1
ANOVA
PSY 211
4-14-09
TERM PAPERS
DUE THURSDAY (4/16)
INCLUDE SPSS OUTPUT
INCLUDE 1 COPY OF ALL SOURCES
A. Types of Analyses (some review)
 Continuous variables: Values are numbers that
have meaning, such as rating scales or other
numeric measurements (IQ, 9-pt attractiveness
scale, height)
 Categorical variables: Groups, any numbers
assigned are meaningless (gender, ethnicity,
experimental condition, treatment group)
Independent
Variable(s)
Dependent
Variable
Correlation (r)
One
continuous
One
continuous
Multiple
Regression (R)
Multiple
continuous
One
continuous
Between-group
t-test (t)
Categorical
(2 categories)
One
continuous
ANOVA (F)
Categorical
(2+ categories)
One
continuous
Categorical
Categorical
Type
Chi square (χ2)
2
B. Review of t-tests
 Two commonly used t-tests
 Between-group t-test: two groups of people are
compared on some continuous outcome variable
 Repeated-measures t-test: same group of
people compared on some outcome variable
across two time points
C. Introduction to ANOVA
 ANOVA = analysis of variance
 Two types
o ANOVA: compares multi-category variable
to a continuous variable; like the betweengroup t-test but can involve 2+ groups
o Repeated-measures ANOVA: examines
how scores change across multiple time
points; like repeated-measures t-test but
can involve 2+ time points
 Thus, ANOVA are like a more powerful t-tests
ANOVA
t-test
Note: If you have
two groups or two
time points,
technically you
could do some type
of t-test or some
type of ANOVA,
but t-tests are
simpler, so we do
them when we can.
3
D. How does it work?
 “Variance” indicates amount of variability or
differences
 Experimental manipulations (e.g. treatment
group) or demographics (e.g. ethnicity) tend to
increase variability – they cause people to get
different scores on some outcome variable (e.g.
anxiety)
 ANOVA examines whether the amount of
variability we obtain is greater than what we’d
expect by chance. If so, there are probably
some interesting group differences
Assume we are examining quality of K-12 education
across several ethnic groups. Just by chance, we
would expect some small differences to show up in
our sample. ANOVA is used to examine whether
there is more variability than what we’d expect by
chance.
E. ANOVA Lingo
 Factor: the categorical independent variable
- Favorite color
 Level: specific value/condition within the factor
- Blue
A researcher wants to compare three treatments
(placebo pill, medication, and surgery) to see which
is most effective in improving heart health.
- What is the IV? DV?
- What is the factor? Levels?
4
F. Hypothesis Testing
 Hypothesis testing with ANOVA can be a bit
peculiar. Let’s examine hypothesis testing,
where the IV has three categories
H0: μ1 = μ2 = μ3 or
At the population level, groups 1, 2, and 3 all
have the same mean on some variable.
10.00
Mean Health
8.00
6.00
4.00
2.00
0.00
Placebo
Medication
Surgery
HA: At the population level, at least one mean differs
from the others.
 HA will be true under a wide variety of conditions…
μ1 ≠ μ2 ≠ μ3 (all means are different)
μ1 = μ3, but μ2 is different (at least one mean differs)
5
8.00
Mean Health
6.00
4.00
2.00
10.00
0.00
Placebo
Medication
Surgery
Placebo
Medication
Surgery
Mean Health
8.00
6.00
4.00
2.00
0.00
8.00
Mean Health
6.00
4.00
2.00
0.00
Placebo
Medication
Surgery
 Alternative hypothesis is true if there are any
mean differences at the population level
6
 Because of sampling error, we always expect
some mild group differences in scores at the
sample level
 Based on the sample size, are the amount of
differences observed in the sample so reliable
that they would be expected to occur at the
population level?
 To determine this, ANOVA relies on the F
statistic
G. Significance Testing with the F Statistic
 F ranges from 0 to ∞
 Measures whether obtained variability is greater
than chance
 When F is small, go with null hypothesis
 When F is large, go with alternative hypothesis
 Like t-tests, the cut score is based on degrees of
freedom, so the critical F value needed for
significance varies from study to study
 As F gets bigger and bigger, the observed
difference is greater and greater; less likely due
to chance
7
For PSY 211, we will not hand-calculate F. We will
learn to understand it conceptually and let SPSS take
care of the grunt work
 Some conceptual definitions of F:
F = variance (differences) between groups
variance (differences) within groups
F = total amount of variability in scores
variability due to chance
Huh?
