Chapter 14 - Repeated Measures Designs
Chapter 14 – Repeated-Measures Designs
[As in previous chapters, there will be substantial rounding in
these answers. I have attempted to make the answers fit with the
correct values, rather than the exact results of the specific
calculations shown here. Thus I may round cell means to two
decimals, but calculation is carried out with many more
decimals.]
14.1Does taking the GRE repeatedly lead to higher scores?
a.Statistical model:
or
ijijijijijijij
XeXe
mptptmpt
=++++=+++
b.Analysis:
(
)
(
)
2
2
2
13520
7811200194933.33
24
total
X
SSX
N
=-=-=
å
å
(
)
(
)
(
)
(
)
2
22
..
.
3[566.67563.33...573.33563.33]363222.221
89,666.67
subj
i
SStXX
=S-
=-++-==
(
)
(
)
(
)
[
]
2
22
2
...
8[552.50563.33(563.75563.33)573.75563.33
]
8226.041808.33
testj
SSnXX
=S-=-+-+-
==
194,933.33189,666.671808.333458.33
errortotalsubjtest
SSSSSSSS
=--
=--=
Source
df
SS
MS
F
Subjects
7
189,666.66
Within subj
16
5266.67
Test session
2
1808.33
904.17
3.66 ns
Error
14
3458.33
247.02
Total
23
194,933.33
14.2Data on first two Test Sessions in Exercise 14.1:
a.Related-sample t test:
Subj
First
Second
Diff
1
550
570
20
2
440
440
0
3
610
630
20
4
650
670
20
5
400
460
60
6
700
680
-20
7
490
510
20
8
580
550
-30
Mean
11.25
(
)
2
90
6500
8
27.999
7
011.25
1.14
27.999
8
D
D
s
D
t
s
-
==
-
===
[t.025(7) = +2.365]
Do not reject H0
b.Repeated-measures ANOVA:
Source
df
SSMSF
Between subj
7
130,793.75
Within subj
8
3250.00
Test session
1
506.25
506.25
1.29ns
Error
7
2743.75
391.96
Total
15
134,185.94
F
=
1
.
29
=
1
.
14
=
t
from
part
a
.
14.3Teaching of self-care skills to severely retarded
children:
Cell means:
Phase
Baseline
Training
Mean
Group:
Exp
4.80
7.00
5.90
Control
4.70
6.40
5.55
Mean
4.75
6.70
5.72
Subject means:
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
Grp
Exp
8.5
6.0
2.5
6.0
5.5
6.5
6.5
5.5
5.5
6.5
Control
4.0
5.0
9.0
3.5
4.0
8.0
7.5
4.5
5.0
5.5
X2 = 1501
X = 229
N = 40
n = 10
g = 2
p = 2
(
)
2
2
2
229
1501189.975
40
total
X
SSX
N
=-=-=
å
å
(
)
(
)
2
2
2
....
2[8.55.72...(5.55.72)]106.475
subjij
SSpXX
=S-
=-++-=
(
)
(
)
(
)
(
)
2
22
.....
28[5.905.725.555.72]1.225
groupk
SSpnXX
=S-
=-+-=
(
)
(
)
(
)
(
)
2
22
.....
210[4.755.726.705.72]38.025
phasej
SSgnXX
=S-
=-+-=
(
)
(
)
(
)
2
22
....
104.805.72...6.405.7239.875
cellsjk
SSnXX
=S-
éù
=-++-=
ëû
39.87538.0251.2250.925
PGcellsphasegroup
SSSSSSSS
=--=--=
Source
df
SS
MS
F
Between Subj
19
106.475
Groups
1
1.125
1.125
0.19
Ss w/in Grps
18
105.250
5.847
Within Subj
20
83.500
Phase
1
38.025
38.025
15.26*
P x G
1
0.625
0.625
0.25
P x Ss w/in Grps
18
44.850
2.492
Total
39
189.975
*
p
<
.
05
[
F
.
05
(
1
,
18
)
=
4
.
41
]
There is a significant difference between baseline and training,
but there are no group differences nor a group x phase
interaction.
14.4Independent t test on data in Exercise 14.3:
a.Difference scores (Training - Baseline)
Exper.
1
2
-1
2
7
1
3
-1
3
5
Control
2
0
2
3
-2
4
3
1
4
0
2
2
2
2
2.2 6.1778 10
1.7 3.7889 10
2.21.7
6.17783.7889
0.50
1010
EEE
EEE
EC
C
E
EC
Xsn
Xsn
XX
t
s
s
nn
===
===
-
-
==
+=
+
[t.025(9) = +2.262]
Do not reject H0
b.
t
2
=
.
50
2
=
.
25
=
F
for
P
x
G
interaction
.
c.The t is a test on whether the baseline vs. training
difference is the same for both groups. This is a test of an
interaction, not a test of overall group differences.
14.5Adding a No Attention control group to the study in Exercise
14.3:
Cell means:
Phase
Baseline
Training
Total
Group
Exp
4.8
7.0
5.90
Att Cont
4.7
6.4
5.55
No Att Cont
5.1
4.6
4.85
Total
4.87
6.00
5.43
Subject means:
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
Group:
Exp
8.5
6.0
2.5
6.0
5.5
6.5
6.5
5.5
5.5
6.5
Att Cont
4.0
5.0
9.0
3.5
4.0
8.0
7.5
4.5
5.0
5.0
No Att Cont
3.5
5.0
7.0
5.5
4.5
6.5
6.5
4.5
2.5
3.0
2
2026 326 60 10 3 2
XXNngp
======
åå
X = 326
N = 60
n = 10
g = 3
p = 2
(
)
2
2
2
326
2026254.7333
60
total
X
SSX
N
=-=-=
å
å
(
)
(
)
2
2
2
....
2[8.55.43...(3.05.43)]159.733
subjij
SSpXX
=S-
=-++-=
(
)
(
)
(
)
(
)
(
)
2
222
.....
28[5.905.435.555.434.855.43]11.433
groupk
SSpnXX
=S-
=-+-+-=
(
)
(
)
(
)
(
)
2
22
.....
310[4.875.436.005.43]19.267
phasej
SSgnXX
=S-
=-+-=
(
)
(
)
(
)
2
22
....
104.805.43...4.605.4352.333
cellsjk
SSnXX
=S-
éù
=-++-=
ëû
51.33319.26711.43320.633
PGcellsphasegroup
SSSSSSSS
=--=--=
Source
df
SSMSF
Between subj
29
159.7333
Groups
2
11.4333
5.7166
1.04
Ss w/ Grps
27
148.300
5.4926
Within subj
30
95.0000
Phase
1
19.2667
19.2667
9.44*
P * G
2
20.6333
10.3165
5.06*
P * Ss w/Grps
27
55.1000
2.0407
Total
59
254.733
*
p
<
.
05
[
F
.
05
(
1
,
27
)
=
4
.
22
;
F
.
05
(
2
,
27
)
=
3
.
36
]
b.Plot:
B
B
J
J
H
H
Baseline
Training
4
4.5
5
5.5
6
6.5
7
Phase
B
Exp
J
Att Cont
H
No Att Cont
c.There seems to be no difference between the Experimental and
Attention groups, but both show significantly more improvement than
the No Attention group.
14.6Summarization of stories by adult and children good and poor
readers:
Cell Means for Age * Readers * Items:
Adults
Items:
Setting
Goal
Disp
Mean
Good Readers
6.20
6.0
5.0
5.73
Poor Readers
5.40
4.8
2.0
4.07
Mean
5.80
5.40
3.50
4.90
Children
Items:
Setting
Goal
Disp
Mean
Good Readers
5.80
5.60
3.00
4.80
Poor Readers
3.00
2.40
1.20
2.20
Mean
4.40
4.00
2.10
3.50
Cell Means for Age * Readers:
Adults
Children
Mean
Good Readers
5.73
4.80
5.27
Poor Readers
4.07
2.20
3.13
Mean
4.90
3.50
4.20
Cell Means for Age * Items:
Adults
Children
Mean
Setting
5.80
4.40
5.10
Goal
5.40
4.00
4.70
Disposition
3.50
2.10
2.80
Mean
4.90
3.50
4.20
Cell Means for Reader * Items:
Good Readers
Poor Readers
Mean
Setting
6.00
4.20
5.10
Goal
5.80
3.60
4.70
Disposition
4.00
1.60
2.80
Mean
5.27
3.13
4.20
Subject Means:
Good Adult Readers:
7.00
5.00
5.00
7.00
4.67
Good Children Readers:
4.00
6.33
6.00
4.33
3.33
Poor Adult Readers:
5.33
3.00
4.67
3.00
4.33
Poor Children Readers:
2.00
1.00
3.33
3.33
1.33
(
)
2
2
2
252
1312253.600
60
total
X
SSX
N
=-=-=
å
å
(
)
(
)
2
2
2
....
