ST 790, Homework 1
Spring 2017
1. Consider the individual-specific model (2.13) for the dental study data given on page 38 of
the notes,
Yij = β0i + β1i tij + eij ,
i = 1, ... , m, j = 1, ... , ni = 4, where the tij = 8, 10, 12, 14.
(a) Suppose we assume that
β0i = β0,B + b0i ,
β0i = β0,G + b0i ,
β1i = β1,B
β1i = β1,G
if i is a boy
if i is a girl,
where E(b0i |gi ) = 0, var(b0i |gi ) = D, gi is the gender of child i ( = 0 for girl, = 1 for boy), and
b0i are independent across i. Suppose further that eij are independent of each other and of
b0i for all (i, j) with E(eij |gi ) = 0 and var(eij |gi ) = σ 2 .
Give expressions for var (Yij |gi ), cov(Yij , Yij 0 |gi ), and corr(Yij , Yij 0 |gi ).
(b) Comment on the similarities and differences between the model in (a) and the model
(3.1) underlying the univariate repeated measures analysis of variance methods in Chapter
3.
(c) Now assume that
β0i = β0,B + b0i ,
β0i = β0,G + b0i ,
β1i = β1,B + b1i ,
β1i = β1,G + b1i ,
if i is a boy
if i is a girl,
where now, with b i = (b0i , b1i )T , E(b i |gi ) = 0, var(b i |gi ) = D, where D is a nonnegative
definite, unstructured matrix that is the same for all gi , and eij are independent of each other
and of b i for all (i, j) with E(eij |gi ) = 0 and var(eij |gi ) = σ 2 .
Give expressions for var (Yij |gi ), cov(Yij , Yij 0 |gi ), and corr(Yij , Yij 0 |gi ).
(d) Comment on the suitability of the various models in Section 2.5 for representing/approximating
the overall pattern of correlation implied by your results in (a) and (c).
2. Nonlinear models. On page 43 of the notes, we indicate that the distinction between subjectspecific and population-averaged models is critical when the models involved are nonlinear.
In this problem, you will examine this in a simple situation with no within- or among-individual
covariates.
Suppose that the the inherent trajectory for individual i is represented as
µi (t) = β0i exp(−β1i t),
(1)
where
β0i = β0 + b0i ,
β1i = β1 + b1i ,
β = (β0 , β1 )T ,
b i = (b0i , b1i )T ,
where β is a fixed parameter vector, and
D11 D12
b i ∼ N (0, D), D =
, D11 > 0, D22 > 0,
D12 D22
1
(2)
(3)
where D is a positive definite covariance matrix. In (1)-(3), assume β1 >> 0 so that pr(β1i ≤
0) is negligible. In (1), then, while µi (t) depends on β0i and thus b0i in a linear fashion, it
depends on the individual-specific decay rate β1i and thus b1i in a nonlinear fashion.
(a) In (2.7), we represented µi (t) as
µi (t) = µ(t) + Bi (t),
where µ(t) is the overall population mean response at time t and E{Bi (t)} = 0, so that
E{µi (t)} = µ(t). Under the model in (1)–(3), find µ(t) and Bi (t).
(b) Based on (a), do you think it is possible to interpret β1 as both the “typical” rate of decay
in the population and the rate of decay leading to the “typical” or overall population mean
response? Explain your answer.
(c) Now suppose that the decay rates β1i vary negligibly in the population, which we can
represent by letting D22 = 0; note that, if the β1i do not vary, the covariance D12 is no longer
well defined and can be taken to be 0 (we will discuss this situation further in Chapter 6).
Under these conditions, give expressions for µ(t) and Bi (t).
(d) Based on (c), do you think it is possible to interpret β1 as both the “typical” rate of decay
in the population and the rate of decay leading to the “typical” or overall population mean
response? Explain your answer.
3. Consider the situation in Chapter 3, where Yh`j is the response on individual h in group ` at
time j, h = 1, ... , r` ; ` = 1, ... , g; and j = 1, ... , n. The mean response for group ` at time j is
E(Yh `j) = µ`j . Suppose g = 3 and n = 4.
(a) If the pattern of change of mean response is the same in each group, write down a set of
conditions that the µ`j must satisfy (in terms of the µ`j ).
(b) Defining the matrix M as in (3.17), write down matrices C and U in terms of which the
null hypothesis that the pattern of change is the same in each group can be written as
H0 : CMU = 0.
Verify that if the conditions you gave in (a) hold, then H0 is true.
4. Protein diets and growth of chicks. In the file chicks.dat on the class website, you will find
data from an experiment carried out to compare the effects of four different protein diets on
the growth of newly hatched chicks. The investigators randomly assigned each of 45 eggs
to one of the four diets. On the day the eggs hatched (birth of the chicks), the chicks were
started on their assigned diets. The weight of each chick (grams) was ascertained 2, 4, 6,
12, and 20 days after birth.
The data set has the following columns:
1
2
3
4
weight (g)
days since birth
chick ID number
diet (coded as 1, 2, 3, 4)
The investigators were interested in the following questions:
(i) Are chick weights over time different under the four diets?
(ii) Is weight gain (i.e., pattern of change of weight) different among the four diets?
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Using methods in Chapter 2 and 3 of the notes, carry out analyses to address these questions and write a brief report summarizing what you did and the results. Be sure to describe
how you interpreted and formalized the questions of interest. Comment on how confident
you feel about the reliability of the inferences and conclusions.
Please turn in your code and output along with your report (you can edit the output to include
only the portions that pertain directly to your report and embed it in your report if you like).
5. Treatment of children exposed to lead. In the file tlc.dat on the class website, you will
find data from a random sample of 90 children who were enrolled in the Treatment of Lead
Exposed Children (TLC) Trial. The TLC trial was a randomized, placebo-controlled enrolling
over 1800 children with blood lead levels of 20-44 µg/dL. The children were randomly assigned to receive oral chelation treatment with succimer (coded as 1), the active agent, or a
placebo (coded as 0). Blood lead levels were obtained from each child at baseline (week 0),
week 1, week 4, and week 6. Succimer combines with lead in the bloodstream, so that the
lead is excreted by the kidneys with the succimer, thereby serving as a treatment for lead
poisoning.
The data set has the following columns:
1
2
3-6
child ID
treatment, coded as above
blood lead level (µg/dL) at weeks 0, 1, 4, and 6
The investigators were interested in the following questions:
(i) Does succimer lower blood lead levels more effectively than placebo?
(ii) Is the pattern of change in blood lead levels different for succimer relative to placebo?
Using methods in Chapter 2 and 3 of the notes, carry out analyses to address these questions and write a brief report summarizing what you did and the results. Be sure to describe
how you interpreted and formalized the questions of interest. Comment on how confident
you feel about the reliability of the inferences and conclusions.
Please turn in your code and output along with your report (you can edit the output to include
only the portions that pertain directly to your report and embed it in your report if you like).
Note: These data are in the “wide” format of one record per individual. To reconfigure
them in the “long” format of one record per observation in R, you can use the reshape()
function, melt() from the reshape2 package, or a variety of other tools. For example, using
reshape(),
long.data <- reshape(wide.data,varying=c("week0","week1","week4","week6"),
v.names="lead",idvar="id",times=c(0,1,4,6),timevar="week",direction="long")
long.data <- long.data[order(long.data$id,]
# reorder
Examples of doing this in SAS are on the course website.
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