.
ST 732, HOMEWORK 2, SPRING 2007
1. Suppose that we have a situation in which units have been randomized into q = 4 groups and
each unit is observed at the same n = 3 times. As in the notation in the notes, let j be the
mean for the th group at the jth t

CHAPTER 1
ST 732, M. DAVIDIAN
1
Introduction and Motivation
1.1
Purpose of this course
OBJECTIVE: The goal of this course is to provide an overview of statistical models and methods that are useful in the analysis of longitudinal data; that is, data in th

.
ST 732, HOMEWORK 5, SPRING 2007
1. Recall the lead exposure study from Homework 3, Problem 3. Consider model (1) in the
statement of that problem, which is repeated here for convenience: Let Yij denote the jth
lead level measurement on the ith child at

.
ST 732, HOMEWORK 5, SOLUTIONS, SPRING 2007
1. We obtain
Yij = (0 + 0a ai + 0g gi + 0ag ai gi ) + (k + ka ai + kg gi + kag ai gi )tij + (b0i + b1i tij + eij ).
Because all of b0i , b1i and eij have mean zero, is it straightforward to see that
E(Yij ) = (

.
ST 732, HOMEWORK 6, SOLUTIONS, SPRING 2007
1. (a) (i) For such a subject, tj = 1. The expected tumor response E(Yj ) for such a subject
(which is the same as the probability that the subject develops a new tumor under these
conditions, P (Yj = 1), is
e0

.
ST 732, HOMEWORK 3, SPRING 2007
1. A study was conducted in which m = 40 devices were randomized to be operated under 4
dierent sets of conditions, 10 devices per set of conditions. A response reecting performance level of such devices was measured on e

.
ST 732, HOMEWORK 6, SPRING 2007
1. A study was conducted to compare two treatments for patients with bladder cancer. Each of
the n = 100 subjects recruited into the study had recently had surgery to remove the tumor;
at baseline, each was then randomize

.
ST 732, HOMEWORK 4, SOLUTIONS, SPRING 2007
1. (a) Here, the dimension of i is (2 1), so Z i
this matrix becomes
1
1
Zi = 1
1
1
is (ni 2) matrix in general. When ni = 5,
ti1
ti2
ti3
ti4
ti5
,
so that sweeping the jth row of Z i down i yields the expres

.
ST 732, HOMEWORK 3, SOLUTIONS, SPRING 2007
1. (a) The model that seems reasonable to capture the possibility of curvature of the mean
proles as in criterion (ii) is one that allows the mean as a function of time in each group to
be quadratic in time. Fu

.
ST 732, HOMEWORK 4, SPRING 2007
1. Consider a straight line model for individual behavior as in Equation (9.1) of the notes, which
for unit i is of the form
Yij = 0i + 1i tij + eij ,
(1)
where Yij is the random variable representing the observation that

.
ST 732, HOMEWORK 1, SPRING 2007
The rst few exercises are meant to familiarize you with some operations that we will summarize
using matrix notation throughout the course. Use of SAS to carry out the analyses we will discuss
requires familiarity with th

.
ST 732, HOMEWORK 2, SOLUTIONS, SPRING 2007
1. (a) We have
M =
11
21
31
41
12
22
32
42
13
23
33
43
.
(b) We have a = (1, 0, 0, 0), so that a M = (11 , 12 , 13 ).
i
i
(c) We have
=
1
2
3
4
1
2
3
( )11
( )12
( )13
( )21
( )22
( )23
( )31
( )32
( )33
( )41

.
ST 732, HOMEWORK 1, SOLUTIONS, SPRING 2007
1. (a) First, it is clear that, using the result at the top of p. 35, E(c1 Y1 + c2 Y2 ) = c1 1 + c2 2 .
Thus, using (3.2), var(c1 Y1 + c2 Y2 ) = Ecfw_(c1 Y1 + c2 Y2 c1 1 c2 2 )2 . This may be rewritten
as, usin