3[7.004.20...(1.334.20)]164.933
subjijki
SSiXX
=S-
=-++-=
(
)
(
)
(
)
(
)
(
)
2
22
.......
325[4.904.203.504.20]29.400
agej
SSirnXX
=S-
=-+-=
(
)
(
)
(
)
(
)
(
)
2
22
.......
235[5.274.203.134.20]68.267
readeri
SSainXX
=S-
=-+-=
(
)
(
)
(
)
(
)
(
)
2
22
......
355.734.20...2.204.20100.933
cellsARij
SSinXX
=S-
éù
=-++-=
ëû
100.93329.40068.2673.267
ARcellsARagereader
SSSSSSSS
=--=--=
(
)
(
)
(
)
(
)
(
)
2
22
2
.......
225[5.104.204.704.20(2.804.20)]60.400
itemj
SSarnXX
=S-
=-+-+-=
(
)
(
)
(
)
(
)
2
22
......
255.804.20...2.104.2089.800
cellsAIjk
SSrnXX
=S-
éù
=-++-=
ëû
89.80029.40060.4000.00
AIcellsAIageitem
SSSSSSSS
=--=--=
(
)
(
)
(
)
(
)
2
22
.......
256.004.20...1.604.20129.600
cellsRIik
SSanXX
=S-
éù
=-++-=
ëû
129.60068.26760.4000.933
RIcellsRIreaderitem
SSSSSSSS
=--=--=
(
)
(
)
(
)
2
22
.....
56.204.20...1.204.20170.800
cellsARIijk
SSnXX
=S-
éù
=-++-=
ëû
170.80029.40068.26760.4003.2670.0000.933
8.533
ARIcellsARIagereaderitemARAIRI
SSSSSSSSSSSSSSSS
=------
=------=
14.7From Exercise 14.6:
a.Simple effect of reading ability for children:
(
)
(
)
(
)
(
)
2
22
35[4.803.502.203.50]50.70
RatCRatCC
SSinXX
=S-
=-+-=
50.70
50.70
1
RatC
RatC
RatC
SS
MS
df
===
Because we are using only the data from Children, it would be
wise not to use a pooled error term. The following is the relevant
printout from SPSS for the Between-subject effect of Reader.
Tests of Between-Subjects Effects
a
Measure: MEASURE_1
Transformed Variable: Average
367.500
1
367.500
84.483
.000
50.700
1
50.700
11.655
.009
34.800
8
4.350
Source
Intercept
READERS
Error
Type III Sum
of Squares
df
Mean Square
F
Sig.
AGE = Children
a.
b.Simple effect of items for adult good readers:
(
)
(
)
(
)
(
)
2
222
5[6.205.736.005.735.005.73]4.133
IatAGIatAGAG
SSnXX
=S-
=-+-+-=
Again, we do not want to pool error terms. The following is the
relevant printout from SPSS for Adult Good readers. The difference
is not significant, nor would it be for any decrease in the df if
we used a correction factor.
Tests of Within-Subjects Effects
Measure: MEASURE_1
Sphericity Assumed
4.133
2
2.067
3.647
.075
4.533
8
.567
Source
ITEMS
Error(ITEMS)
Type III Sum
of Squares
df
Mean Square
F
Sig.
14.8Within-groups covariance matrices for the data in Exercise
14.10:
S
ˆ
within
AG
=
1
.
70
1
.
25
1
.
00
1
.
25
2
.
50
1
.
25
1
.
00
1
.
25
1
.
00
S
ˆ
within
CG
=
1
.
70
1
.
90
1
.
25
1
.
90
3
.
30
1
.
50
1
.
25
1
.
50
1
.
00
S
ˆ
within
AP
=
1
.
30
1
.
10
0
.
75
1
.
10
1
.
70
1
.
00
0
.
75
1
.
00
1
.
00
S
ˆ
within
CP
=
2
.
00
2
.
00
0
.
25
2
.
00
2
.
80
0
.
40
0
.
25
0
.
40
0
.
70
S
ˆ
pooled
=
1
.
6750
1
.
5625
0
.
8125
1
.
5625
2
.
5750
1
.
0375
0
.
8125
1
.
0375
0
.
9250
14.9It would certainly affect the covariances because we would
force a high level of covariance among items. As the number of
responses classified at one level of Item went up, another item
would have to go down.
14.10Cigarette smoking quitting techniques:
a.Analysis:
Cell Means for Group * Time * Place:
Pre
Post
Home
Work
Home
Work
Mean
Taper
6.80
6.00
5.80
3.60
5.55
Immediate
7.00
6.20
5.80
4.80
5.95
Aversion
7.00
6.20
4.80
2.40
5.10
Mean
6.93
6.13
5.47
3.60
5.53
Means Group * Time:
Means Group * Place:
Pre
Post
Mean
Home
Work
Mean
Taper
6.40
4.70
5.55
6.30
4.80
5.55
Immediate
6.60
5.30
5.95
6.40
5.50
5.95
Aversion
6.60
3.60
5.10
5.90
4.30
5.10
Mean
6.53
4.53
5.53
6.20
4.87
5.53
Time * Place:
Pre
Post
Total
Home
6.93
5.47
6.20
Work
6.13
3.60
4.87
Tot
6.53
4.53
5.53
Subject * Time:
Pre
6.5
4.5
7.5
8.0
5.5
7.5
5.0
6.5
7.5
6.5
8.5
4.0
7.0
6.0
7.5
Post
5.0
3.5
5.5
5.5
4.0
6.5
4.5
5.5
5.5
4.5
4.5
2.5
4.0
2.5
4.5
Mean
5.75
4.00
6.50
6.75
4.75
7.00
3.75
6.00
6.50
5.50
6.50
3.25
5.50
4.25
6.00
Subject * Place:
Home
6.5
5.0
7.5
7.0
5.5
7.5
5.0
6.5
7.0
6.0
7.0
3.5
6.0
6.0
7.0
Work
5.0
3.0
5.5
5.5
4.0
6.5
4.5
5.5
6.0
5.0
6.0
3.0
5.0
2.5
5.0
(
)
2
2
2
332
2024186.933
60
total
X
SSX
N
=-=-=
å
å
(
)
(
)
(
)
(
)
2
2
2
.......
22[5.755.53...(6.005.53)]69.433
subjl
SStpXX
=S-
=-++-=
(
)
(
)
(
)
(
)
(
)
(
)
2
222
...
....
225[5.555.535.955.535.105.53]7.233
groupi
SStpnXX
=S-
=-+-+-=
(
)
(
)
(
)
(
)
(
)
2
22
.......
325[6.535.534.535.53]60.000
timej
SSgpnXX
=S-
=-+-=
(
)
(
)
(
)
2
22
......
(2)6.505.53...4.505.53143.933
cellsTSijl
SSpnXX
=S-
éù
=-++-=
ëû
143.93360.00069.43314.500
TScellsTStimesubj
SSSSSSSS
=--=--=
(
)
(
)
(
)
(
)
2
22
......
256.405.53...3.605.5375.133
cellsGTij
SSpnXX
=S-
éù
=-++-=
ëû
75.1337.23360.0007.900
GTcellsGTgrouptime
SSSSSSSS
=--=--=
(
)
(
)
(
)
(
)
(
)
2
22
.......
325[6.205.534.875.5326.667
placek
SSgtnXX
=S-
=-+-=
(
)
(
)
(
)
2
22
......
26.505.53...5.005.53104.933
cellsPSkl
SStXX
=S-
éù
=-++-=
ëû
104.93326.66769.4338.833
PScellsPSplacesubj
SSSSSSSS
=--=--=
(
)
(
)
(
)
(
)
2
22
.......
256.305.53...4.305.5335.333
cellsGPik
SStnXX
=S-
éù
=-++-=
ëû
35.3337.23326.6671.433
GPcellsGPgroupplace
SSSSSSSS
=--=--=
(
)
(
)
(
)
(
)
2
22
.......
356.935.53...3.605.5390.933
cellsTPjk
SSgnXX
=S-
éù
=-++-=
ëû
90.93360.00026.6674.267
TPcellsTPtimeplace
SSSSSSSS
=--=--=
186.933
cellsTPStotal
SSSS
==
186.93360.00026.66769.43314.5008.8334.26
73.233
TPScellsTPStimeplacesubjTSPSTP
SSSSSSSSSSSSSSSS
=------
=------=
(
)
(
)
(
)
2
22
.....
