Repeated measures analyses and contrasts (Profile analysis):
Example Questions and Outline Model Answers
Question 1
Theory suggests that the acute phase of schizophrenia is characterised by a deficit in the process of disambiguating stimuli (and their associated
responses) using the so-called “cognitive context”. In chronic schizophrenia, this specific deficit is no longer prominent, and patients are
generally poor at cognitive tasks. To test the theory, groups of schizophrenic patients in either the acute (group=1) or chronic (group=2) phase of
their illness were compared with a matched healthy control group (group=3). The subjects were each tested on 3 versions of the continuous
performance test (Tasks 1 to 3). By varying the nature of the task, Task 3 was intended to measure the contextual disambiguation process and
Tasks 1 and 2 were control tasks. Performance for all tasks was scored on a common scale. It was decided to carry out a repeated-measures
analysis of these data with “Task” as the within-subjects factor and “Subject Group” as a between-subjects factor.
(a) Inspect the resulting SPSS output (Part I) and discuss the findings in relation to the above theory, explaining which statistics one
should report. (70% of marks for question)
An a priori interaction contrast analysis was employed to test the specific pattern of data implied by the theory. An SPSS syntax file was
constructed to create a repeated-measures contrast comparing the critical context task (task 3) with the average of the other 2 tasks (1 and 2).
Three between-subjects contrasts were also constructed: healthy controls vs. the schizophrenic patient groups combined; acute vs chronic
schizophrenia; healthy controls vs. acute schizophrenia. The syntax file (Part II) shows the LMATRIX subcommand for one of the contrasts.
(b) Write down the LMATRIX commands needed for the other two between-subjects contrasts and comment on which two of the 3
between-subjects contrasts constitute an orthogonal pair. (10% of marks for question)
(c) Inspect the printout (Part III) which gives the details of the interactions between the repeated-measures contrast and each of the
between-subjects contrasts. Comment on the results obtained. (20% of marks for question)
Printout for Question 1
Part I – Means and ANOVA/MANOVA printout
Re port
Subjec t Group
healthy controls
patient s with chronic
sc hizophrenia
patient s with ac ute
sc hizophrenia
Mean
N
St d. Deviat ion
Mean
N
St d. Deviat ion
Mean
N
St d. Deviat ion
Continuous
performanc e
tes t 1
8.2477
30
2.4177
4.9046
27
2.0441
8.2623
27
Continuous
performanc e
tes t 2
8.8645
30
2.5839
5.8279
27
2.4725
8.1677
27
Continuous
performanc e
tes t 3
6.4648
30
1.5540
3.6577
27
1.3910
4.3607
27
2.9383
1.6898
1.8105
Multivariate Tests c
Effect
TESTTYPE
TESTTYPE * GROUP
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Value
.478
.522
.917
.917
.124
.876
.142
.141
F
36.661 a
36.661 a
36.661 a
36.661 a
2.683
2.742a
2.798
5.712b
Hypothesis
df
2.000
2.000
2.000
2.000
4.000
4.000
4.000
2.000
Error df
80.000
80.000
80.000
80.000
162.000
160.000
158.000
81.000
a. Exact s tatis tic
b. The statistic is an upper bound on F that yields a lower bound on the significance level.
c.
Design: Intercept+GROUP
Within Subjects Des ign: TESTTYPE
Sig.
.000
.000
.000
.000
.033
.030
.028
.005
Ma uchly's Test of Sphericityb
Measure: MEA SURE_1
Epsilona
Approx .
Mauchly's
Chi-Squa
Greenhous Huynh-Fe Lower-bo
W ithin Subject s Effect
W
re
df
Sig.
e-Geis ser
ldt
und
TE STTYPE
.732
24.966
2
.000
.789
.821
.500
Tests t he null hypothes is that the error c ovarianc e matrix of the orthonormalized transformed dependent variables is
proport ional to an identity matrix .
a. May be us ed to adjust t he degrees of freedom for the averaged tes ts of signific ance. Correc ted tests are
dis play ed in the Tests of W ithin-Subjects Effects table.
b.
