Lecture 10:
One Way ANOVA
Between Subjects: Practice!
Laura McAvinue
School of Psychology
Trinity College Dublin
Example 1: ANOVA by hand
• Our research interest is the treatment of social anxiety
• We would like to evaluate different therapies for social anxiety
• We took a sample of 15 people suffering from social anxiety &
randomly assigned them to three groups
– Placebo
– Cognitive Behavioural Therapy (CBT)
– Gestalt Therapy
• Did any of the treatments significantly improve social anxiety?
• Are any of the means significantly different?
Steps
1. Sum of Squares
2. Degrees of
Freedom
3. Mean Square
Mean
Group A
Group B
Group C
Placebo
CBT
Gestalt
0
4
1
1
3
2
3
6
2
1
3
0
0
4
0
1
4
1
4. F Ratio
5. P Value
What is the grand mean of these observations?
2
SSTotal
∑ (Xij – Grand mean)2
(0-2)2
4
+ (1-2) 2
1
+ (3-2) 2
1
+ (1-2) 2
1
+ (0-2) 2
4
+ (4-2) 2
+ (3-2) 2
+ (6-2) 2
+ (3-2) 2
+ (4-2) 2
4
+ (1-2) 2
1
1
+ (2-2) 2
0
16
+ (2-2) 2
0
1
+ (0-2) 2
4
4
+ (0-2) 2
4
= 46
SSBetween
n∑ (Group mean – Grand mean)2
5 ∑ (1-2) 2 + (4-2) 2 + (1-2) 2
5 ( 1 + 4 + 1)
5 (6)
= 30
SSwithin
∑ (Xij – Group meanj)2
SSplacebo (0-1) 2
1
SSCBT
(4-4) 2
0
SSgestalt (1-1) 2
0
+ (1-1) 2
0
+ (3-4) 2
1
+ (2-1) 2
1
+ (3-1) 2
4
+ (6-4) 2
4
+ (2-1) 2
1
= 16
+ (1-1) 2
0
+ (3-4) 2
1
+ (0-1) 2
1
+(0-1) 2
1
=6
+ (4-4) 2
0
=6
+ (0-1) 2
1
=4
Degrees of Freedom
Dftotal
N–1
15 – 1
14
Dfbetween
K–1
3–1
2
K (n – 1)
3 (5 – 1)
12
Dfwithin
Mean Square
MSbetween
SSbetween / dfbetween
30 / 2
15
MSwithin
SSwithin / dfwithin
16 / 12
1.33
F Ratio
F Ratio
MSbetween / MSwithin
15 / 1.33
11.278
Is F > 1? Yes!
Is F big enough to reject Ho?
Compare your F value to the F distribution
Dfnumerator
Dfdenominator
= Dfbetween
= Dfwithin
=2
= 12
F Ratio
• What is the critical value of F when  = .05?
– 3.88
• What is the critical value of F when  = .01?
– 6.93
• Is your F value greater than the critical values?
– Yes!
• Can you reject Ho? At what alpha level?
– Yes! At P < .01
Example 2: ANOVA by computer
Enter the data into SPSS...
Group
Social Anxiety
1 (Placebo)...
0
1
1
1
3
1
1
1
0
2 (CBT)...
4
2
3
2
6
2
3
Run the ANOVA
• Analyse, Compare Means, One Way ANOVA
– Dependent List: Social Anxiety
– Factor:
Group
– Options:
Descriptives
Homogeneity of variance test
Means plot
Examine the means plot
4
Are the means
more or less
the same or
does one seem
a little
different?
Mean of socialanx
3.5
3
2.5
2
1.5
1
placebo
CBT
Therapy
Gestalt
Examine the test for Homogeneity of Variance
Is Levine’s statistic
significant?
Test of Homogeneity of Variances
socialanx
Levene
Statistic
.000
df1
df2
2
12
Sig.
1.000
Can we assume
homogeneity of
variance among
groups?
Examine the ANOVA table
ANOVA
socialanx
Between Groups
Within Groups
Total
Sum of
Squares
30.000
16.000
46.000
df
2
12
14
Mean Square
15.000
1.333
F
11.250
•Is it similar to the one you created?
•What is the p value?
•Is it statistically significant?
•What can you conclude from this ANOVA?
At least one mean is significantly different from the others
Sig.
.002
Multiple Comparisons
Multiple Comparisons
Dependent Variable: socialanx
Tukey HSD
(I) Therapy
placebo
CBT
Gestalt
Bonferroni
placebo
CBT
Gestalt
(J) Therapy
CBT
Gestalt
placebo
Gestalt
placebo
CBT
CBT
Gestalt
placebo
Gestalt
placebo
CBT
Mean
Difference
(I-J)
-3.000*
.000
3.000*
3.000*
.000
-3.000*
-3.000*
.000
3.000*
3.000*
.000
-3.000*
Std. Error
.730
.730
.730
.730
.730
.730
.730
.730
.730
.730
.730
.730
Sig.