56.805.53...2.405.53108.933
cellsGTPijk
SSnXX
=S-
éù
=-++-=
ëû
108.9337.23360.00026.6677.9001.4334.2671
.433
GTPgrouptimeplaceGTGPTP
SSSSSSSSSSSSSS
=-----
=------=
Source
df
SS
MS
F
Between subj
14
69.433
Group
2
7.233
3.617
0.70
Ss w/in grp
12
62.200
5.183
Within subj
45
116.500
Time
1
60.000
60.000
109.09*
TxS
14
14.500
GxT
2
7.900
3.950
7.18*
GxTw/in grps
12
6.600
0.550
Place
1
26.667
26.667
43.24*
PxS
14
8.833
GxP
2
1.433
0.717
1.16
GxPw/in grps
12
7.400
0.617
TxP
1
4.267
4.267
28.44*
TxPxS
14
3.233
TxPxG
2
1.433
0.717
4.78*
TxPxSw/in grp
12
1.800
0.150
Total
59
186.933
*p < .05 [F.05(1,12) = 4.75; F.05(2,12) = 3.89]
b.There is a significant decrease in desire from Pre to Post and
a significant reduction at Work relative to at Home. There is also
a Time by Place interaction, with a greater Place difference after
treatment. The Time by Group interaction is the real test of our
hypothesis, showing that the Pre-Post difference depends on the
treatment group, with the greatest difference in the Aversion
condition. The 3-way interaction shows that the Time by Group
interaction itself interacts with Place.
14.11Plot of results in Exercise 14.10:
B
B
J
J
H
H
Pre
Post
3
3.5
4
4.5
5
5.5
6
6.5
7
Time
B
Taper
J
Immediate
H
Aversion
14.12I will look at the Group × Time interaction by looking at
the simple effect of Time for each Group. For a more complex
design, we should run separate analyses for each group, to avoid
problems with sphericity. However, with only two levels of the
repeated measures variables, we have only one off-diagonal
covariance, so we don’t have a problem with constant covariances. I
will test each by the same test term as was used to test the Group
× Time interaction, MSGxTxSs w/in groups (The pattern of
significance would not change with separate analyses. The Fs if we
used separate analyses would be 44.46, 18.78, and 51.43,
respectively.)
(
)
(
)
(
)
(
)
2
11
22
.....
256.405.554.705.5514.45
timeattaperj
SSpnXX
=S=
éù
=-+-=
ëû
14.45
14.45
1
14.45
26.27*
0.55
tatt
timeattaper
tatt
tatt
timeattaper
error
SS
MS
df
MS
F
MS
===
===
(
)
(
)
(
)
(
)
2
22
22
.....
256.605.955.305.958.45
timeatimmedj
SSpnXX
=S=
éù
=-+-=
ëû
8.45
8.45
1
8.45
15.36*
0.55
tati
timeatimmed
tati
tati
timeatimmed
error
SS
MS
df
MS
F
MS
===
===
(
)
(
)
(
)
(
)
2
33
22
256.605.103.605.1045.00
...
..
timeataverj
SSpnXX
=S=
éù
=-+-=
ëû
15.00
45.00
1
45.00
81.82*
0.55
tata
timeataver
tata
tati
timeatimmed
error
SS
MS
df
MS
F
MS
===
===
*
p
<
.
05
[
F
.
05
(
1
,
12
)
=
4
.
75
]
These SSSimple Effects sum to the same total as SSTime and SSG *
T:
14.45 + 8.45 + 45.00 = 67.90 = 60.00 + 7.90
Each of the methods led to a significant reduction in desire to
smoke. If we then look at the effect of Group at Post:
(
)
(
)
(
)
(
)
(
)
2
122
222
.....
25[4.704.535.304.533.604.53]14.867
groupatpost
SSpnXX
=S-
=-+-+-=
/
//
/
/
14.867
7.433
2
66.2006.600
3.033
1212
GxTxSswingrps
groupatpost
groupatpost
groupatpost
SswingrpsGxTxSswingrps
wincell
Sswingrps
SS
MS
df
SSSS
MS
dfdf
===
+
+
===
++
/
7.433
2.45
3.033
GatP
groupatpost
wincell
MS
F
MS
===
This would not be significant even for the maximum possible
value of f ', meaning that we do not have data to allow us to
recommend one method over the other. (If we had run a separate
analysis on just the Posttest data, the corresponding F would have
been 4.33, with p = .038.)
14.13Analysis of data in Exercise 14.5 by BMDP:
a.Comparison with results obtained by hand in Exercise 14.5.
b.The F for Mean is a test on H0: = 0.
c.MSw/in Cell is the average of the cell variances.
14.14An analysis of data in Exercise 14.6 by SPSS as if it were
a factorial:
From Exercise 14.6:
/ */
/
/ */
64.00018.800
1.725
1632
SswingroupsISswngroups
wincell
SswingroupsISswngroups
MSMS
MS
dfdf
+
+
===
++
This equals the MSResidual in the SPSS printout.
14.15Source column of summary table for 4-way ANOVA with
repeated measures on A & B and independent measures on C &
D.
Source
Between Ss
C
D
CD
Ss w/in groups
Within Ss
A
AC
AD
ACD
A x Ss w/in groups
B
BC
BD
BCD
B x Ss w/in groups
AB
ABC
ABD
ABCD
AB x Ss w/in groups
Total
14.16Analysis of Foa et al. (1991) data
All three groups decreased over time, but the Supportive
Counseling group decreased the least and the interaction was
significant.
14.17Using the mixed models procedure on data from Exercise
14.16
If we assume that sphericity is a reasonable assumption, we
could run the analysis with covtype(cs). That will give us the
following, and we can see that the F’s are the same as they were in
our analysis above.
However, the correlation matrix below would make us concerned
about the reasonableness of a sphericity assumption. (This matrix
is collapsed over groups, but reflects the separate matrices well.)
Therefore we will assume an autoregressive model for our
correlations.
These F values are reasonably close, but certainly not the
same.
14.18Standard analysis with missing data:
Notice that we have lost considerable degrees of freedom and our
F for Group is no longer significant
14.19Mixed model analysis with unequal size example.
Notice that we have a substantial change in the F for Time,
though it is still large.
14.20Analysis of Stress data:
Source
df
SS
MS
F
Pillai F
Prob
Between subj
97
137.683
Gender
1
7.296
7.296
5.64*
Role
1
8.402
8.402
6.49*
G * R
1
0.298
0.298
<1
Ss w/in Grps
94
121.687
1.294
Within subj
97
87.390
Time
1
1.064
1.064
1.23*
1.23
0.2700
T*G
1
0.451
0.451
<1
0.52
0.4720
T*R
1
0.001
0.001
<1
0.00
0.9708
T*G*R
1
4.652
4.652
5.38*
5.38
0.0225
T*Ss w/in grps
94
81.222
0.864
Total
194
103.386
*p < .05
The univariate and multivariate F values agree because we have
only two levels of each independent variable.
14.21Everitt’s study of anorexia:
a.SPSS printout on gain scores:
Tests of Between-Subjects Effects
Dependent Variable: GAIN
614.644
a
2
307.322
5.422
.006
732.075
1
732.075
12.917
.001
614.644
2
307.322
5.422
.006
3910.742
69
56.677
5075.400
72
4525.386
71
Source
Corrected Model
Intercept
TREAT
Error
Total
Corrected Total
Type III Sum
of Squares
df
Mean Square
F
Sig.
R Squared = .136 (Adjusted R Squared = .111)
a.
b.SPSS printout using pretest and posttest:
Tests of Within-Subjects Effects
Measure: MEASURE_1
Sphericity Assumed
366.037
1
366.037
12.917
.001
307.322
2
153.661
5.422
.006
1955.371
69
28.339
Source
TIME
TIME * TREAT
Error(TIME)
Type III Sum
of Squares
df
Mean Square
F
Sig.
c.The F comparing groups on gain scores is exactly the same as
the F for the interaction in the repeated measures design.
d.
TREAT: 1.00 Cognitive Behavioral
PRETEST
100
90
80
70
60
POSTTEST
110
100
90
80
70
EMBED StaticEnhancedMetafile
TREAT: 2.00 Control
PRETEST
100
90
80
70
POSTTEST
100
90
80
70
TREAT: 3.00 Family Therapy
PRETEST
100
90
80
70
POSTTEST
110
100
90
80
70
The plots show that there is quite a different relationship
between the variables in the different groups.
e.Treatment Group = Control
One-Sample Statistics
a
26
-.4500
7.9887
1.5667
GAIN
N
Mean
Std. Deviation
Std. Error Mean
Treatment Group = Control
a.
One-Sample Test
a
-.287
25
.776
-.4500
-3.6767
2.7767
GAIN
t
df
Sig. (2-tailed)
Mean
Difference
Lower
Upper
95% Confidence Interval
of the Difference
Test Value = 0
Treatment Group = Control
a.
This group did not gain significantly over the course of the
study. This suggests that any gain we see in the other groups
cannot be attributed to normal gains seen as a function of age.
f.Without the control group we could not separate gains due to
therapy from gains due to maturation.
14.22When multiple respondents come from the same family, their
data are not likely to be independent. We act as if we have 98
different respondents, but in fact we do not have 98 independent
respondents, which is important. If we had complete data from each
family we could treat Spouse and Patient as a repeated measures
variable—it is a "within-family" variable. Alternatively, we could
delete data so as to have only one respondent per family. In this
situation, numerous studies have shown that there is a remarkably
small degree of dependence between members of the same family, and
many people would ignore the problem entirely.
