2. (12 marks) In a study of 85 individuals, food questionnaires were used to measure the change in polyunsaturated fatty acids (PUFA) intake from the beginning to the end of the study. In addition researchers measured a genetic marker in the TNF-αgene; for this analysis, we will take the two values of this TNF-αvariable to be A- or A+. The researchers also collected information on gender, age and BMI. The primary question of interest is whether TNF-αmodifies the effect of PUFA on HDL.
(a) (1 mark) Let TNFa be a variable with value 1 for A+ and 0 for A- TNF-αstatus and
let gender be 1 for males and 0 for females. State a linear model for mean change in HDL that includes the predictors PUFA, TNFa, PUFA×TNFa interaction, and effects for gender, age and BMI.
(b) (1 mark) A summary of the fitted model is:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.037531 0.196980 0.191 0.8494
pufa 0.019143 0.009245 2.071 0.0417
TNFa 0.074433 0.034128 2.181 0.0322
gender -0.008988 0.031793 -0.283 0.7781
age -0.003895 0.002156 -1.806 0.0747
bmi 0.006777 0.004332 1.565 0.1217
pufa:TNFa -0.024575 0.017844 -1.377 0.1724
Write a short sentence to describe why we would even consider removing gender from the model (i.e., why we would not just be happy with the above model from
the outset).
(c) (2 marks) Below is a summary of the model fit without gender. Using our 10% rule-of-thumb, is there any evidence that gender confounds the PUFA×TNFa effect?
If so why, if not why not?
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.024467 0.190364 0.129 0.8981
pufa 0.018841 0.009130 2.064 0.0423
TNFa 0.074850 0.033897 2.208 0.0301
age -0.003783 0.002107 -1.795 0.0765
bmi 0.006835 0.004301 1.589 0.1161
pufa:TNFa -0.024496 0.017737 -1.381 0.1712
(d) (2 marks) The researchers decided to use the model with PUFA, TNFa, PUFA×TNFa, age and BMI for inference of PUFA×TNFa interaction. What are the degrees of
freedom for the t-distribution whose critical value we need to construct confidence intervals for the interaction? What upper tail probability should you use for a 95%
confidence interval?
(e) (1 mark) The appropriate critical value is 1.96. Construct a 95% confidence interval for the interaction.
(f) (2 marks) State the null hypothesis and a two-sided alternative hypothesis for testing whether TNFa modifies the effect of PUFA on HDL.
(g) (1 mark) From the computer output, what is the p-value for this test?
: 
and the alternative hypothesis: 
