http: www.utsc.utoronto.ca bors HomoVariance.ppt

Homogeneity of Variance
Pooling the variances doesn’t make sense when we cannot assume all of the sample
Variances are estimating the same value.

For two groups:
Levene (1960): replace all of the individual scores with either
then run a t-test

Given: 1. Random and independent samples
2. Both samples approach normal distributions
Then: F is distributed with (n-large-1) and (n-small-1) df.

Null Hypothesis:

E (S

Alternate Hypothesis:



2

l a rg e

)  E (S


l arg e



or

( y ij  y j ) 2

y1  y2
2 M S error
n

t 

F - test

( y ij  y j )

2
s m a ll

2
s m a ll

)

F 

S
S

2
l a rg e
2
s m a ll

K independent groups:
Hartley: If the two maximally different variances are NOT significantly different,
Then it is reasonable to assume that all k variances are estimating the population variance.
The average differences between pairs will be less than the difference between the smallest
And the largest variance.

S
E 
S
Thus:

2
l a rg e
2
s m a ll


 S A2 
  E  2

SB


F 

F m ax

A and B are randomly selected pairs.

2

S

l a rg e

S

2
s m a ll

will NOT be distributed as a normal F.

(k groups, n-1) df

Then, use

F m ax

Table to test

Null Hypothesis:



Alternate Hypothesis:

2



1



2
j

2
2



   
2

2
i

. . . 





Data Transformation: When Homogeneity of Variance is violated
Looking at the correlation between the variances (or standard deviations)
And the means or the squared means.


r 

( x  x )( y  y )
n  1
S xS y

b) Use square root transformation
c) Use logarithmic transformation
d) Use reciprocal transformation