14.23t = -0.555. There is no difference in Time 1 scores between
those who did, and did not, have a score at Time 2.
b.If there had been differences, I would worried that people did
not drop out at random.to answer.
14.24Intraclass correlation:
(
)
(
)
(
)
(
)
1/
82.574.08
82.5724.08370.124.08/20
78.49
.85
82.578.166.30
subjectsIxS
subjectsJxSjudgeJxS
MSMS
IC
MSjMSjMSMSn
-
=
+-+-
-
=
++-
==
++
The remainder of this exercise raises some questions that anyone
interested in the reliability of their data (and we all should be)
needs to be prepared
14.25Differences due to Judges play an important role.
14.26I would leave the variability due to Judge out of my
calculations entirely.
14.27If I were particularly interested in differences between
subjects, and recognized that judges probably didn’t have a good
anchoring point, and if this lack was not meaningful, I would not
be interested in considering it.
14.28The fact that the “parent” who supplies the data changes
from case to case simply adds additional variability to our data,
and this variability is confounded with differences between
children
14.29Strayer et al. (2006)
b.Contrasts on means:
Because the variances within each condition are so similar, I
have used MSerror(within) as my error term. The means are 776.95,
778.95, and 849.00 for Baseline, Alcohol, and Cell phone
conditions, respectively..
2
12
13
23
2
12
*
13
*
23
ˆ
ˆ
776.95778.952
ˆ
776.95849.0072.05
ˆ
778.95849.0070.5
216303.709
28.551
40
2/28.5510.07
72.05/28.5512.52
70.05/28.5512.45
ierror
vs
vs
vs
ierror
vs
vs
vs
t
aMS
n
aMS
den
n
t
t
t
y
y
y
y
=
S
=-=
=-=
=-=
S
´
===
==
==
==
Both Baseline and Alcohol conditions show poorer performance
than the cell phone condition, but, interestingly, the Baseline and
Alcohol conditions do not differ from each other.
14.30 Study by Teri et al. (1997):
The following are the results from SPSS. As we would expect, the
interaction is significant. In fact, the F for the interaction is
exactly equal to the F for the Group effect on Change in Exercise
11-37, and the sum of squares for both the interaction and the
error term are exactly half of what they were in Exercise
11-37.
Tests of Within-Subjects Effects
Measure:MEASURE_1
Source
Type III Sum of Squares
df
Mean Square
F
Sig.
PrePost
Sphericity Assumed
164.360
1
164.360
32.557
.000
Greenhouse-Geisser
164.360
1.000
164.360
32.557
.000
Huynh-Feldt
164.360
1.000
164.360
32.557
.000
Lower-bound
164.360
1.000
164.360
32.557
.000
PrePost * Group
Sphericity Assumed
195.695
3
65.232
12.922
.000
Greenhouse-Geisser
195.695
3.000
65.232
12.922
.000
Huynh-Feldt
195.695
3.000
65.232
12.922
.000
Lower-bound
195.695
3.000
65.232
12.922
.000
Error(PrePost)
Sphericity Assumed
343.285
68
5.048
Greenhouse-Geisser
343.285
68.000
5.048
Huynh-Feldt
343.285
68.000
5.048
Lower-bound
343.285
68.000
5.048
Tests of Between-Subjects Effects
Measure:MEASURE_1
Transformed Variable:Average
Source
Type III Sum of Squares
df
Mean Square
F
Sig.
Intercept
26031.127
1
26031.127
911.469
.000
Group
42.771
3
14.257
.499
.684
Error
1942.048
68
28.560
Chapter 15 - Multiple Regression
15.1Predicting Quality of Life:
a.All other variables held constant, a difference of +1 degree
in Temperature is associated with a difference of –.01 in perceived
Quality of Life. A difference of $1000 in median Income, again all
other variables held constant, is associated with a +.05 difference
in perceived Quality of Life. A similar interpretation applies to
b3 and b4. Since values of 0.00 cannot reasonably occur for all
predictors, the intercept has no meaningful interpretation.
b.
Y
ˆ
=
5
.
37
-
.
01
(
55
)
+
.
05
(
12
)
+
.
003
(
500
)
-
.
01
(
200
)
=
4
.
92
c.
Y
ˆ
=
5
.
37
-
.
01
(
55
)
+
.
05
(
12
)
+
.
003
(
100
)
-
.
01
(
200
)
=
3
.
72
15.2A difference of +1 standard deviation in Temperature is
associated with a difference of ‑.438 standard deviations in
perceived quality of life, while a difference of +1 standard
deviation in Income is associated with about three quarters of a
standard deviation difference in perceived Quality of Life. A
similar interpretation can be made for the other variables, but in
all cases it is assumed that all variables are held constant except
for the one in question.
15.3The F values for the four regression coefficients would be
as follows:
F
1
=
b
1
s
b
1
2
=
-
0
.
438
0
.
397
2
=
1
.
22
F
2
=
b
2
s
b
2
2
=
0
.
762
0
.
252
2
=
9
.
14
F
3
=
b
3
s
b
3
2
=
0
.
081
0
.
052
2
=
2
.
43
F
4
=
b
4
s
b
4
2
=
-
0
.
132
0
.
025
2
=
27
.
88
I would thus delete Temperature, since it has the smallest F,
and therefore the smallest semi-partial correlation with the
dependent variable.
15.4Predicting Job Satisfaction:
a.
Y
ˆ
=
.
605
Respon
-
.
334
NumSup
+
.
486
Envir
+
.
070
Yrs
+
1
.
669
b.[.624 -.311 .514 .063]
15.5a.Envir has the largest semi-partial correlation with the
criterion, because it has the largest value of t.
b.The gain in prediction (from r = .58 to R = .697) which we
obtain by using all the predictors is more than offset by the loss
of power we sustain as p became large relative to N.
15.6Adjusted R2 for the data in Exercise 15.4:
est
R
*
2
=
1
-
(
1
-
R
2
)
(
N
-
1
)
(
N
-
p
-
1
)
=
1
-
(
1
-
.
4864
)
(
14
)
(
15
-
4
-
1
)
=
.
2810
15.7As the correlation between two variables decreases, the
amount of variance in a third variable that they share decreases.
Thus the higher will be the possible squared semi-partial
correlation of each variable with the criterion. They each can
account for more previously unexplained variation.
15.8The more highly two predictor variables are intercorrelated,
the more "substitutability" there is between them. Thus for one
data set variable X1 may receive the greater weight, but for a
second data set variable X2 may happen to "substitute" for X1,
leaving X1 with only a minor role to play.
15.9The tolerance column shows us that NumSup and Respon are
fairly well correlated with the other predictors, whereas Yrs is
nearly independent of them.
15.10For the data in Exercise 15.4:
Y
ˆ
=
.
605
Respon
-
.
334
NumSup
+
.
486
Envir
+
.
070
Yrs
+
1
.
669
Satisfaction
Y
ˆ
2
3.26
2
2.85
3
5.90
3
4.63
5
4.18
5
8.15
6
4.73
6
5.59
6
6.91
7
5.99
8
7.86
8
5.62
8
5.93
9
6.86
9
8.55
µ
µ
µ
(
)
(
)
0.1234
2.426
1.693
2.864
2.846
.697
2.4261.693
Y
Y
YY
YY
s
s
s
rR
=
=
=
===
15.11Using Y and
µ
Y
from Exercise 15.10:
(
)
2
ˆ
1
42.322
4.232 (also calculated by BMDP in Exerc
ise 15.4)
1541
residual
YY
MS
Np
-
=
--
==
--
å
15.12The effect of sample size on the multiple correlation,
using random data:
For N = 15, R2 = .173
For N = 10, R2 = .402
For N = 6, R2 = .498 Notice that the correlation increases as N
- p decreases.
For N = 5, R2 = 1.000 Here N is equal to the number of
variables.
For N = 4 the matrix is singular and there is no solution.
15.13Adjusted R2 for 15 cases in Exercise 15.12:
R
2
0
.
1234
=
.
173
est
R
*
2
=
1
-
(
1
-
R
2
)
(
N
-
1
)
(
N
-
p
-
1
)
=
1
-
(
1
-
.
173
)
(
14
)
(
15
-
4
-
1
)
=
-
.
158
Since a squared value cannot be negative, we will declare it
undefined. This is all the more reasonable in light of the fact
that we cannot reject H0:R* = 0.
15.14Using the first three variables from Exercise 15.4:
a.The squared semi-partial correlation is .32546 – .31654 =
.00892.
The squared partial correlation is .00892/(1 – .32546) =
.01305.
b.Venn diagram illustrating squared partial and semi-partial
correlations for Satisfaction predicted by Number Supervised,
partialling out Responsibility.