Examples:
F = differences in IQ across SES groups
differences in IQ within SES groups
F = differences in liberalism across music genres
differences in liberalism within music genres
Basically, we are checking to see whether the
differences across groups are larger than the chance
or meaningless differences we can observe within
groups
8
Example with Data:
Where the null hypothesis is supported…
Health scores across groups
7 different people per group
Placebo
Medication
1
1
1
2
2
3
3
3
5
5
6
6
8
8
Surgery
2
2
2
3
5
6
8
Health scores across groups
7 different people per group
Placebo
Medication
1
1
1
2
2
3
3
3
5
5
6
6
8
8
Surgery
2
2
2
3
5
6
8
Health scores across groups
7 different people per group
Placebo
Medication
1
1
1
2
2
3
3
3
5
5
6
6
8
8
Surgery
2
2
2
3
5
6
8
Differences within
each group are
chance differences
(bottom of fraction
for F equation)
Differences across
groups (top of
fraction for F
equation)
9
F = total variability / chance variability
 Here, total variability and chance variability are
about the same, so F will be near one
 Why? Any number divided by itself equals one
o 100/100 = 1
o 2.76/2.76 = 1
o 0.38/0.38 = 1
 When the total amount of variability (across
groups) is about the same as the chance
variability (within groups), F will be small
What if there was a treatment effect?
10
Example with Data:
Where the alternative hypothesis is supported…
Health scores across groups
7 different people per group
Placebo
Medication
1
4
1
4
1
4
1
4
2
5
2
5
2
5
Surgery
8
8
8
9
9
9
9
Health scores across groups
7 different people per group
Placebo
Medication
1
4
1
4
1
4
1
4
2
5
2
5
2
5
Surgery
8
8
8
9
9
9
9
Health scores across groups
7 different people per group
Placebo
Medication
1
4
1
4
1
4
1
4
2
5
2
5
2
5
Surgery
8
8
8
9
9
9
9
Differences within
each group are
chance differences
(bottom of fraction
for F equation)
Differences across
groups (top of
fraction for F
equation)
11
F = total variability / chance variability
 Here, total variability is much greater than
chance variability, so F will be large
 Why? Any big number divided by a much
smaller number yields a big number
o 100/10 = 10
o 2.76/0.21 = 13.14
o 0.38/0.11 = 3.45
 As the total amount of variability (across groups)
gets bigger than chance variability (within
groups), F will get bigger
 Thus, F increases as group differences account
for more and more of the variability
 F is small (near 1.0) when most of the variability
is due to chance
Distribution of F scores has a different
shape than the z or t distributions.
Also, the critical F value for determining
significance depends on sample size and
the number of groups.
12
H. Post-hoc tests
 The F test can be used to tell us whether the null
hypothesis should be rejected
 Weakness: F test tells us that at least one of the
groups has a significantly different mean, but
doesn’t tell us which group(s) significantly differ
 This is about as useful as saying “one of the
treatment conditions had some effect”
 Obviously, we want to know which condition or
conditions differed, and whether scores were
significantly higher or lower
 Post hoc test: Post hoc is Latin for “after this.”
The F test tells us whether any groups differ,
and the post hoc test tells test us how the
groups differ (e.g. whether surgery is better than
medication, or vice versa)
 There are many post hoc tests. They all differ
slightly in how they calculate error terms; some
are liberal, some conservative in judging which
groups significantly differ
13
5
Mean 76. Pro-Bush
4
3
2
1
0
9/11
Oil
Spread
Democracy
28. Reason for the War
Mean 40. Cigarettes Smoked per Day
3
2.5
2
1.5
1
0.5
0
Shame
Anger
Sadness
22. Negative Emotion (Childhood)
Anxiety
14
Mean 103. Extraversion
7
6
5
4
3
Friends
Family
Love
Success
19. Top Life Priority
 We’ll use the LSD (Least Significant Difference)
post hoc test
 The Bonferroni and Scheffe tests are also
commonly used
 The LSD test is very liberal; it will pick up on any
major group differences, but has the highest risk
of Type I errors
 Other tests are more conservative about labeling
group differences as significant, and nerds fight
about which test is best to use
Different kind of LSD
15
I. Examples Using SPSS
Is top life priority (friends, family, love, or success)
related to extraversion (9-point rating)?
