Chapter 6
The One Way Layout
Introduction
In Chapter 4,
we were interested in the difference of location parameters (medians) of two populations,
using independent random samples from these populations.
In this chapter
this is extended to k ≥ 3 populations. Hence we test the hypothesis of difference in
location parameters of 3 or more populations (also called
treatments
) using independent random samples
from each of these populations.
EXAMPLE
: In a study of the effect of glucose on insulin release, 12 identical specimens of pancreatic
tissue were divided into three groups of four specimens each. Three levels (Low, Medium, High) of
glucose concentration were randomly assigned to the three groups and each specimen with a group was
treated with its assigned level of glucose. Given below are the amounts of insulin released by the tissue
specimens. Does the data indicate that there is a difference in the three concentrations of glucose?
Concentration
Low
Medium
High
1.59
3.36
3.92
1.73
4.01
4.82
3.64
3.49
3.87
1.97
2.89
5.39
*
Note that in this example, twelve specimens were
randomly
assigned
to the three groups and each
group had a different treatment, i.e., we
"did something"
to the items.
*
The resulting groups of observations for each glucose concentration are treated as
independent
random sample from
three different populations.
In general we have the following setup:
Data:
A random sample of size n
j
is drawn from the j
th
treatment group (i.e., j
th
population), for each of the
k treatments (populations). Let X
ij
denote the i
th
observation sampled from the j
th
treatment group.
Thus,
i = 1, 2, …, n
j
and j = 1, 2, …, k.
Assumptions
•
A1
All N = n
1
+ n
2
+ … + n
k
observations are mutually independent of one another.
•
A2
All k treatments (populations) have continuous distributions, with df F
j
(t).
•
A3
The treatment effects are
additive
, that is,
,
for i = 1, 2, …, n
j
; j = 1, 2, …, k.
These three assumptions are equivalent to what is known as “one-way lay out model” in experimental
design, which assumes normal distributions of the populations. Here we are not making the normality
assumption. In both cases, a random sample of n
j
observations are selected from the j
th
treatment group
(population). Each group receives a different treatment and we compare the effects of each treatment with
each other. In some cases, one of the groups is called the “control group” which receives no treatment.
Then we compare each treatment with the control.
Hypotheses: Ho: τ
1
= τ
2
= … = τ
k
vs. Ha: The τ
j
’s are not all equal.