NumSup
Respon
R
0(2.1)
2
R
2
0.12
15.15Using the first three variables from Exercise 15.4:
a.Figure comparable to Figure 15.1:
b.
µ
Y
= 0.4067Respon + 0.1845NumSup + 2.3542
The slope of the plane with respect to the Respon axis (X1) =
.4067
The slope of the plane with respect to the NumSup axis (X2) =
.1845
The plane intersects the Y axis at 2.3542
15.16Predicting percentage of low-birthweight live births from
Vermont Health Planning statistics:
a.
R
0
.
2
=
.
6215
R
0
.
25
=
.
7748
R
0
.
2
54
=
.
8181
b.
r
2
0
(
5
.
2
)
=
r
2
05
.
2
(
1
-
R
2
0
.
2
)
=
(
-
.
59063
2
)
(
1
-
.
3862
)
=
.
2141
R
2
0
.
25
=
R
2
0
.
2
+
r
2
0
(
5
.
2
)
=
.
3862
+
.
2141
=
.
6003
=
.
7748
2
15.17It has no meaning in that we have the data for the
population of interest (the 10 districts).
15.18The gain in R2 is not sufficient to offset the gain in p
relative to N.
15.19It plays a major role through its correlation with the
residual components of the other variables.
15.20For the data in Exercise 15.16:
Y
µ
Y
1
µ
Y
2
6.1
34.2
4.48
7.1
37.3
5.12
7.4
43.9
5.57
6.3
36.6
4.74
6.5
41.0
5.22
5.7
29.6
3.87
6.6
33.0
4.51
8.1
43.0
5.40
6.3
30.6
4.32
6.9
41.3
5.48
µ
Y
1 = 1X2 + 1X4 – 3X5
µ
Y
2 = .10446X2 + .189720X4 – .29372X5
Using the approximate regression coefficients the correlation
between Y and
µ
Y
would be 0.793 instead of the 0.818 which would be calculated
from the regression of Y and the
µ
Y
obtained using the actual regression equation.
15.21Within the context of a multiple-regression equation, we
cannot look at one variable alone. The slope for one variable is
only the slope for that variable when all other variables are held
constant. The percentage of mothers not seeking care until the
third trimester is correlated with a number of other variables.
15.22Create set of data illustrating leverage, distance, and
influence.
15.23Create set of data examining residuals.
15.24Modeling depression
DepressT
=
0
.
614
PVLoss
-
0
.
164
SuppTotl
-
0
.
106
AgeAtLoss
+
59
.
961
R
2
=
.
2443
F
.
05
(
3
,
131
)
=
14
.
115
Both PVLoss and SuppTotl are significant predictors, but
AgeAtLoss is not.
15.25Rerun of Exercise 15.24 adding PVTotal.
b.The value of R2 was virtually unaffected. However, the
standard error of the regression coefficient for PVLoss increased
from 0.105 to 0.178. Tolerance for PVLoss decreased from .981 to
.345, whereas VIF increased from 1.019 to 2.900. (c) PVTotal should
not be included in the model because it is redundant with the other
variables.
15.26Vulnerability as a function of social support and age at
loss.
PVLoss
=
.
041
SuppTotl
+
.
164
AgeAtLos
+
17
.
578
R
2
=
.
0203
;
F
=
1
.
42
;
F
.
05
(
2
,
137
)
=
3
.
06
The relationship is not significant.
Neither of the predictors is significant.
15.27Path diagram showing the relationships among the variables
in the model.
SuppTotl
PVLoss
AgeAtLoss
DepressT
-.2361
.0837
.4490
.1099
-.0524
15.28The direct effect of SuppTotl is represented by the arrow
that goes from SuppTotl to DepressT. This is a semi-partial
relationship because PVLoss was in the model to begin with. The
indirect effect of SuppTotl is the path that runs from SuppTotl to
PVLoss and then runs from PVLoss to DepressT. The expected change
in DepressT for a one standard deviation increase in SuppTotl (all
other things equal) is the sum of the direct and indirect effects
as measured by the standardized coefficients. In this case it is
–.2361 + (.0837)(.4490) = –.1985. (Keep in mind that the indirect
path is far from significant, and it is discussed here just to
illustrate a point.)
15.29Regression diagnostics.
Case # 104 has the largest value of Cook's D (.137) but not a
very large Studentized residual (t = –1.88). When we delete this
case the squared multiple correlation is increased slightly. More
importantly, the standard error of regression and the standard
error of one of the predictors (PVLoss) also decrease slightly.
This case is not sufficiently extreme to have a major impact on the
data.
15.30Adding error to a predictor.
As we add error to a predictor we should expect to see that the
standard error of that predictor will increase and its significance
decrease. For my particular example the standard error actually
decreased slightly. The most noticeable results in this example
were a substantial decrease in the multiple correlation
coefficient, and a corresponding increase in the residual
variance.
15.31Logistic regression using Harass.dat:
The dependent variable (Reporting) is the last variable in the
data set.
I cannot provide all possible models, so I am including just the
most complete. This is a less than optimal model, but it provides a
good starting point. This result was given by SPSS.
Block 1: Method = Enter
Omnibus Tests of Model Coefficients
35.442
5
.000
35.442
5
.000
35.442
5
.000
Step
Block
Model
Step 1
Chi-square
df
Sig.
Model Summary
439.984
.098
.131
Step
1
-2 Log
likelihood
Cox & Snell
R Square
Nagelkerke
R Square
Classification Table
a
111
63
63.8
77
92
54.4
59.2
Observed
No
Yes
REPORT
Overall Percentage
Step 1
No
Yes
REPORT
Percentage
Correct
Predicted
The cut value is .500
a.
Variables in the Equation
-.014
.013
1.126
1
.289
.986
-.072
.234
.095
1
.757
.930
.007
.015
.228
1
.633
1.007
-.046
.153
.093
1
.761
.955
.488
.095
26.431
1
.000
1.629
-1.732
1.430
1.467
1
.226
.177
AGE
MARSTAT
FEMIDEOL
FREQBEH
OFFENSIV
Constant
Step 1
a
B
S.E.
Wald
df
Sig.
Exp(B)
Variable(s) entered on step 1: AGE, MARSTAT, FEMIDEOL, FREQBEH,
OFFENSIV.
a.
From this set of predictors we see that overall LR = 35.44,
which is significant on 5 df with a p value of .0000 (to 4 decimal
places). The only predictor that contributes significantly is the
Offensiveness of the behavior, which has a Wald of 26.43. The
exponentiation of the regression coefficient yields 0.9547. This
would suggest that as the offensiveness of the behavior increases,
the likelihood of reporting decreases. That’s an odd result. But
remember that we have all variables in the model. If we simply
predicting reporting by using Offensiveness, exp(B) = 1.65, which
means that a 1 point increase in Offensiveness multiplies the odds
of reporting by 1.65. Obviously we have some work to do to make
sense of these data. I leave that to you.
15.32Predicting Reporting from Marital Status:
I requested this because both variables are a dichotomy and it
is easy to see the odds ratios. If we set this up as a contingency
table using the CrossTabs procedure, we get the following from
SPSS:
MARSTAT * REPORT Crosstabulation
Count
84
84
168
90
85
175
174
169
343
Married
Not Married
MARSTAT
Total
No
Yes
REPORT
Total
Chi-Square Tests
.070
b
1
.791
.024
1
.876
.070
1
.791
.829
.438
.070
1
.792
343
Pearson Chi-Square
Continuity Correction
a
Likelihood Ratio
Fisher's Exact Test
Linear-by-Linear Association
N of Valid Cases
Value
df
Asymp. Sig.
(2-sided)
Exact Sig.
(2-sided)
Exact Sig.
(1-sided)
Computed only for a 2x2 table
a.
0 cells (.0%) have expected count less than 5. The minimum
expected count is 82.78.
b.
For married women, the odds of reporting are 84/84 = 1.00
For unmarried women, the odds of reporting are 85/90 = .944
The odds ratio is 1.059, which means that you are 1.059 times
more likely to report the offense if you are married. Put the other
way around, you are .944/1.00 = .944 times more likely to report
the offensive behavior if you are unmarried. Since the odds are
less than 1.00, being unmarried means that you are less likely to
report.
If we run the logistic regression, we obtain:
Classification Table
a
90
84
51.7
85
84
49.7
50.7
Observed
No
Yes
REPORT
Overall Percentage
Step 1
No
Yes
REPORT
Percentage
Correct
Predicted
The cut value is .500
a.
Variables in the Equation
-.057
.216
.070
1
.791
.944
.057
.344
.028
1
.868
1.059
MARSTAT
Constant
Step 1
a
B
S.E.
Wald
df
Sig.
Exp(B)
Variable(s) entered on step 1: MARSTAT.
a.
Notice that the exp(B) = .9444, which is exactly what you
obtained above for the odds ratio given that you are unmarried.
