STAT 510
Homework 11
Due Date: 11:00 A.M., Wednesday, April 20
1. Consider a generic repeated measures experiment like the experiment on strength training programs
that we considered in class. Suppose there are three treatments indexed by i = 1, 2, 3 with ni subjects
indexed by j = 1, . . . , ni for the ith treatment group. Suppose the response of interest is measured
at t time points for each subject. Let yijk be the response for treatment i, subject j, and time point k
(i = 1, 2, 3; j = 1, . . . , ni ; k = 1, . . . , t). For all i and j, let
y ij = (yij1 , . . . , yijt )0 .
Suppose all y ij are mutually independent of one another and that, for all i and j,
y ij ∼ N (µi , W ),
where µi = (µi1 , . . . , µit )0 and W is some unknown t × t positive definite and symmetric matrix.
Let
y = (y 011 , . . . , y 01n1 , y 021 , . . . , y 02n2 , y 031 , . . . , y 03n3 )0 , and let β = (µ01 , µ02 , µ03 )0 .
(a) Use Kronecker product notation to specify a matrix X so that E(y) = Xβ.
(b) Use Kronecker product notation to specify Var(y) = Σ in terms of W .
(c) Find a simplified expression for (X 0 Σ−1 X)−1 .
(d) Find a simplified expression for (X 0 Σ−1 X)−1 X 0 Σ−1 .
(e) Find a simplified expression for (X 0 Σ−1 X)−1 X 0 Σ−1 y.
(f) Give a simplified expression for the BLUE of µ1
(g) Suppose W = σ 2 U where U is a known positive definite and symmetric matrix and σ 2 is an
unknown positive real number. Then our model for y can be written in the form of the Aitken
model where y = Xβ + and ∼ N (0, σ 2 V ). Provide an expression for V in terms of U .
(h) Under the conditions of part (g), find a simplified expression for the REML estimator of σ 2 .
2. Consider the model fit on slide 40 of slide set 24 on repeated measures analysis in R. Suppose the
data were sorted so that rows of the dataset were first ordered by programs, then by subject IDs within
programs, and then by time (as in the setup of problem 1). The model fit can be written in the form
y = Xβ + Zu + e with our usual notational conventions.
(a) Provide expressions for Z and u in terms of days 2, 4, 6, . . . , 14 and the bij random vectors
defined in the annotated version of the course notes.
(b) Provide expressions for G = Var(u) and R = Var(e) in terms of the variance parameters
defined in the annotated version of the course notes.
(c) Provide the REML estimate of Σb defined in the annotated version of the course notes.
(d) Provide an estimate of the covariance between the strength measurement of a given subject on
day 2 and the strength measurement for that same subject on day 10.
(e) Provide the equation of the predicted quadratic function for the subject with ID 56 (i.e., Subj=56).
(f) Make a plot of strength measurement vs. day for subject 56. In addition to plotting the 7 data
points for that subject, include the predicted quadratic curve for subject 56 on the plot as well
as the estimated mean quadratic curve for subjects assigned to the same program as subject 56.
Use a dashed curve for the subject-specific quadratic function and a solid curve for the estimated
mean quadratic function.
3. In 1846, a group of pioneers traveling west became stranded in the eastern Sierra Nevada mountains.
By the time the last survivor was rescued in the spring of 1847, 40 of 87 members in the original group
had died from starvation and exposure to extreme cold. The group became known as the Donner Party.
The dataset
http://www.public.iastate.edu/∼dnett/S510/Donner.txt
contains the age, sex, and status (survived or died) of the members of the group that were 15 years of
age or older. Conduct an analysis of this data set to determine how age and sex are associated with
the probability of survival. Support your answer with appropriate tests and confidence intervals. State
your conclusions in ways that will be easily interpretable by nonstatisticians.
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