Week 6-1
Week 6
Analysis of Variance
(ANOVA)

Week 6-2
Week 6 - Learning Objectives
Recognize situations in which to use analysis of
variance
Understand different Analysis of Variance
(ANOVA) designs
Perform a single-factor hypothesis test and
interpret results
Conduct and interpret post-hoc multiple
comparisons procedures

Week 6-3
Parametric Hypothesis Testing
Single sample
Two independent samples
Two dependent samples or paired samples
More than two independent samples ***

Week 6-4
Chapter Overview
Analysis of Variance (ANOVA)
F-test
Tukey-
Kramer
test
One-Way
ANOVA

Week 6-5
ANOVA
AN
alysis
O
f
VA
riance
One-Way ANOVA/
Completely
Randomized Design
Two-Way ANOVA/
ANOVA for Randomized
Block Design
Not discussed in
this chapter

Week 6-6
General ANOVA Setting
Investigator controls one or more independent
variables
Called
factors
(or treatment variables)
Each factor contains two or more
levels.
Observe effects on the dependent variable
Response to levels of independent variable
Experimental design: the plan used to collect the
data

Week 6-7
Completely Randomized Design
Experimental units (subjects) are assigned
randomly to treatments
Subjects are assumed homogeneous
Only one factor or independent variable
With two or more treatment levels
Analyzed by one-factor analysis of variance
(one-way ANOVA)

Week 6-8
The Assumptions of One-Way
ANOVA Hypothesis Testing
the populations are normally distributed.
samples are randomly or independently
selected.
sample sizes must be large enough.
Populations have the same variances (each
group has equal population variance).
Before we test, we must assume that

Week 6-9
The Steps in One-Way ANOVA
Hypothesis Testing
Step 1: State the null and alternative hypothesis
Step 2: State the significance level.
Step 3: State the statement of decision rule to determine
the rejection region(s).
Step 4: Find the critical value of the test from the
statistical table.
Step 5: Determine the appropriate statistical technique
and the test statistic to use.
Step 6: Make the comparison between test statistics value
and critical value
to make statistical decision.
Step 7: Making conclusion.

Week 6-10
The Steps in One-Way ANOVA
Hypothesis Testing (continue)
Additional steps (more information):
Find the p-value from statistical table based
on test statistic value.
Conduct the
Tukey-Kramer
test.

Week 6-11
Step 1: State the null and
alternative hypothesis
H
0
:
m
1
=
m
2
=
m
3
= … =
m
k
All population means are equal
There is no treatment effect (no variation in means among
groups)
H
1
: At least one of
m
i
is not equal; i = 1, 2, 3,…, k
Not all population means are equal
There is a treatment effect
Does not mean that all population means are different
(some pairs may still be the same statistically)

Week 6-12
Step 1 (Continue)
All Means are the same:
The Null Hypothesis is True
(No Treatment Effect)
3
2
1
μ
μ
μ
H
0
:
m
1
=
m
2
=
m
3
= … =
m
k

Week 6-13
Step 1 (Continue)
i
m
At least one mean is different:
The Null Hypothesis is NOT true
(Treatment Effect is present)
3
2
1
μ
μ
μ
3
2
1
μ
μ
μ
or
H
1
: At least one of
m
i
is not equal;
i
= 1, 2, 3,…, k