Newsom
USP 634 Data Analysis I
Spring 2013
1
Within-Subjects ANOVA
General Comments
As with the rationale for the paired t-test, the within-subjects ANOVA is used under similar
circumstances. The within-subjects ANOVA, however, is more general than the paired t-test in that it
also can be used with more than two repeated measures. The within-subjects ANOVA is appropriate
for repeated measures designs (e.g., pretest-posttest designs), within-subjects experimental designs,
matched designs, or multiple measures. It is sometimes difficult to understand that two dependent
measures (e.g., the DV at pretest and posttest) function as two “levels” of the independent variable.
So, for instance, in the case of the pretest-posttest design the independent variable is “time.”
Within-subjects designs have advantages over between-subjects designs, because, in general, they
have greater power to detect significance. The fact that each participant serves as his or her own
control (or there is a related other used as a control) leads to the advantage of eliminating variance
due specifically to individual differences. Thus, the error term used in within-subjects ANOVA is a
more precise one, because individual differences have been removed from it. The separate estimation
of variance due to individual differences is explicit in the sum of squares for subject (SSS). Most of the
other general procedures are similar to what we have done before.
Definitional Formulas
The only new quantity in these formulas is SSS, which is the sum of squares for subject. This is
computed by finding the mean score for each case (averaging across rows of the data). SSs
represents individual variation. Individual variation is thus a function of variation of average scores for
an individual around the average of all scores for the sample, SST.
Source
A
S
AxS
SS
=
SS A n∑ (Y. j − Y.. )
=
SS S a ∑ (Yi. − Y.. )
=
SS
AxS
df
2
a −1
2
n −1
∑∑ (Y − Y
i.
− Y. j + Y.. )
or SS AxS = SST − SS A − SS S
T
=
SST
∑ (Y
ij
− Y.. )
2
2
( a − 1)( n − 1)
( a )( n )
MS
F
SS A
df A
SS
MS S = S
df s
SS
MS AxS = AxS
df AxS
FA =
MS A =
MS A
MS AxS
Newsom
USP 634 Data Analysis I
Spring 2013
2
Example
Consider a hypothetical example in which non-native English speaking students are tested before and
after the implementation of an “immersion” approach to teaching proficiency in English to eight
students. Thus, we test students on a language usage scale (say with values from 1-10) during the
traditional bilingual education program and then after the conversion to immersion. 1
Yij − Y..
(Y
.5
-.5
2.5
1.5
2.5
-.5
-1.5
-.5
.25
.25
6.25
2.25
6.25
.25
2.25
.25
Student Bilingual
1
2
3
4
5
6
7
8
4
3
6
5
6
3
2
3
ij
− Y.. )
Immersion
2
2
3
4
5
4
3
2
1
Y.1 = 4
Yij − Y..
(Y
-1.5
-.5
.5
1.5
.5
-.5
-1.5
-2.5
2.25
.25
.25
2.25
.25
.25
2.25
6.25
ij
− Y.. )
2
Yi.
Yi. − Y..
(Y
3
3
5
5
5
3
2
2
-.5
-.5
1.5
1.5
1.5
-.5
-1.5
-1.5
.25
.25
2.25
2.25
2.25
.25
2.25
2.25
i.
− Y.. )
∑ (Y
Y.2 = 3
i.
2
− Y.. ) =
12
2
Y.. = 3.5
2
2
SS A= n∑ (Y. j − Y.. ) = 8 ( 4 − 3.5 ) + ( 3 − 3.5 ) = 4


2
SS S = a ∑∑ (Yi. − Y.. ) = 2 [.25 + .25 + 2.25 +  + 2.25 + 2.25] = 24
2
SS AxS = SST − SS A − SS S = 32 − 24 − 4 = 4
SST =
32
( 4 − 3.5) + ( 3 − 3.5) +  + ( 2 − 3.5) + (1 − 3.5) =
∑ (Yij − Y.. ) =
2
2
2
Source
A (Language
Program)
SS
4
df
S
24
n −1 = 8 −1 = 7
AxS
4
1) (1)(=
7)
( a − 1)( n −=
T
32
Fcrit ( df=
1, df AxS=
A
2
a −1 = 2 −1 = 1
2 )( 8 )
( a=
)( n ) (=
7=
) 5.59
MS
F
SS A 4
= = 4
df A 1
SS S 24
MS=
= = 3.4
S
df s
7
SS AxS 4
MS AxS=
= = .57
df AxS 7
=
FA
MS A=
7
2
MS A
4
= = 7.02
MS AxS .57
16
Language usage scores were significantly higher when students received bilingual education,
F(1,7)=7.01, p<.05. The average difference between language scores when students were in the
bilingual versus the immersion program was 1.00.
1
As discussed earlier in the course, this kind of pretest-posttest design with only one group is subject to many potential threats to internal
validity, so it is not a very good research design.
Newsom
USP 634 Data Analysis I
Spring 2013
3
Analyze General Linear Model Repeated Measures
SPSS Output
General Linear Model
Within-Subjects Factors
Measure: MEASURE_1
langprog
1
2
Dependent
Variable
biling
immerse
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
langprog
Error(langprog)
Type III Sum
of Squares
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
df
1
1.000
1.000
1.000
7
7.000
7.000
7.000
Mean Square
4.000
4.000
4.000
4.000
.571
.571
.571
.571
F
7.000
7.000
7.000
7.000
Sig.
.033
.033
.033
.033
Tests of Within-Subjects Contrasts
Measure: MEASURE_1
Source
langprog
Error(langprog)
langprog
Linear
Linear
Type III Sum
of Squares
4.000
4.000
df
1
7
Mean Square
4.000
.571
F
7.000
Tests of Between-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average
Source
Intercept
Error
Type III Sum
of Squares
196.000
24.000
df
1
7
Mean Square
196.000
3.429
F
57.167
Sig.
.000
Sig.
.033