De scri ptives
103. Extravers ion
Friends
Family
Love
Success
Total
N
23
188
56
12
279
Mean
4.83
5.38
5.50
6.00
5.39
St d.
Deviation
1.825
2.110
2.098
1.595
2.067
St d.
Error
.381
.154
.280
.461
.124
95% Confidenc e
Int erval for Mean
Lower
Upper
Bound
Bound
4.04
5.62
5.08
5.69
4.94
6.06
4.99
7.01
5.14
5.63
Minimum
1
1
2
4
1
ANOVA
103. Extraversion
Between Groups
W ithin Groups
Total
Sum of
Squares
12.464
1175.730
1188.194
df
3
275
278
Mean Square
4.155
4.275
F
.972
Sig.
.406
1) Check the descriptives
2) Look at the p-value “Sig”
3) If p < .05, the result is significant, so look at the
post-hoc test (later in Output). If p < .05, the result is
not significant, so ignore any additional Output.
- ANOVA uses two different types of degrees of
freedom (reference numbers) to determine the shape
of the F-distribution, and thus the critical F value.
One is based on sample size; the other on the
number of groups.
- F is used to determine the p-value
Life priority did not significantly predict level of
extraversion, F(3,275) = 0.97, p = .41.
Maximum
7
9
9
9
9
16
The relationship between eyewear and GPA using
ANOVA.
De scri ptives
41. GPA (High School)
Glasses
Contac t Lenses
Neither
Total
N
93
85
101
279
Mean
3.141
3.493
3.448
3.359
St d.
Deviation
.6604
.4154
.5013
.5578
St d.
Error
.0685
.0451
.0499
.0334
95% Confidenc e
Int erval for Mean
Lower
Upper
Bound
Bound
3.005
3.277
3.403
3.583
3.349
3.546
3.293
3.425
Minimum
1.3
2.3
1.7
1.3
Maximum
4.0
4.0
4.0
4.0
ANOVA
41. GPA (High School)
Between Groups
W ithin Groups
Total
Sum of
Squares
6.742
79.752
86.494
df
2
276
278
Mean Square
3.371
.289
F
11.666
Sig.
.000
This time there is a significant effect, so we need to
examine the post-hoc test to see which groups
reliably differ.
Multiple Comparisons
Dependent Variable: 41. GPA (High School)
LSD
(I) 29. Lens es
Glasses
Contact Lenses
Neither
(J) 29. Lens es
Contact Lenses
Neither
Glasses
Neither
Glasses
Contact Lenses
Mean
Difference
(I-J)
-.3521*
-.3067*
.3521*
.0454
.3067*
-.0454
Std. Error
.0807
.0773
.0807
.0791
.0773
.0791
*. The mean difference is s ignificant at the .05 level.
Sig.
.000
.000
.000
.566
.000
.566
95% Confidence Interval
Lower Bound Upper Bound
-.511
-.193
-.459
-.155
.193
.511
-.110
.201
.155
.459
-.201
.110
17
Multiple Comparisons
Dependent Variable: 41. GPA (High School)
LSD
(I) 29. Lens es
Glasses
Contact Lenses
Neither
(J) 29. Lens es
Contact Lenses
Neither
Glasses
Neither
Glasses
Contact Lenses
Mean
Difference
(I-J)
-.3521*
-.3067*
.3521*
.0454
.3067*
-.0454
Std. Error
.0807
.0773
.0807
.0791
.0773
.0791
Sig.
.000
.000
.000
.566
.000
.566
95% Confidence Interval
Lower Bound Upper Bound
-.511
-.193
-.459
-.155
.193
.511
-.110
.201
.155
.459
-.201
.110
*. The mean difference is s ignificant at the .05 level.
 The post-hoc test is basically like running t-tests
for all possible comparisons.
 Blue box: Compared glasses to contacts. The
mean difference is .35, which is significant.
 Red box: Compared glasses to neither (no
corrective lenses). The mean difference is .31,
which is significant.
 Green box: Compared contact lenses to neither.
The mean difference is .045, which is not
significant.
APA-style:
Eyewear was significantly related to GPA, F(2,276) =
11.67, p < .001. The results of a post-hoc LSD test
indicated that people who wore glasses had reliably
lower grades than people who wore contacts or no
lenses. The contact and no lens groups did not
significantly differ. Thus, wearing glasses was related
to lower GPA.