That was the point of the exercise. Notice also how the
classification table here matches with the one in the CrossTabs
procedure, except that the rows and columns are now labeled
“Observed” and “Predicted.”
15.33It may well be that the frequency of the behavior is tied
in with its offensiveness, which is related to the likelihood of
reporting. In fact, the correlation between those two variables is
.20, which is significant at p < .000. (I think my explanation
would be more convincing if Frequency were a significant predictor
when used on its own.)
15.34Malcarne’s data on distress in cancer patients,
a.Predicting Distress2 from Distress1 and BlamPer
Both predictors play a significant role in Distress2
b.We need to include Distress1 in our prediction because it is
very reasonable to assume that initial distress would relate to
later distress, and we want to control for that effect when looking
at the effect of BlamPer.
15.35BlamPer and BlamBeh are correlated at a moderate level (r =
.52), and once we condition on BlamPer by including it in the
equation, there is little left for BlamBeh to explain.
15.36I want students to think about what it means when we speak
of “capitalizing on chance.” They also should think about the fact
that stepwise regression is a very atheoretical way of going about
things, and perhaps theory should take more of a role.
15.37Make up an example.
15.38They should see no change in the interaction term when they
center the data, but they should see important differences in the
“main effects” themselves. Have them look at the matrix of
intercorrelations of the predictors.
15.39This should cause them to pause. It is impossible to change
one of the variables without changing the interaction in which that
variable plays a role. In other words, I can’t think of a sensible
interpretation of “holding all other variables constant” in this
situation.
15.40Testing mediation with Jose’s data.
(
)
(
)
(
)
(
)
(
)
222222222222
.478.022.321.017.017.022
0.000150.0123
.478.321
12.47
.012
ab
ab
abbaab
ab
sssss
z
s
bb
bb
bb
bb
=+-=+-
==
===
15.41Analysis of results from Feinberg and Willer (2011).
The following comes from using the program by Preacher and
Leonardelli referred to in the chapter. I calculated the t values
from the regression coefficients and their standard errors and then
inserted those t values in the program. You can see that the
mediated path is statistically significant regardless of which
standard error you use for that path.
15.42Guber’s results:
This is a computer analysis question and there is no fixed set
of answers.
Chapter 16 - Analyses of Variance and Covariance as General
Linear Models
16.1Eye fixations per line of text for poor, average, and good
readers:
a.Design matrix, using only the first subject in each group:
X
=
1
0
-
1
0
1
-
1
b.Computer exercise:
R
2
=
.
608
SS
reg
=
57
.
7333
SS
residual
=
37
.
2000
c.Analysis of variance:
X
1 = 8.2000
X
2 = 5.6
X
3 = 3.4
X
. = 5.733
n1 = 5n2 = 5
n3 = 5
N = 15
X = 86X 2 = 588
(
)
(
)
(
)
(
)
22
2
2
222
.
()86
58894.933
15
5[8.20005.7335.65.7333.45.733]
57.733
94.93357.73337.200
total
groupj
errortotalgroup
X
SSX
N
SSnXX
SSSSSS
S
=S-=-=
=S-=-+-+-
=
=-=-=
Source
df
SS
MS
F
Group
2
57.733
28.867
9.312*
Error
12
37.200
3.100
Total
14
94.933
*
p
<
.
05
[
F
.
05
(
2
,
12
)
=
3
.
89
]
16.2Continuing with the data in Exercise 16.1:
a.Treatment effects:
EMBED Equation.DSMT4
123'
11.1
22.2
8.2 5.6 3.4 5.733
8.25.7332.467
5.65.7330.133
XXXX
XXb
XXb
a
a
====
=-=-==
=-=-=-=
b.
EMBED Equation.DSMT4
22
57.733
.608
94.933
treatment
total
SS
R
SS
h
====
16.3Data from Exercise 16.1, modified to make unequal ns:
R
2
=
.
624
SS
reg
=
79
.
0095
SS
residual
=
47
.
6571
Analysis of variance:
X
1 = 8.2000
X
2 = 5.8571
X
3 = 3.3333
.
X
= 5.7968
n1 = 5n2 = 7n3 = 9N = 21X = 112X 2 = 724
(
)
(
)
(
)
(
)
2
2
2
2
222
.
()
112
724126.6666
21
5[8.20005.796875.85715.796893.33335.7968
]
79.0095
126.666679.009547.6571
total
groupjj
errortotalgroup
X
SSX
N
SSnXX
SSSSSS
=-=-=
=-=-+-+-
=
=-=-=
S
å
å
Source
df
SS
MS
F
Group
2
79.0095
39.5048
14.92*
Error
18
47.6571
2.6476
Total
20
126.6666
*
p
<
.
05
[
F
.
05
(
2
,
18
)
=
3
.
55
]
16.4Continuing with the data in Exercise 16.3:
a.Treatment effects:
123.
11.1
22.12
.0
8.2 5.8571 3.3333 5.7968
8.25.79682.4032
5.85715.79680.0603
5.7968
XXXX
XXb
XXb
Xb
a
a
====
=-=-==
=-=-==
==
b.
22
79.0095
.624
126.6666
treatment
total
SS
R
SS
h
====
16.5Relationship between Gender, SES, and Locus of Control:
a.Analysis of Variance:
SES
Low
Average
High
Mean
Gender
Male
12.25
14.25
17.25
14.583
Female
8.25
12.25
16.25
12.250
Mean
10.25
13.25
16.75
13.417
X = 644X 2 = 9418n = 8N = 48
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
22
2
2
22
2
222
2
...
...
..
()644
9418777.6667
48
38[14.58313.41712.25013.417]
65.333
28[10.2513.41713.2513.41716.7513.417]
338.6667
8[12.251
total
genderi
SESj
cellsij
X
SSX
N
SSsnXX
SSgnXX
SSnXX
S
=S-=-=
=S-=-+-
=
=S-=-+-+-
=
=S-=-
(
)
(
)
22
3.417...16.2513.417]422.6667
422.666765.3333338.666718.6667
777.6667422.6667355.0000
GScellsgenderSES
errortotalcells
SSSSSSSS
SSSSSS
++-=
=--=--=
=-=-=
Source
df
SS
MS
F
Gender
1
65.333
65.333
7.730*
SES
2
338.667
169.333
20.034*
G x S
2
18.667
9.333
1.104
Error
42
355.000
8.452
Total
47
777.667
*
p
<
.
05
[
F
.
05
(
1
,
42
)
=
4
.
08
;
F
.
05
(
2
,
42
)
=
3
.
23
]
b.ANOVA summary table constructed from sums of squares
calculated from design matrix:
(
)
(
)
(
)
(
)
(
)
(
)
,,,
,,,
,,,
422.6667357.333365.333
422.666784.0000338.667
422.6667404.00018.667
777.667
G
regareg
S
regareg
GS
regareg
totalY
SSSSSS
SSSSSS
SSSSSS
SSSS
abbbab
abbaab
abbab
=-=-=
=-=-=
=-=-=
==
The summary table is exactly the same as in part a (above).
16.6
SSSES = SSreg
a.This is because we have equal ns, and therefore the variables
are orthogonal—that is they do not account for overlapping portions
of the variance.
b.This will not be true with unequal ns.
16.7The data from Exercise 16.5 modified to make unequal ns:
(
)
(
)
(
)
(
)
,,
(,,)
,)
(,,)
,
(,,)
,
750.1951458.7285291.467
458.7285398.713560.015
458.7285112.3392346.389
458.7285437.633821
errorY
reg
Greg
reg
reg
reg
reg
reg
SSSSSS
SSSSSS
SSSSSS
SSSSSS
abab
abab
bab
abab
aab
abab
ab
=-=-=
=-=-=
=-=-=
=-=-=
.095
Source
df
SS
MS
F
Gender
1
60.015
60.015
7.21*
SES
2
346.389
173.195
20.80*
G x S
2
21.095
10.547
1.27
Error
35
291.467
8.328
Total
40
*
p
<
.
05
[
F
.
05
(
1
,
35
)
=
4
.
12
;
F
.
05
(
2
,
35
)
=
3
.
27
]
16.8
SSSES ≠ SSreg()
346.389 ≠ 379.3325
The two values are not the same because (as pointed out in
Exercise 16.6) they will not agree when there are unequal ns. In
Exercise 16.7 some of the variation accounted for by SES was shared
with Gender and the interaction, and thus was not included in SSSES
from Exercise 16.5.
16.9Model from data in Exercise 16.5:
1
.
1667
A
1
-
3
.
1667
B
1
-
0
.
1667
B
2
+
0
.
8333
AB
11
-
0
.
1667
AB
12
+
13
.
