Lab3: writing up results and ANOVAs
with within and between factors
1
Should be able to answer:
• What are the independent and dependent
variables?
• Are the conditions of application met?
– Compound symmetry?
– Sphericity?
2
Conditions of application
• Normality:
– fully balanced design, all subjects in all conditions,
all cells filled so probably safe
• Compound symmetry:
– Smallest covariance: .147, Largest covariance: 39
• Sphericity:
– p>.05, fail to reject hypothesis that pairwise
variances differ
3
Should be able to answer:
• Are there main effects?
– Contrasts: to learn about others, p. 371
4
Summary of results
• A significant main effect of coil was observed, suggesting that the fMRI
signal varies for coils with different numbers of channels (F1,11)=37,
p<.001), while collapsing across (or irrespective to) acceleration level.
Means reveal that signal was greater for the 32 channel as predicted.
• A significant main effect of acceleration was found, suggesting that MRI
signals differ for different levels of acceleration (F2,22=13.6, p<.001), when
collapsing across coils.
• Contrasts revealed that acceleration of a factor of 2 or 3 both differenced
significantly from no acceleration (2factor: F1,11=6.1, p<.05; 3factor:
F1,11=57.1, p<.001). In addition, a significant linear contrast was observed
(F1,11=57.1, p<.001) with a non-significant quadratic contrast (F1,11=0.0,
p>.1), suggesting that signal changes linearly with acceleration. Graphs
reveal that the linear relationship is such that MRI signals decrease with
increasing acceleration. This is consistent with our hypothesis.
• When reporting F, need degrees of freedom, first one is for
the factor and is number of levels -1, second one is for error,
which is (number of subjects -1) X (number of levels-1)
5
Should be able to answer:
• Are there main effects?
• Interactions?
6
Summary of results
• However!! There was a significant interaction of coil and
acceleration (F2,22=9.9, p<.01).
• We therefore investigated the effect of acceleration
separately for each coil.
• 12 channel: There is a significant main effect of acceleration
using the GG (F1.25, 13.8=18.1, p<.001).
Contrasts reveal that acceleration of a factor of 2
(F1,11=11.3, p<.01) and 3 (F1,11=131.1, p<.001) both differ
from no acceleration.
• 32 channel: There is again a significant main effect of
acceleration (F2,22=4.1, p<.05). However, contrasts revealed
that only acceleration of factor 3 differed significantly from
no acceleration (F1,11=7.9, p<.05). This suggests 32 channel
coil is less affected by acceleration than 12 channel, as
hypothesized.
7
Interpreting interactions via contrasts
45
45
40
40
35
35
30
30
25
12ch
20
32ch
15
25
10
5
5
0
0
a2
32ch
15
10
a0
12ch
20
a0
a3
8
Writing an abstract
• Background
• Objective
• Methods. May include:
–
–
–
–
–
(Sample size calculation / power)
Instruments
Procedure
Sample description (or may be in results)
Analysis (or may be in results)
• Results
• Discussion
9
Abstract format
Background: Advances in MRI hardware have led to coils with greater numbers of channels, while
software improvements have allowed MR data to be collected faster.
Objective: Here, we set out to test whether more channels are better, and how MRI acceleration
techniques might affect signal strength.
Methods: Twelve subjects underwent fMRI with both a 12 (12ch) and a 32 channel (32ch) coil.
For each coil, three levels of acceleration were tested (none, 2factor and 3factor). Average MR
signal was extracted and used as the dependent measure in a repeated measures ANOVA.
Results: There was a main effect of number of channels (F1,11)=37, p<.001), resulting from greater
signal from the 32ch versus the 12ch. There was a main effect of acceleration (F2,22=13.6,
p<.001), with signal decreasing as acceleration increased. In addition, there was an interaction
between number of channels and acceleration (F2,22=9.9, p<.01). To investigate the interaction,
repeated measure ANOVAs were performed for the simple effects. They revealed a significant
main effect of acceleration for both the 12ch, where the Greenhouse-Geisser correction was
used (F1.25, 13.8=18.1, p<.001), and for the 32ch (F2,22=4.1, p<.05). Simple contrasts revealed that
for the 12ch, signals decreased significantly for both 2factor (F1,11=11.3, p<.01) and 3factor
(F1,11=131.1, p<.001) acceleration. However, for the 3ch, signals decreased significantly for only
the highest level of acceleration (F1,11=7.9, p<.05).
Discussion: This suggests that signals are greater when collected with more channels, and as the
amount of acceleration increases, signals decrease. This was particularly true for the 12ch data.
10
Next up: mixed design ANOA
• What if you have both within and between
subjects factors?
• No worries, ANOVA can handle it 
• What are between subject factors?
11
Mixed design: conditions of application
1. Normality within each factor level or group
–
Robust to violations as long as fully factorial -> no levels
missing (like having only 2 of the three levels of
acceleration for 32 channel data or group).
This can also be tested via histograms and tests for
normality (see chapter 4??)
–
2. Homogeneity of variance:
–
Replaced by compound symmetry or sphericity in RM
ANOVA
But now need to check for between subject factors with
Levene’s test
–
•
•
Tests the null hypothesis that the error variance of the
dependent variable is equal across groups -> want nonsig
need to be careful because can be positive from small deviations
with large sample sizes
12
Practical differences
• Must define between subject variables
13
Practical differences
• Can use posthoc tests to investigate
differences: Scheffe
14
Practical differences
• Making plots
15
Practical differences
• Levene’s test
16
New output
17
Now you try!
• Expansion of MRI methods study from last week.
• Again tested two different MRI coils (12 channel
and 32 channel coils), and 3 levels of acceleration
(a0,a2,a3). Same hypotheses as before:
– Signal should be larger for 32 than 12 channel coil
– Signal should decrease with increasing levels of
acceleration
• In addition, half of the subjects (12) were
scanned with a resolution of 2mm (voxels
2x2x2mm) and half (12) with 3mm.
– Hypothesize that large voxels result in more signal.
18
Should be able to answer:
• What are the independent and dependent
variables? Are they within or between?
• Are the conditions of application met?
– Sphericity?
– Homogeneity of variance?
• Are there main effects?
• Interactions: investigate with simple effects and
the contrasts
• What can you conclude: try writing up an abstract
19