18
[Examples from old data; skip in class]
Is Favorite Fast Food Place related to ability to Delay
Gratification?
 Arby’s, BK, McD’s, T-Bell, Wendy’s, Other
 Delay of Gratification (patience): 1=low, 7=high
 Correlation? T-test? ANOVA?
 Output is fairly easy to understand!
De scri ptives
87. Delaying Gratification
Arby's
Burger King
Mc Donald's
Taco Bell
W endy 's
Ot her
Total
N
54
26
37
120
55
34
326
Mean
4.556
4.346
4.216
4.417
4.364
4.500
4.411
St d.
Deviation
1.8801
1.6719
1.8125
1.6783
1.8193
1.7965
1.7532
St d.
Error
.2558
.3279
.2980
.1532
.2453
.3081
.0971
95% Confidenc e
Int erval for Mean
Lower
Upper
Bound
Bound
4.042
5.069
3.671
5.021
3.612
4.821
4.113
4.720
3.872
4.855
3.873
5.127
4.220
4.602
Minimum
1.0
1.0
1.0
1.0
1.0
1.0
1.0
ANOVA
87. Delaying Gratification
Between Groups
W ithin Groups
Total
Sum of
Squares
3.038
995.882
998.920
df
5
320
325
Mean Square
.608
3.112
F
.195
Sig.
.964
 Notice that all of the means are about the same.
No major differences, so F is small, and there
are no statistically significant differences
Maximum
7.0
7.0
7.0
7.0
7.0
7.0
7.0
19
 There are no significant differences, so no post
hoc test is needed. If the F-value was
significant, what would the post hoc tests tell us?
 Ability to delay gratification was unrelated to
favorite fast food place, F(5, 320) = 0.20, ns.
Thus, customers are probably equally patient
across fast food locations.
Is sleeping position related crying frequency?
Sleeping positions: Back, Side, Stomach, Fetal
Crying Frequency: Compared to others I cry a lot…
1 = Disagree – 7 = Agree
De scri ptives
85. Cry ing
Back
Side
St omach
Fetal Position
Total
N
22
127
82
95
326
Mean
1.864
2.480
2.695
3.168
2.693
St d.
Deviation
1.5825
1.6849
1.7263
1.8022
1.7535
St d.
Error
.3374
.1495
.1906
.1849
.0971
95% Confidenc e
Int erval for Mean
Lower
Upper
Bound
Bound
1.162
2.565
2.184
2.776
2.316
3.074
2.801
3.536
2.502
2.884
 Step 1: Look at the means for each group.
Who cries the most?
Who cries the least?
ANOVA
85. Cry ing
Between Groups
W ithin Groups
Total
Sum of
Squares
42.350
956.975
999.325
df
3
322
325
Mean Square
14.117
2.972
F
4.750
Sig.
.003
Minimum
1.0
1.0
1.0
1.0
1.0
Maximum
7.0
7.0
7.0
7.0
7.0
20
 Step 2: Check for significance.
What is the F value?
Is it statistically significant (reliable)?
Post Hoc Tests
Multiple Comparisons
Dependent Variable: 85. Crying
LSD
(I) 30. Sleeping Pos ition
Back
Side
Stomach
Fetal Position
(J) 30. Sleeping Pos ition
Side
Stomach
Fetal Position
Back
Stomach
Fetal Position
Back
Side
Fetal Position
Back
Side
Stomach
Mean
Difference
(I-J)
-.6167
-.8315*
-1.3048*
.6167
-.2148
-.6881*
.8315*
.2148
-.4733
1.3048*
.6881*
.4733
Std.
Error
.3981
.4139
.4079
.3981
.2442
.2338
.4139
.2442
.2599
.4079
.2338
.2599
*. The mean difference is s ignificant at the .05 level.
 Step 3: Exam the post hoc results.
Which sleep positions significantly differ from
each other in terms of crying?
Which sleep positions do not differ significantly
from each other?
(More details on next page)
Sig.
