One-way within-subjects ANOVA

Outline:
-- Definitional Formula
-- Computational Formula
-- An example

Definitional Formula
Again, the reason is to eliminate individual differences from our hypothesis testing statistic.

Yesterday with the t-test, we looked at what the within did in terms of the bottom of our conceptual formula

  difference b/t groups  = difference b/t groups
  variability w/i groups = ind. diffs. + measurement error

Allowed us to pull out the ind. diffs.

Letís turn to what this will mean for a within subjects ANOVA,
 We want to be able to do the same thing for the case where we have more than two groups.  The goal is the same and the end result is the same, but the actual process is different.

WHAT IS AN ANOVA?
1.  Partition Variance
2.  Compare Variances

For a one-way between-subjects the parts were:  MSA and MSE

How does this apply here, with a within subject design?
1. Still going to have the part due to IV:  MSA
 But we are going to look more closely at the error component
  due to subjects (MSS), due to measurement error (MSE)
2.  Compare:  part due to IV to part due to error (so just like in t-test, we will remove ind. diffs. from our final statistic).
 
How do we partition variance?
1.  Start at deviations

DevT = DevA + DevS + DevE

Xia - MT = (MA  - MT) + (MS - MT) + (Xia - MA - MS + MT)
DevT is how far the individual score is from the overall mean
DevA is how far the group mean is from the overall mean -- due to IV
DevS is how far the individual mean is from the overall mean -- due to subject
DevE is how far the individual score is from what is expected based on group and subject and total means -- thought of as not measuring accurately

2.  Square the deviations and sum across individuals and groups

SST = SUMaSUMi (Xia - MT)2
SSA = SUMaSUMi (MA  - MT)2
SSS = SUMaSUMi (MS - MT)2
SSE = SUMaSUMi (Xia - MA - MS + MT)2

3.  Divide by the df

dfT  = NT-1
dfA  = A - 1
dfS  = NS -1
dfE  = (A - 1)(NS -1)

4.  Now you have partitioned variance and can fill in the source table
 
Source
SS
df
MS
F
p
w/i factor
.
.
.
.
.
    A
SSA
dfA 
MSA
F = MSA / MSE
.
Subject
 SSA
 dfS
 .
.
.
error
SSE
dfE
MSE
.
.
    total
 SST
 dfT
 .
.
.
 
 

Letís do the example:
 

Within Subjects ANOVA

DV - number of hours taken to read chapters of equivilent lengths
 
 
Hemingway
Wolfe
Faulkner
 
i
Xia
Xia
Xia
Ms
1
2
3
4
3.00
2
4
3
5
4.00
3
8
9
7
8.00
4
6
8
9
7.67
5
3
4
4
3.67
6
3
5
6
4.67
7
5
7
8
6.67
8
1
2
3
2.00
Sum Xia
32
41
46
 
MA
4.00
5.13
5.75
 
 

Within-Subjects ANOVA
 
 
Source
SS
df
MS
F
p
w/i factor
.
.
.
.
.
    Author
12.583
6.292
8.739
<.01
Subject
 104.291
 7
 .
.
.
error
10.084
14
0.720
.
.
    total
 126.958
 23
 .
.
.
 
 

 
 

if Between-Subjects ANOVA
 
 
Source
SS
df
MS
F
p
b/t factor
.
.
.
.
.
    Author
12.583
6.292
1.155
ns
error
 114.375
 21
5.446
.
.
total
1266.958
23
.
.
.