Design: Int ercept+GROUP
W ithin Subject s Design: TE STTYPE
Te sts of W ithi n-Subje cts Effects
Measure: MEASURE_1
Source
TESTTYPE
TESTTYPE * GROUP
Error(TESTTYPE)
Sphericity Ass umed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Sphericity Ass umed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Sphericity Ass umed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Ty pe III
Sum of
Squares
373.359
373.359
373.359
373.359
54.950
54.950
54.950
54.950
603.790
603.790
603.790
603.790
df
2
1.577
1.643
1.000
4
3.154
3.286
2.000
162
127.753
133.076
81.000
Te sts of Betw een-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average
Ty pe III
Sum of
Source
Squares
df
Int ercept 10714. 63
1
GROUP
414.739
2
Error
531.198
81
Mean
Square
10714. 63
207.370
6.558
F
1633.826
31.621
Sig.
.000
.000
Mean
Square
186.680
236.723
227.254
373.359
13.738
17.420
16.723
27.475
3.727
4.726
4.537
7.454
F
50.087
50.087
50.087
50.087
3.686
3.686
3.686
3.686
Sig.
.000
.000
.000
.000
.007
.012
.011
.029
Part II – Syntax Commands
GLM
cptest1 cptest2 cptest3 BY group
/WSFACTOR = testtype 3
/METHOD = SSTYPE(3)
/CRITERIA = ALPHA(.05)
/MMATRIX = "context vs control" cptest1 -1 cptest2 -1 cptest3 2
/LMATRIX = "schizophrenic patients vs controls" group 1 1 -2
/WSDESIGN = testtype
/DESIGN = group .
Part III – Results for Contrast-by-Contrast Interactions
Contra st Results (K Matri x)a
Contrast
L1
Contrast Estimate
Hy pothesiz ed Value
Difference (Est imat e - Hypothes ized)
St d. Error
Sig.
95% Confidenc e Interval
for Difference
Transformed
Variable
contex t vs
control
2.760
0
Lower Bound
Upper Bound
a. Based on t he user-specified contrast coeffic ient s (L') matrix:
sc hizophrenic patients vs c ontrols
2.760
2.647
.300
-2. 507
8.028
Test Results
Transformed Variable: context vs control
Sum of
Mean
Source
Squares
df
Square
Contrast
36.737
1
36.737
Error
2736.829
81
33.788
F
1.087
Sig.
.300
Contra st Resul ts (K Ma trix)a
Contrast
L1
Contrast Es timate
Hy pothesiz ed V alue
Difference (Estimate - Hypot hesiz ed)
St d. Error
Sig.
95% Confidence Interval
for Differenc e
Lower Bound
Upper Bound
Transformed
Variable
contex t vs
control
4.292
0
4.292
1.582
.008
1.144
7.439
a. Based on t he user-s pec ified cont rast coefficients (L') matrix : ac ute vs
chronic pat ients
Te st Results
Transformed Variable: cont ext vs c ontrol
Sum of
Mean
Source
Squares
df
Square
Contrast
248.638
1
248.638
Error
2736.829
81
33.788
F
7.359
Sig.
.008
Contra st Results (K Matri x)a
Contrast
L1
Transformed
Variable
contex t vs
control
3.526
0
Contrast Estimate
Hy pothesiz ed Value
Difference (Est imat e - Hypothes ized)
St d. Error
Sig.
95% Confidenc e Interval
for Difference
3.526
1.542
.025
.458
6.594
Lower Bound
Upper Bound
a. Based on t he user-specified contrast coeffic ient s (L') matrix: pat ients
with ac ute schizophrenia vs controls
Test Results
Transformed Variable: context vs control
Sum of
Mean
Source
Squares
df
Square
Contrast
176.672
1
176.672
Error
2736.829
81
33.788
F
5.229
Sig.