.122
.045
.002
.122
.380
.003
.045
.380
.069
.002
.003
.069
95% Confidence
Interval
Lower
Upper
Bound
Bound
-1.400
.167
-1.646
-.017
-2.107
-.502
-.167
1.400
-.695
.266
-1.148
-.228
.017
1.646
-.266
.695
-.985
.038
.502
2.107
.228
1.148
-.038
.985
21
Multiple Comparisons
Dependent Variable: 85. Crying
LSD
(I) 30. Sleeping Pos ition
Back
Side
Stomach
Fetal Position
(J) 30. Sleeping Pos ition
Side
Stomach
Fetal Position
Back
Stomach
Fetal Position
Back
Side
Fetal Position
Back
Side
Stomach
Mean
Difference
(I-J)
-.6167
-.8315*
-1.3048*
.6167
-.2148
-.6881*
.8315*
.2148
-.4733
1.3048*
.6881*
.4733
Std.
Error
.3981
.4139
.4079
.3981
.2442
.2338
.4139
.2442
.2599
.4079
.2338
.2599
Sig.
.122
.045
.002
.122
.380
.003
.045
.380
.069
.002
.003
.069
*. The mean difference is s ignificant at the .05 level.
Comparing Back to Side (pink box), the Mean
Difference is small and non-significant.
Comparing Back to Fetal Position (blue box), the
mean difference in crying is much larger. The
negative sign indicates that Back sleepers cry less
than Fetal Position sleeper. The Sig. value (p-value)
shows that this is statistically significant (reliable).
Note that the post hoc table allows us to compare any
two groups with each other (kind of like a big table of
t-tests all at once), and some of the information is
redundant (see gray box).
 Step 4: Write up the results in APA format.
Sleeping positions is related to crying frequency, F(3,
322) = 4.75, p < .05. People who sleep in the fetal
position cry the most, and people who sleep in their
back cry the least.
95% Confidence
Interval
Lower
Upper
Bound
Bound
-1.400
.167
-1.646
-.017
-2.107
-.502
-.167
1.400
-.695
.266
-1.148
-.228
.017
1.646
-.266
.695
-.985
.038
.502
2.107
.228
1.148
-.038
.985
22
J. Repeated-Measures ANOVA
 ANOVA is similar to a between-group t-test
 Repeated-measures ANOVA is similar to a
repeated-measures t-test, but with more time
points
10
9
8
7
6
Pain 5
4
3
2
1
0
Intake
Discharge
Follow-up
Ignore the dashed
lines. The red line
shows a repeatedmeasures ANOVA
for life satisfaction
across 10 different
time points among
people diagnosed
with a disability.
23
Comparison Group
High Schizotypy
Two different
repeated-measures
ANOVAs: The
comparison group
(blue line) has a nonsignificant ANOVA,
whereas the
Schizotypy group
(red line) has a
significant ANOVA.
 Example of hypothesis testing, where a
researcher is examining how the dependent
variable (e.g. pain score) differs across three
time points:
H0: μ1 = μ2 = μ3 or
At the population level, average scores will
not differ across any time point.
10
9
8
7
6
5
4
3
2
1
0
Intake
Discharge
Follow-up
The subscripts
1, 2, and 3
signify different
time points.
24
HA: At the population level, the mean for at least one
time point will significantly differ from the others
 Note: HA will be true under a wide variety of
conditions…
μ1 ≠ μ2 ≠ μ3 (all means are different)
μ1 = μ3, but μ2 is different (at least one mean differs)
10
9
8
7
6
5
4
3
2
1
0
10
9
8
7
6
5
4
3
2
1
0
Intake
Discharge
Follow-up
10
9
8
7
6
5
4
3
2
1
0
Intake
Discharge
Follow-up
10
9
8
7
6
5
4
3
2
1
0
Intake
Discharge
Follow-up
Intake
Discharge
Follow-up
 Alternative hypothesis is true if any of the means
significantly (reliably) differ from the others
 Just like ANOVA uses the F statistic to test
whether there are between-group differences,
the repeated-measures ANOVA uses the F
statistic to examine whether means differ across
time points
F = total amount of variability in scores
variability due to chance
25
Repeated-Measures ANOVA
Similarities to
 Rationale for the F-test is the
ANOVA
same: compare total variability
to the amount expected by
chance
 Post-hoc tests are the same
Differences
from ANOVA
 Controls for some individual
differences, so tends to be
more powerful
 Smaller sample needed to
obtain significant results
K. SPSS Examples for Repeated-Measures ANOVA
 Repeated-measures ANOVA is a bit
complicated to run in SPSS, so it is not a
requirement of PSY 211
 See the following web site if you would ever like
to run one:
http://academic.reed.edu/psychology/RDDAweb
site/spssguide/anova.html
 Reading and understanding the Output is less
difficult, so I will give you some Output to
interpret
26
27
Descriptive Statistics
Mean
7.8500
5.3000
5.4500
Intake
Discharge
Follow-up
Std. Deviation
1.56525
2.36421
2.06410
N
20
20
20
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
factor1
Error(factor1)
Sphericity Assumed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Type III Sum
of Squares
81.900
81.900
81.900
81.900
56.767
56.767
56.767
56.767
df
2
1.348
1.414
1.000
38
25.607
26.860
19.000
Pa irw ise Com parisons
Mean Square
40.950
60.768
57.933
81.900
1.494
2.217
2.113
2.988
F
27.412
27.412
27.412
27.412
Post hoc test
Measure: MEA SURE_1
(I) fact or1
1
2
3
(J) fact or1
2
3
1
3
1
2
Mean
Difference
(I-J)
St d.
