Name
PH1821 Midterm exam
March 2, 1998
Page 10
PH 1821 Midterm Exam
Date Exam Begins:Wednesday, February 28, 2000
Date Exam End:Monday, March 3, 2000
Instructions:
The exam consists of one questions. Answer each part of this question, using any textbook, library, or computing resources available. Although you may discuss the questions with either me or the Teaching Assistant, you must not discuss the exam with any fellow student.
There is enough paper provided for you to write your answers to these questions on the exam sheets themselves, especially if you use both sides of the pages. However, you may of course submit additional pages if you need to.
Don't attempt solutions when you are tired, hungry, preoccupied, rushed, or angry. Let this exam reflect your best effort attempt to solve these problems when you have your very best judgement. Stay sharp eyed, careful, confident, and disciplined.
Two Stage Quadratic Regression
Consider the following attempt to assess the hetergeneity of of HDL cholesterol over time. Each of n patients has a sequence of repeated HDL cholesterol (i.e. the “good cholesterol”) at a sequence of time points. The number of time points varies from subject to subject. For the ith individual, nI measurements are obtained. Let these measurements for the ith patient (xi1, yi1), (xi2, yi2), (xi3, yi3), … (xin, yin). For each subject the following model is fitted
E(yij) = β1xij + β2xij2
which you will recognize as a “through the origin” quadratic model. Upon the conclusion of each of these n quadratic model computations a second level of regression is executed. For this second level, each subject “contributes” both the estimate of β1 and β2 to the new y vector. The purpose of this second stage of regression is to obtain a summary estimator of each of β1 and β2 and to carry out the 1) two hypothesis tests for β1 and β2 to each be zero and 2) β1 = -0.005 β2
A) Using linear algebra, find the formula for the estimate b1 of β1 and b2 of β2 for the ith patient. Find the variance inflation factors for each of these parameter estimates (10 points)
B) Write the model out for the second stage of regression in matrix notation. Be explicit about the tuple of the dependent variable vector, the elements of the design matrix X, and the dimention and elements of V, the variance-covariance matrix of the dependent vector y.
(20 points)
C) Using the linear algebra of generalized weighted least squares, find the least squares estimates of the parameters β1 and β2. Find the variance inflation factors for these estimates (40 points)
D) Find the explicit formula for the correlation between b1 and b2 from the second stage regression (10 points)
E) Write the formulas for testing each of the hyptheses that (1) H0: β1 = 0 and (2) H0: β2 = 0 (each should be tested against a two sided alternative (10 points)
F) Write the formulas for testing each of the hyptheses that H0: β1 = -0.005 β2 (10 points)