Chapter 15
Analysis of Variance
©
Framework for One-Way Analysis
of Variance
Suppose that we have independent samples of n1, n2, . . ., nK
observations from K populations. If the population means are
denoted by 1, 2, . . ., K, the one-way analysis of variance
framework is designed to test the null hypothesis
H 0 : 1   2     K
H1 :  i   j
For at least one pair i ,  j
Sample Observations from Independent
Random Samples of K Populations
(Table 15.2)
1
x11
x12
.
.
.
x1n1
POPULATION
2
...
x21
...
x22
...
.
.
.
x2n2
...
K
xK1
xK2
.
.
.
xKnK
Sum of Squares Decomposition for
One-Way Analysis of Variance
Suppose that we have independent samples of n1, n2, . . ., nK
observations from K populations. Denote by x1 , x2 ,, xK the K
group sample means and by x the overall sample mean. We
define the following sum of squares:
K
Within - Groups : SSW  
i 1
K
nj
2
(
x

x
)
 ij i
j 1
Between - Groups : SSG   ni ( xi  x ) 2
i 1
K
Total : SST  
i 1
nj
2
(
x

x
)
 ij
j 1
where xij denotes the jth sample observation in the ith group.
Then
SST  SSW  SSG
Sum of Squares Decomposition for
One-Way Analysis of Variance
(Figure 15.2)
Within-groups
sum of squares
Total sum of squares
Between-groups
sum of squares
Hypothesis Test for One-Way
Analysis of Variance
Suppose that we have independent samples of n1, n2, . . ., nK
observations from K populations. Denote by n the total sample
size so that
n  n1  n2   nK
We define the mean squares as follows:
SSW
Within - Groups : MSW 
nK
SSG
Between - Groups : MSG 
K 1
The null hypothesis to be tested is that the K population means
are equal, that is
H 0 : 1  2     K
Sum of Squares Decomposition for
Two-Way Analysis of Variance
Suppose that we have a sample of observations with xij denoting
the observation in the ith group and jth block. Suppose that
there are K groups and H blocks, for a total of n = KH
observations. Denote the group sample means by xi (i  1,2,, K,)
the block sample means by x j ( j  1,2,, H )and the overall
sample mean by x.
We define the following sum of squares:
K
Total : SST  
i 1
K
H
2
(
x

x
)
 ij
j 1
Between - Groups : SSG  H  ( xi  x ) 2
i 1
H
Within - Blocks : SSB  K  ( x j  x ) 2
j 1
Sum of Squares Decomposition for
Two-Way Analysis of Variance
(continued)
K
ERROR : SSE  
i 1
H
2
(
x

x

x

x
)
 ij i  j
j 1
SST  SSG  SSB  SSE
Hypothesis Test for Two-Way
Analysis of Variance
Suppose that we have a sample observation for each group-block
combination in a design containing K groups and H blocks.
xij    Gi  B j  Eij
Where Gi is the group effect and Bj is the block effect.
Define the following mean squares:
SSG
Between - Groups : MSG 
K 1
SSB
Between - Blocks : MSB 
H 1
SSE
Error : MSE 
( K  1)( H  1)
We assume that the error terms ij in the model are independent of
one another, are normally distributed, and have the same variance
Hypothesis Test for Two-Way
Analysis of Variance
(continued)
A test of significance level  of the null hypothesis H0 that the K
population group means are all the same is provided by the
decision rule
MSG
Reject H 0 if
 FK 1,( K 1)( H 1),
MSE
A test of significance level  of the null hypothesis H0 that the H
population block means are all the same is provided by the
decision rule
MSB
Reject H 0 if
 FH 1,( K 1)( H 1),
MSE
Here F v1,v2,  is the number exceeded with probability  by a
random variable following an F distribution with numerator
degrees of freedom v1 and denominator degrees of freedom v2
General Format of Two-Way Analysis
of Variance Table
(Table 15.9)
Source of
Variation
Sums of
Squares
Degrees of
Freedom
Mean
Squares
F Ratios
Between
groups
SSG
K–1
SSG
MSG 
K 1
MSG
MSE
Between
blocks
SSB
H–1
MSB 
SSB
H 1
MSB
MSE
Error
SSE
(K – 1)(H – 1)
MSE 
Total
SST
n-1
SSE
( K  1)( H  1)
Sum of Squares Decomposition for Two-Way
Analysis of Variance: Several Observations per Cell
Suppose that we have a sample of observations on K groups and
H blocks, with L observations per cell. Then, we define the
following sums of squares and associated degrees of freedom:
Total : SST   ( xijl  x ) 2
i
j
KHL  1
l
K
Between - Groups : SSG  HL  ( xi  x ) 2
i 1
H
Between - Blocks : SSB  KL ( x j   x ) 2
K 1
H 1
j 1
K
H
Interactio n : SSI  L
2
(
x

x

x

x
)
 ij i  j
i 1
j 1
Error : SSE   ( xijl  xij ) 2
i
( K  1)( H  1)
j
l
KH ( L  1)
Sum of Squares Decomposition for
Two-Way Analysis of Variance with
More than One Observation per Cell
(Figure 15.12)
Within-groups
sum of squares
Between-groups
sum of squares
Total sum of squares
Interaction
sum of squares
Error
sum of squares
Key Words
 Hypothesis Test for One-Way Analysis of Variance
 Hypothesis Test for Two-Way Analysis of Variance
 Interaction
 Kruskal-Wallis Test
 One-Way Analysis of Variance
 Randomized Block Design
 Sum of Squares Decomposition for One-Way
Analysis of Variance
 Sum of Squares Decomposition for Two-Way
Analysis of Variance
 Two-Way Analysis of Variance: Several
Observations per Cell