2.550*
2.400*
-2. 550*
-.150
-2. 400*
.150
E rror
.489
.387
.489
.244
.387
.244
a
Sig.
.000
.000
.000
.545
.000
.545
95% Confidenc e Interval for
a
Difference
Lower Bound Upper Bound
1.526
3.574
1.591
3.209
-3. 574
-1. 526
-.660
.360
-3. 209
-1. 591
-.360
.660
Based on estimated marginal means
*. The mean differenc e is significant at the .05 level.
a. Adjust ment for multiple comparisons: Least Signific ant Differenc e (equivalent to no
adjustments).
Sig.
.000
.000
.000
.000
28
Red Box: Overview of descriptive statistics for each
time point
Green Box: These are the degree of freedom (df)
values. Remember, the critical t value depended on
the df value. The critical F value depends on two df
values, one which is related to group size, and one
which is related to sample size. These values help
determine the critical F value, used to determine
whether a finding will be significant, and they are used
in the APA-style write-up too.
Blue Box: F value and p value. If p is less than .05
(p<.05), the finding is significant, indicating that there
is at least one reliable mean difference across time
points.
Orange Box: These values indicate the mean
differences across two time points. For example, the
first indicates a mean differences of 2.55 points
between time 1 (Intake) and time 2 (Discharge). The
next number indicates that there is a 2.40 point
difference between time 1 (Intake) and time 3
(Follow-up). Stars (*) indicate a statistically significant
(reliable) difference.
Pink Box: These are the precise p values for the post
hoc tests. If p<.05, the difference is statistically
significant. All mean differences in the Orange Box
that had a star also have a p<.05.
APA Style Write-up:
Pain scores significantly differed across time, F(2, 38)
= 27.41, p < .05. A post hoc LSD test indicated that
pain scores at discharge and follow-up were
significantly lower than at intake.
29
A non-significant result….
30
Descriptive Statistics
Mean
7.8000
7.3000
7.2000
Intake
Discharge
Follow-up
Std. Deviation
1.70448
1.65752
1.82382
N
20
20
20
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
factor1
Error(factor1)
Sphericity Assumed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Type III Sum
of Squares
4.133
4.133
4.133
4.133
81.867
81.867
81.867
81.867
df
2
1.613
1.741
1.000
38
30.653
33.075
19.000
Mean Square
2.067
2.562
2.374
4.133
2.154
2.671
2.475
4.309
F
.959
.959
.959
.959
Pa irw ise Com parisons
Measure: MEA SURE_1
(I) fact or1
1
2
3
(J) fact or1
2
3
1
3
1
2
Mean
Difference
(I-J)
.500
.600
-.500
.100
-.600
-.100
St d. E rror
.516
.520
.516
.332
.520
.332
a
Sig.
.344
.263
.344
.766
.263
.766
95% Confidenc e Interval for
a
Difference
Lower Bound Upper Bound
-.579
1.579
-.489
1.689
-1. 579
.579
-.594
.794
-1. 689
.489
-.794
.594
Based on estimated marginal means
a. Adjust ment for multiple comparisons: Least Signific ant Differenc e (equivalent to no
adjustments).
Pain scores did not significantly differ across time,
F(2, 38) = 0.96, ns. Thus, the treatment was
unsuccessful.
Sig.
.392
.377
.383
.340