Satterthwaite’s Approximation for
Degrees of Freedom
Art Instruction Effect on Reading
Development
J.C. Mills (1973), “The Effect of Art Instruction Upon a Reading Development Test: An Experimental
Study with Rural Appalachian Children,” Studies in Art Education, Vol. 14, #3, pp.4-8
Setting
• Comparing Means from 2 Normal Distributions
• Small Samples (Computer Packages Solve for any sizes)
• Distributions have Possibly Different Variances
N(50,100)
N(60,225)
0
20
40
60
80
100
120

Y11 ,...,Y1n1 ~ N 1 ,  12
ni
Yi 
j 1
S 
2
i
ni

 Y 2  1   2 
1 1
 
n
 1 n2 
i  1,2
2
p
2
~ N 0,1
 n1  n2  2 S p2 


2
S p2



n1  n2  2
2
S

n1  1S12  n2  1S 22

n1  n2  2
W
n1  n2  2S p2

2
~  n21  n2  2

Z W




 Y 1  Y 2  1   2  




1
1

 2    

Z
 n1 n2   Y 1  Y 2  1   2 



~ t n1  n2  2
2
W
Sp
1
2 1

S

p
 n n 
dfW
2
2 
 1








Y 1  Y 2  1   2 
 P  t / 2,n1  n2  2 
 t / 2,n1  n2  2   1  


1 1
S p2   


n
n
1
2





Y1  Y2 
~  n2i 1
 2 
W

dfW
 12   22   2
2
ni  1Si2



ni  1
 2 

Y i ~ N  i ,
n
i 

1
ij  Y i
j 1

Y21 ,...,Y2 n2 ~ N  2 ,  22
 Y
ni
 Yij
Y
Z

Case 1 – Variances are Equal


Case 2 – Variances are Unequal - I

Y11 ,...,Y1n1 ~ N 1 ,  12
ni  1Si2
 i2
Z
Y
1
~


Y21 ,...,Y2 n2 ~ N  2 ,  22

 12   22
Y1  Y2 
 ni  1S i2 
 ni  1Si2 
2 i4
2
2
2
 E
  ni  1, V 
  2ni  1  E Si   i , V Si 
2
2


ni  1
i
i




 
2
ni 1

 Y 2  1   2 
 
~ N 0,1

 



 n1 n2 
P roblem: ReplacingDenominator with est imatedvariances, consider:
2
1
2
2
 S12 S 22 
  
W *  n1 n2 

which is NOT a chi - square divided by its degrees of freedom.
dfW *   12  22 



n
n
2 
 1
W 
W  2
Aside : W ~ 2  E W    , V W   2  E    1, V   
 
  
  S12 S 22     12  22 
     


n
n
n
n
W*
W*

1
2  
2 
 1
  E   12


E 


1
V

2

 df 
2
2
2
2
2










W
*
 dfW * 




 1  2   1  2 
 1  2 



 n

  1 n2    n1 n2 
 n1 n2 
 1 2 14
1 2 24  2
 2
 2

n
n

1
n
n

1
2
2
 1 1
 *
Case 2 – Variances are Unequal - II
W*
1
V


2
2
2
df
 W *   1  2 



n
n
 1
2 
 1 2 14
1 2 24  2
 2

 2

 n1 n1  1 n2 n2  1   *
  14

 24
  12  22 
2 2
 2




n1  n1  1 n2  n2  1 
n1 n2 
2



 * 
2
2
2
*
  2 n  2  2 n  2 
 1  2 

 1 1  2 2 


 n1  1
 n1 n2 
n2  1 


Replacing the unknown variances with their estimates:
2
 S12 S 22 
  
 n1 n2 
2
2
 2

g
MS
 i i 
^
 i 1
 where: g  1 MS  S 2   n  1
* 

i
i
i
i
i
2
2
n
  S 2 n 2  S 2 n 2 
g
MS


i
i
i
 1 1  2 2  
i
 n1  1
n2  1  i 1


So, we have the approximate degrees of freedom if our denominator were the square root
of the ratio of a chi-square to its degrees of freedom
Example – Art Instruction Effect on Reading
• Experiment to Determine Effect of Art Instruction on
a Reading Development Test
 52 Children Given Baseline Reading Test
 26 Received Art Instruction (Trt), 26 Did not (Control)
 Y=Post-Test – Pre-Test Score
Y T  7.77 ST2  70.49 nT  26
Y C  1.58 SC2  26.00 nC  26
H0 :  
2
T
2
C
H 0 : T   C
H A :  
2
T
2
C
H A : T   C
T .S . : Fobs
ST2
 2  2.71 RR : maxFobs ,1 Fobs   F.025 , 25, 25  2.23
SC
T .S . : tobs 
YT Y C
ST2 SC2

nT nC

7.77  (1.58)
 4.85
70.49 26.00

26
26
2
 70.49 26 
 

^
13.77
26
26 

 *

 41.23
 70.49 262 26.00 262  0.33





25
25


RR : tobs  t.025 , 41.23  2.020