.004
1.000
.004
.004
1.000
.004
.004
1.000
.004
.004
1.000
.004
95% Confidence Interval
Lower Bound
Upper Bound
-4.95
-1.05
-1.95
1.95
1.05
4.95
1.05
4.95
-1.95
1.95
-4.95
-1.05
-5.03
-.97
-2.03
2.03
.97
5.03
.97
5.03
-2.03
2.03
-5.03
-.97
*. The mean difference is significant at the .05 level.
•According to the Bonferroni & Tukey posthoc tests, which means are
significantly different from the others?
Example 3: T tests v ANOVAs
Software/ Kevin Thomas/ ANOVA data set
Analyse the ‘age’ & ‘adherence’ variables using an independent
samples t test & an ANOVA
Independent Samples Test
Levene's Test for
Equality of Variances
F
adherence
Equal variances
assumed
Equal variances
not assumed
Sig.
.000
1.000
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-3.873
8
.005
-3.00000
.77460
-4.78622
-1.21378
-3.873
8.000
.005
-3.00000
.77460
-4.78622
-1.21378
ANOVA
adherence
Between Groups
Within Groups
Total
Sum of
Squares
22.500
12.000
34.500
df
1
8
9
Mean Square
22.500
1.500
F
15.000
Sig.
.005
Similar results?
T2 = ?
• Example 4: Research example
– Eysenck (1974) used three groups to investigate the impact of
levels of processing (Craik & Lockhart, 1972) on incidental
learning - learning in the absence of expectation that the material
will need to be recalled later
• Group 1 - count number of letters in each word - lowest level
of processing
• Group 2 - think of an adjective that might be used to describe
the word
• Group 3 - form a vivid image of the word
– What are Ho & H1?
H o:
There is no difference between the groups
Level of processing has no effect on recall
H1 :
At least one group is significantly different
As level of processing increases, incidental memory
increases
Run the ANOVA
ANOVA dataset: Group & Recall variables
ANOVA
recall
Between Groups
Within Groups
Total
Sum of
Squares
209.067
268.400
477.467
•What is the F value?
•Is F > 1?
•Is it statistically significant?
•What can we conclude?
df
2
27
29
Mean Square
104.533
9.941
F
10.516
Sig.
.000
Conclude:
•At least one of the means is
significantly different from the
others
•Level of processing does
significantly affect incidental recall
Which means are different?
14.00
13.00
Mean of recall
12.00
What does the
means plot
suggest?
11.00
10.00
9.00
8.00
7.00
counting
adjective
group
imagery
Which means are different?
What do the posthoc tests suggest?
Multiple Comparisons
Dependent Variable: recall
Tukey HSD
(I) group
counting
adjective
imagery
Bonferroni
counting
adjective
imagery
(J) group
adjective
imagery
counting
imagery
counting
adjective
adjective
imagery
counting
imagery
counting
adjective
Mean
Difference
(I-J)
-4.00000*
-6.40000*
4.00000*
-2.40000
6.40000*
2.40000
-4.00000*
-6.40000*
4.00000*
-2.40000
6.40000*
2.40000
*. The mean difference is significant at the .05 level.
Std. Error
1.41002
1.41002
1.41002
1.41002
1.41002
1.41002
1.41002
1.41002
1.41002
1.41002
1.41002
1.41002
Sig.
.022
.000
.022
.223
.000
.223
.026
.000
.026
.301
.000
.301
95% Confidence Interval
Lower Bound
Upper Bound
-7.4960
-.5040
-9.8960
-2.9040
.5040
7.4960
-5.8960
1.0960
2.9040
9.8960
-1.0960
5.8960
-7.5990
-.4010
-9.9990
-2.8010
.4010
7.5990
-5.9990
1.1990
2.8010
9.9990
-1.1990
5.9990
Effect Size
Calculate Eta squared
Calculate Omega squared
SSbe tween  (k 1)MS withi n
SSbe twee n
2
 
 
SStotal  MS withi n
SStotal
2
209.06 /
477.46
= .44
209.09 (3 1)  9.94 189.18

 .39
477.46 9.94
487.4
• Fully reporting the analysis
– A one-way (or one-factor) analysis of variance
(ANOVA) compared the mean number of words
recalled across three groups who processed the words
differently: count, adjective or imagery. With an alpha
level of .05, the analysis was statistically significant,
F(2,27) = 10.516, p< .001. A Tukey HSD test indicated
that the mean of 7 (SD =1.83) for the count group was
significantly different to that of the adjective (M=11,
SD=2.5) and imagery (M=13.4, SD=4.5) groups. The
difference between the adjective and imagery groups
was not significant (p > 10).