 95% confidence interval for the expected value of Y corresponding to X1 = 30 and X2 = 15.
(b) Suppose that a regression model with two regressors X1 and X2 which have been centred is being fitted. Show that
where ρ is the correlation coefficient between X1 and X2 and SX_iX_j is the corrected sum of products (or squares if i = j) of Xi and Xj.
(c) Use the result in part (d) to find the variance inflation factors of the least squares estimates of the regression coefficients, b^1 and b^2, and show that the standard errors of b^1 and b^2 tend to infinity as ρ tends to -1 or 1. What does this indicate about regression analysis if there is collinearity?