.025
Question1: MODEL ANSWER
(again a pretty short answer can get full marks)
(a) Part I output:
Means Report -- The data appear to support the theory quite closely. The healthy controls and acute schizophrenics produce very similar
mean performance on the two control tasks (test1 and test2; with mean scores around 8). The critical task (test3) at which acute
schizophrenic patients are predicted to have specific difficulties is associated with lower performance in all groups, but is more impaired
in the acute schizophrenics than the controls (means 4.4 and 6.5 respectively). The chronic schizophrenic patients are worse on all tests
than either of the other 2 groups but show a similar relative profile across the tasks to that of the healthy controls, thereby not showing
the specific deficit apparently present in the acute schizophrenics. (Students may illustrate this answer with an appropriately annotated
profile plot of means across groups.)
Multivariate Tests -- this is the summary of the MANOVA (or “profile analysis”) findings and shows both a significant main effect
across tests (Testtype) and a significant TesttypexGroup interaction. The interaction is in keeping with theory and the specific pattern of
means it implies (discussed above). It does not matter which MANOVA statistic one reports for Testype (all have same P value) or the
interaction (all significant).
Sphericity test -- This tests the often violated sphericity assumption/requirement of repeated measures ANOVA (also known as
homogeneity of covariance -- but not “homogeneity of variance” or “homogeneity of variance-covariance matrices”). It is important to
test for sphericity because: [1] there are more than 2 levels (more than 1 df) for the repeated-measures factor; [2] sphericity is oftenviolated in repeated-measures designs; [3] and non-sphericity can seriously bias the resulting ANOVA statistics. As the sphericity test is
significant this means that sphericity is not present in the data.
Tests of Within-Subjects Effects -- As sphericity is not present in the data, ANOVA cannot be safely used uncorrected. (ie do not report
“Sphericity Assumed statistics”). As the violation of the sphericity assumption is quite severe (p<0.001) it is also not safe to use either
the “Huyn-Feldt” or “Greenhouse-Geisser” corrections to the ANOVA statistics (i.e. they do not remove the biases created by nonsphericity).
[An impressive student might additionally say: One might have decided in advance of doing the analysis that sphericity was not likely to
have been obtained and go for the most conservative “lower-bound” adjusted ANOVA statistic; this preserves Type I error rates more
effectively than the G-G of H-F corrections in the face of non-sphericity.]
Choice of Statistic to Report In this situation (sphericity strongly violated) one can choose to report the MANOVA (profile analysis)
statistics rather than the ANOVA results, because the MANOVA stats do not depend require sphericity (other additional assumptions
required by MANOVA are less likely to be violated and are less serious when violated unless group sizes are grossly unequal). Profile
analysis is appropriate because the DVs are all measured on the same scale. Alternatively one could investigate the predictions of the
hypothesis via a priori single df contrasts, which (because they are 1 df only) cannot violate sphericity (this approach was taken in the
later parts of the question).
(b) LMATRIX = “acute vs chronic schizophrenic” group 1 -1 0 (if the final zero is missing this is OK)
LMATRIX = “acute schizophrenic vs control ” group 1 0 -1
(any other number except 1 is also OK, although the zeros have to be zero)
The orthogonal pair is formed of the contrast between controls and patients combined and the contrast between the 2 schizophrenic
groups.
(c) The repeated measures contrast is specified (and is constructed appropriately for the theory) by the MMATRIX command. The first
interaction of a between-subjects contrast with this repeated-measures contrast shows that there is no significant difference between the
controls and the schizophrenic patient groups combined in terms of the difference in performance between the critical contextual task (3)
and the other two control tasks combined (1+2). The second betweenXrepeated contrastXcontrast interaction shows that the acute
patients are significantly different from the chronic patients in terms of the task contrast concerned. The final contrast interaction shows
that the acute patients are also significantly different from the controls in terms of the task contrast. When coupled with the inspection of
means (to see the direction of the effects), these results are thus exactly in line with the theory: acute schizophrenics have a reduction in
performance on the critical context dependent task (3) relative to control tasks (1 and 2), and this is significantly greater than the
corresponding reduction shown by the either of the other two groups. Schizophrenics in general (i.e. both acute and chronic patients
combined) do not show this specific relative deficit (compared with healthy subjects), and so the key performance profile is specific to
the patients in the acute stage of the illness.