4167
Means:
SES (B)
Low
Avg
High
Gender (A)
Male
12.25
14.25
17.25
14.583
Female
8.25
12.25
16.25
12.250
¶
¶
0
111
112
223
11
11114
12
121
ˆ
..13.4167intercept
ˆ
..14.58313.41671.1667
ˆ
..10.2513.41673.1667
ˆ
..13.2513.41670.1667
..12.2514.58310.2513.14670.8337
Xb
AXb
BXb
BXb
ABABXb
ABA
m
a
b
b
ab
ab
====
=-=-==
=-=-=-=
=-=-=-=
=--+=--+==
=-
25
..14.2514.58313.2513.14670.1667
BXb
-+=--+=-=
16.10Model from data in Exercise 16.7:
1
.
2306
A
1
-
3
.
7167
B
1
+
0
.
3500
B
2
+
0
.
4778
AB
11
+
0
.
5444
AB
12
+
13
.
6750
Means:
SES (B)
Weighted
Means
Unweighted
Means
Low
Avg
High
Gender (A)
Male
11.617
15.800
17.250
15.105
14.906
Female
8.250
12.250
16.833
12.045
12.444
Weighted Means:
9.714
13.615
17.070
13.463
Unweighted:
9.958
14.025
17.043
13.675
With unequal ns we need to use the unweighted means in order to
reproduce the values found by the regression model.
¶
¶
0
111
112
223
11
11114
12
1212
ˆ
..13.475intercept
ˆ
..14.90613.6751.231
ˆ
..9.95813.6753.717
ˆ
..14.02513.6750.350
..11.66714.9069.95813.6750.478
..
Xb
AXb
BXb
BXb
ABABXb
ABABX
m
a
b
b
ab
ab
====
=-=-==
=-=-=-=
=-=-==
=--+=--+==
=--+=
5
15.80014.90614.02513.6750.544
b
--+==
16.11Does Method III really deal with unweighted means?
Means:
B1
B2
weighted
unweighted
A1
4
10
8.5
7.0
A2
10
4
8.0
7.0
weighted
8.0
8.5
8.29
unweighted
7.0
7.0
7.0
The full model produced by Method 1:
1111
ˆ
0.00.03.07.0
YABAB
=+-+
Effects calculated on weighted means:
¶
0
111
112
11
11113
ˆ
..8.29intercept
ˆ
..8.508.290.21
ˆ
..8.008.290.29
..4.008.508.008.294.21
Xb
AXb
BXb
ABABXb
m
a
b
ab
===¹
=-=-=¹
=-=-=¹
=--+=--+=-¹
Effects calculated on unweighted means:
¶
0
111
112
11
11113
ˆ
..7.00=intercept
ˆ
..7.007.000.00
ˆ
..7.007.000.00
..4.007.007.007.003.00
Xb
AXb
BXb
ABABXb
m
a
b
ab
===
=-=-==
=-=-==
=--+=--+=-=
These coefficients found by the model clearly reflect the
effects computed on unweighted means. Alternately, carrying out the
complete analysis leads to SSA = SSB = 0.00, again reflecting
equality of unweighted means.
16.12Venn diagram representing the sums of squares in Exercise
16.5:
SS(total)
16.13Venn diagram representing the sums of squares in Exercise
16.7:
SES
Sex
SxS
SS(total)
SS(error)
16.14SAS printout for the data in Exercise 16.7:
The SAS System 13:05 Saturday, November 18, 2000
The GLM Procedure
Dependent Variable: dv
Source DF Type I SS Mean Square F Value Pr > F
A 1 95.4511028 95.4511028 11.46 0.0018
B 2 342.1827043 171.0913522 20.55 <.0001
A*B 2 21.0946481 10.5473241 1.27 0.2944
Source DF Type II SS Mean Square F Value Pr > F
A 1 58.3013226 58.3013226 7.00 0.0121
B 2 342.1827043 171.0913522 20.55 <.0001
A*B 2 21.0946481 10.5473241 1.27 0.2944
Source DF Type III SS Mean Square F Value Pr > F
A 1 60.0149847 60.0149847 7.21 0.0110
B 2 346.3892120 173.1946060 20.80 <.0001
A*B 2 21.0946481 10.5473241 1.27 0.2944
Source DF Type IV SS Mean Square F Value Pr > F
A 1 60.0149847 60.0149847 7.21 0.0110
B 2 346.3892120 173.1946060 20.80 <.0001
A*B 2 21.0946481 10.5473241 1.27 0.2944
16.15Energy consumption of families:
a.Design matrix, using only the first entry in each group for
illustration purposes:
X
=
1
.
.
.
0
.
.
.
-
1
0
.
.
.
1
.
.
.
-
1
58
.
.
.
60
.
.
.
75
75
.
.
.
70
.
.
.
80
b.Analysis of covariance:
SS
reg
(
a
,
cov
,
a
c
)
=
2424
.
6202
SS
reg
(
a
,
cov
)
=
2369
.
2112
SS
residual
=
246
.
5221
=
SS
error
There is not a significant decrement in SSreg and thus we can
continue to assume homogeneity of regression.
SS
reg
(
a
)
=
1118
.
5333
SS
cov
=
SS
reg
(
a
,
cov
)
-
SS
reg
(
a
)
=
2369
.
2112
-
1118
.
5333
=
1250
.
6779
SS
reg
(
cov
)
=
1716
.
2884
SS
A
=
SS
reg
(
a
,
cov
)
-
SS
reg
(
cov
)
=
2369
.
2112
-
1716
.
2884
=
652
.
9228
Source
df
SS
MS
F
Covariate
1
1250.6779
1250.6779
55.81*
A (Group)
2
652.9228
326.4614
14.57*
Error
11
246.5221
22.4111
Total
14
2615.7333
*
p
<
.
05
[
F
.
05
(
1
,
11
)
=
4
.
84
;
F
.
05
(
2
,
11
)
=
3
.
98
]
16.16Exercise 16.15 expanded into a two-way analysis of
covariance.
a.Analysis of covariance:
First we will test for homogeneity of regression:
(
)
2
,,,cov,
(,,,cov,)
2
,,,cov
,,,cov
.8931
4288.5572
.8931
4283.9008
512.79916
c
regbc
reg
residual
R
SS
R
SS
SS
ababab
abaab
abab
abab
=
=
=
=
=
There is a nonsignificant decrement in attributable variation
(in fact, R2 is still the same to four decimal places!), so we will
take our second model as our full model.
(
)
(
)
(
)
(
)
(
)
(
)
(
)
,,cov
,,,cov,,cov
,,cov
,,,cov,,cov
,,cov
4111.2036
4283.90084111.2036172.6972
3163.0287
4283.90083163.02871120.8721
4275.6550
reg
B
regreg
reg
A
regreg
reg
AB
SS
SSSSSS
SS
SSSSSS
SS
SSSS
aab
ababaab
bab
ababbab
ab
=
=-=-=
=
=-=-=
=
=
(
)
(
)
(
)
(
)
(
)
,,,cov,,cov
,,
cov
,,,cov,,
4283.90084275.65508.2458
1979.1000
4283.90081979.10002304.8008
regreg
reg
regreg
SS
SS
SSSSSS
ababab
abab
abababab
-=-=
=
=-=-=
Source
df
SS
MS
F
Covariate
1
2304.8008
2304.8008
103.37*
A (Group)
2
1120.8721
560.4361
25.14*
B (Meter)
1
172.6972
172.6972
7.75*
AB
2
8.2458
4.1229
<1
Error
11
512.7992
22.2956
Total
14
4796.7000
*
p
<
.
05
[
F
.
05
(
1
,
11
)
=
4
.
84
;
F
.
05
(
2
,
11
)
=
3
.
98
]
b.Conclusions: After adjusting for last year’s usage, there are
significant differences between the time-of-day groups in terms of
usage and between those who could check on current usage (the
‘metered’ group) and those who could not. In particular, Group 3
appears to use more electricity than the other two groups, and
those with meters use less than those without meters. (A more
precise statement would require the calculation of adjusted means
as in Exercise 16.17.) There is no interaction between the two
independent variables.
16.17Adjusted means for the data in Exercise 16.16:
(The order of the means may differ depending on how you code the
group membership and how the software sets up its design matrix.
But the numerical values should agree.)
1211121
ˆ
7.90990.87862.40220.56670.13110.72606.37
40
YAABABABC
=-+-++++
Y
ˆ
11
=
-
7
.
9099
(
1
)
+
0
.
8786
(
0
)
-
2
.
4022
(
1
)
+
0
.
5667
(
1
)
+
0
.
1311
(
0
)
+
0
.
7260
(
61
.
3333
)
+
6
.
3740
=
41
.
1566
Y
ˆ
12
=
-
7
.
9099
(
1
)
+
0
.
8786
(
0
)
-
2
.
4022
(
-
1
)
+
0
.
5667
(
-
1
)
+
0
.
1311
(
0
)
+
0
.
7260
(
61
.
3333
)
+
6
.
3740
=
44
.
8276
Y
ˆ
21
=
-
7
.
9099
(
0
)
+
0
.
8786
(
1
)
-
2
.
