Question

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?

(h) (2 marks) For fixed age and BMI, what is the estimated effect of a one-unit change in PUFA for a subject who is A- at TNF-α? For fixed age and BMI, what is the estimated effect of a one-unit change in PUFA for a subject who is A+ at TNF-α.