An experiment involved a quantitative analysis of factors found in high-density lipoprotein (HDL) in a sample of

Question:

An experiment involved a quantitative analysis of factors found in high-density lipoprotein (HDL) in a sample of human blood serum. Three variables thought to be predictive of or associated with HDL measurement (Y) were the total cholesterol (X1) and total triglyceride (X2) concentrations in the sample, plus the presence or absence of a certain sticky component called sinking pre-beta, or SPB (X3), which was coded as 0 if absent and 1 if present. The data obtained are shown in the table and the accompanying computer results.

a. Test whether X1, X2, or X3 alone significantly helps in predicting Y.
b. Test whether X1, X2, and X3 taken together significantly help to predict Y.
c. Test whether the true coefficients of the product terms X1X3 and X2X3 are simultaneously zero in the model containing X1, X2, and X3 plus these product terms. State the null hypothesis in terms of a multiple partial correlation coefficient. If this test is not rejected, what can you conclude about the relationship of Y to X1 and X2 when X3 equals 1, as compared with when X3 equals 0?
d. Using Î± = .05, test whether X3 is associated with Y after the combined contribution of X1 and X2 is taken into account. State the appropriate null hypothesis in terms of a partial correlation coefficient. What does your result, together with your answer to part (c), tell you about the relationship of Y with X1 and X2 when SPB is present as compared with when it is absent?
e. How would you determine whether X1, X2, or both X1 and X2 need to be retained in the model to control for confounding and possibly to enhance precision? Assume that no interaction occurs and that the study variable of interest is X3.
f. Based on the information provided, can confounding of X1 and/or X2 be assessed in evaluating the relationship of X3 to Y? Explain.