Satterthwaite`s Approximation for Degrees of Freedom
Download
Report
Transcript Satterthwaite`s Approximation for Degrees of Freedom
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 1S12 n2 1S 22
n1 n2 2
W
n1 n2 2S 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 1Si2
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 1Si2
i2
Z
Y
1
~
Y21 ,...,Y2 n2 ~ N 2 , 22
12 22
Y1 Y2
ni 1S i2
ni 1Si2
2 i4
2
2
2
E
ni 1, V
2ni 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 : maxFobs ,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
70.49 262 26.00 262 0.33
25
25
RR : tobs t.025 , 41.23 2.020