4022
(
1
)
+
0
.
5667
(
0
)
+
0
.
1311
(
1
)
+
0
.
7260
(
61
.
3333
)
+
6
.
3740
=
49
.
5095
Y
ˆ
22
=
-
7
.
9099
(
0
)
+
0
.
8786
(
1
)
-
2
.
4022
(
-
1
)
+
0
.
5667
(
0
)
+
0
.
1311
(
-
1
)
+
0
.
7260
(
61
.
3333
)
+
6
.
3740
=
54
.
0517
Y
ˆ
31
=
-
7
.
9099
(
-
1
)
+
0
.
8786
(
-
1
)
-
2
.
4022
(
1
)
+
0
.
5667
(
-
1
)
+
0
.
1311
(
-
1
)
+
0
.
7260
(
61
.
3333
)
+
6
.
3740
=
54
.
8333
Y
ˆ
32
=
-
7
.
9099
(
-
1
)
+
0
.
8786
(
-
1
)
-
2
.
4022
(
-
1
)
+
0
.
5667
(
1
)
+
0
.
1311
(
1
)
+
0
.
7260
(
61
.
3333
)
+
6
.
3740
=
61
.
0333
(We enter 61.3333 for the covariate in each case, because we
want to estimate what the cell means would be if the observations
in those cells were always at the mean of the covariate.)
16.18Analysis of difference scores for data in Exercise
16.16:
Source
df
SS
MS
F
Group
2
1086.4667
543.2333
15.50*
Meter
1
197.6333
197.6333
5.64*
G(M
2
6.0667
3.0333
<1
Error
24
841.2000
35.0500
Total
29
2131.3667
*p < .05
16.19Klemchuk, Bond, & Howell (1990)
16.20Analysis of variance on Epinuneq.dat:
Model Includes:
SSregression
SSresidual
Dose, Interval, DxI
Dose, Interval
Dose, DxI
Interval, DxI
162.39512
150.78858
159.44821
45.29395
226.7619
Source
df
SS
MS
F
Dose
2
117.1112
58.5556
28.92*
Interval
2
2.9469
1.4734
0.73
DI
4
11.60654
2.9016
1.43
Error
112
226.7619
2.0247
Total
120
16.21Analysis of GSIT in Mireault.dat:
Tests of Between-Subjects Effects
Dependent Variable: GSIT
1216.924
a
5
243.385
2.923
.013
1094707.516
1
1094707.516
13146.193
.000
652.727
1
652.727
7.839
.005
98.343
2
49.172
.590
.555
419.722
2
209.861
2.520
.082
30727.305
369
83.272
1475553.000
375
31944.229
374
Source
Corrected Model
Intercept
GENDER
GROUP
GENDER * GROUP
Error
Total
Corrected Total
Type III Sum
of Squares
df
Mean Square
F
Sig.
R Squared = .038 (Adjusted R Squared = .025)
a.
Estimated Marginal Means
GENDER * GROUP
Dependent Variable: GSIT
62.367
1.304
59.804
64.931
64.676
1.107
62.500
66.853
63.826
1.903
60.084
67.568
62.535
.984
60.600
64.470
60.708
.858
59.020
62.396
58.528
1.521
55.537
61.518
GROUP
1
2
3
1
2
3
GENDER
Male
Female
Mean
Std. Error
Lower Bound
Upper Bound
95% Confidence Interval
16.22Analysis of covariance for Mireault’s data:
Tests of Between-Subjects Effects
Dependent Variable: GSIT
1441.155
a
6
240.192
2.844
.010
178328.902
1
178328.902
2111.844
.000
496.225
1
496.225
5.877
.016
303.986
1
303.986
3.600
.059
154.540
2
77.270
.915
.402
325.689
2
162.845
1.928
.147
24319.374
288
84.442
1163091.000
295
25760.529
294
Source
Corrected Model
Intercept
YEARCOLL
GENDER
GROUP
GENDER * GROUP
Error
Total
Corrected Total
Type III Sum
of Squares
df
Mean Square
F
Sig.
R Squared = .056 (Adjusted R Squared = .036)
a.
Estimated Marginal Means
GENDER * GROUP
Dependent Variable: GSIT
61.879
a
1.449
59.026
64.731
64.467
a
1.169
62.165
66.768
62.841
a
2.108
58.692
66.991
62.699
a
1.162
60.412
64.986
61.077
a
1.037
59.036
63.118
58.513
a
1.650
55.265
61.762
GROUP
1
2
3
1
2
3
GENDER
Male
Female
Mean
Std. Error
Lower Bound
Upper Bound
95% Confidence Interval
Evaluated at covariates appeared in the model: YEARCOLL =
2.68.
a.
16.23Analysis of variance on the covariate from Exercise
16.22.
The following is abbreviated SAS output.
General Linear Models Procedure
Dependent Variable: YEARCOLL
Sum ofMean
SourceDFSquaresSquareF ValuePr > F
Model513.34776452.66955292.150.0600
Error292363.00122881.2431549
Corrected Total297376.3489933
R-SquareC.V.Root MSE
YEARCOLL Mean
0.03546641.532581.11497
2.6845638
SourceDFType III SSMean SquareF ValuePr > F
GENDER15.950062995.950062994.790.0295
GROUP20.780704310.390352160.310.7308
GENDER*GROUP22.962723101.481361551.19 0.3052
GENDERGROUPYEARCOLL
LSMEAN
112.27906977
122.53225806
132.68421053
212.88888889
222.85000000
232.70967742
These data reveal a significant difference between males and
females in terms of YearColl. Females are slightly ahead of males.
If the first year of college is in fact more stressful than later
years, this could account for some of the difference we found in
Exercise 16.21.
16.24Analysis of Everitt’s data:
a.Analysis on Gain scores:
Dependent Variable: Gain
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 2 614.643667 307.321833 5.42 0.0065
Error 69 3910.742444 56.677427
Corrected Total 71 4525.386111
R-Square Coeff Var Root MSE Gain Mean
0.135821 272.3858 7.528441 2.763889
Source DF Type III SS Mean Square F Value Pr > F
Group 2 614.6436669 307.3218334 5.42 0.0065
b.Analysis of Post scores ignoring Pre scores:
Dependent Variable: Post
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 2 918.986916 459.493458 8.65 0.0004
Error 69 3665.057528 53.116776
Corrected Total 71 4584.044444
R-Square Coeff Var Root MSE Post Mean
0.200475 8.556928 7.288126 85.17222
Source DF Type III SS Mean Square F Value Pr > F
Group 2 918.9869160 459.4934580 8.65 0.0004
c.Analysis of Post scores with Pre score as covariate
Dependent Variable: Post
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 3 1272.781825 424.260608 8.71 <.0001
Error 68 3311.262620 48.695039
Corrected Total 71 4584.044444
R-Square Coeff Var Root MSE Post Mean
0.277655 8.193027 6.978183 85.17222
Source DF Type III SS Mean Square F Value Pr > F
Group 2 766.2728128 383.1364064 7.87 0.0008
Pre 1 353.7949086 353.7949086 7.27 0.0089
d.The analysis of gain scores and the analysis of covariance ask
similar questions, but they would only agree if the relationship
between Pre and Post had a slope of 1. In general the analysis of
covariance will be better. The analysis of Post scores is
confounded, because we can’t discriminate between effects of the
treatments and any pre-existing group differences. In addition, the
analysis of covariance would generally be better because it would
adjust the error term.
e.Because there is variability in the pretest scores that has
little to do with weight, I would be tempted to remove that from
SStotal to get a more appropriate denominator for h2. If I did that
would be 766.27/(4584.04-353.795) = 766.27/4230.245 = .18, which
strikes me as a far amount of explained variance given all of the
factors that influence weight. (A case might well be made for
leaving in the variation due to pretest scores for the same reasons
given when talking about matched sample t.)
f.When we leave out the Control group we can run a standard
analysis of covariance between the remaining groups adjusted for
the pretest. This gives us adjusted means of 85.70 and 90.198 for
CogBehav and Family groups, respectively. The standard deviations
of the groups are close, and a weighted average of the variances is
72.63, which gives a standard deviation of 8.5225. Then, using
adjusted means,
12
85.7091.198
0.625
8.5225
XX
d
s
-
-
===-
Thus after we adjust for pretest weights, the Family Therapy
group gained about two thirds of a standard deviation more than the
Cognitive Behavior Therapy group.
16.25Everitt compared two therapy groups and a control group
treatment for anorexia. The groups differed significantly in
posttest weight when controlling for pretest weight (F = 8.71, p
< .0001, with the Control group weighing the least at posttest.
When we examine the difference between just the two treatment
groups at posttest, the F does not reach significant, F = 3.745, p
= .060, though the effect size for the difference between means
(again controlling for pretest weights) is 0.62 with the Family
Therapy group weighing about six pounds more than the
Cognitive/Behavior Therapy group. It is difficult to k
