This exercise uses the data Y = ln(days survival) for colon cancer patients receiving supplemental ascorbate to the variables sex (X 1 ), age of patient (X 2 ), and ln(average survival of control...

This exercise uses the data Y = ln(days survival) for colon cancer patients receiving supplemental ascorbate to the variables sex (X₁), age of patient (X₂), and ln(average survival of control group) (X₃).

(a) Complete the analysis of variance for the model using all three variables plus an intercept. Compute the partial sum of squares for each independent variable using the formula

²
_j/c_jj. Demonstrate that each is the same as the sum of squares one obtains by computing Q for the general linear hypothesis that the corresponding β_j
is zero. Compute the standard error for each regression coefficient and the 95% confidence interval estimates.

(b) Does information on the length of survival time of the control group (X₃) help explain the variation in Y? Support your answer with an appropriate test of significance.

(c) Test the null hypothesis that “sex of patient” has no effect on survival beyond that accounted for by “age” and survival of the control group. Interpret the results.

(d) Test the null hypothesis that “age of patient” has no effect on survival beyond that accounted for by “sex” and survival time of the control group. Intrepret the results.

(e) Test the composite hypothesis that β₁
= β₂
= β₃
= 0. From these results, what do you conclude about the effect of sex and age of patient on the mean survival time of patients in this study receiving supplemental ascorbate? With the information available in these data, what would you use as the best estimate of the mean ln(days survival)?