Homework
Assignment #9
Developing Expressions for
the Random Response Vector in a Nested
Study
1. Searle describes a study in
Chapter 3, p44-49. Read this discussion. Following this,
using the notation discussed in class, and assuming the
number of classes in the population is very large (virtually
infinite), and the number of students in each class is also
very large (virtually infinite):
- A. Write a model for a
randomly selected subject (j) from a randomly selected
class (i) . Define the fixed and random effects in the
model, and define parameters for the
population.
- B. Express a model for a
vector of random variables corresponding to the responses
in Table 3.1 (page 47), where the subjects from the first
selected class are listed first, the second listed
second, etc.
- C. Let C denote expectation
with respect to classes, and P denote expectation with
respect to subjects. Write out expressions for the
following:
- i. The expected value with
respect to P of the response vector.
- ii. The variance with
respect to P of the response vector.
- iii. The expected value
with respect to C and P of the response
vector.
- iv. The variance with
respect to C and P of the response vector.
- D. Define a matrix K that when
pre-multiplied by the response vector will result in the
mean response for each class. Using this matrix,
evaluated expressions corresponding to parts i-iv in C of
the vector of class means.
2. Searle describes a similar study in Chapter 3 with
unequal numbers of subjects selected per class. The
resulting data for the study are given in Table 3.2, page
52. Making similar assumptions as in problem 1,
- A. Write a model for a
randomly selected subject (j) from a randomly selected
class (i) . Define the fixed and random effects in the
model, and define parameters for the population. Assume
that the unequal number of subjects selected are the
result of the study design.
- B. Express a model for a
vector of random variables corresponding to the responses
in Table 3.2 (page 52), where the subjects from the first
selected class are listed first, the second listed
second, etc.
- C. Let C denote expectation
with respect to classes, and P denote expectation with
respect to subjects. Write out expressions for the
following:
- i. The expected value with
respect to P of the response vector.
- ii. The variance with
respect to P of the response vector.
- iii. The expected value
with respect to C and P of the response
vector.
- iv. The variance with
respect to C and P of the response vector.
- D. Define a matrix K that when
pre-multiplied by the response vector will result in the
mean response for each class. Using this matrix,
evaluated expressions corresponding to parts i-iv in C of
the vector of class means.
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