STA 4502 Chapter 7, Page 1
Chapter 7
The TwoWay Layout
In chapter 6
, we were interested in the effect of one factor at k levels (k treatments) on the response
variable, hence the name “One –
Way Layout.”
In this chapter
we will look at the effects of two different factors, each of which has 2 or more levels,
hence the name “two way layout.” Our main interest is in the location parameter (population median)
of one factor, called the
treatment
factor,
within
the various levels of the second factor, called the
blocking factor.
In experimental design problems, the subjects are first divided into n homogeneous groups, called
blocks
and then
within
each block they are randomly assigned to one of the
k treatments
. Hence there
are n×k treatmentblock combinations.
Note that the two factors are called
block
and
treatment
.
The factor called
block
has n levels called
blocks. Similarly the second factor is called
treatment
and has k levels called treatments.
Data
: Data are collected on k treatment populations. The structure of the data collection process is that
observations are collected in blocks, by repeating observations on the same or similar experimental
unit.
Blocks are used to reduce the variability (by controlling the effect of other factors) between responses
in order to improve the chances of identifying differences among the treatments, if they exist.
Within the j
th
block there are c
ij
≥ 1 observations collected on treatment j. The total number of
observations in the study is
1
1
n
k
ij
i
j
N
c
. The data may look as in the following table:
Blocks
Treatments
1
2
k
1
11
111
11
c
X
X
12
121
12
c
X
X
1
1 1
1
k
k
kc
X
X
2
21
211
21
c
X
X
22
221
22
c
X
X
2
2 1
2
k
k
kc
X
X
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STA 4502 Chapter 7, Page 2
n
1
11
1
n
n
n c
X
X
2
21
2
n
n
n
c
X
X
1
nk
nk
nkc
X
X
The case where
c
ij
= 1 for all
i = 1, 2, …, n and j = 1, 2, …, k is known as
a randomized
complete block design.
The first few sections of this chapter will deal with
one
observation per treatment in each block.
Test for general alternatives in a Randomized Complete Block Design
(Friedman, Kendall
–
Babington Smith)
Assumptions
A1.
All observations within the i
th
block and j
th
treatment are mutually independent of one
another
A2.
Let F
ij
denote the distribution function for the data in the i
th
block and j
th
treatment. All of
these n×k treatment populations are continuous.
A3.
The effects are additive, i.e.,
ijt
i
j
ijt
X
e
for t = 1, 2, …, c
ij
; i = 1, 2, …, n
and
j = 1, 2, …, k.
Here,
θ represents the overall median response,
β
i
represents the effect of i
th
block,
τ
j
represents the effects of the j
th
treatment and
e
ijt
denote some random noise (error) terms that are independent with the same
continuous distribution that has median zero.
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 Summer '11
 YESILCAY

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