lecture 10: one way anova between subjects: practice! laura mcavinue school of psychology trinity...
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Lecture 10:One Way ANOVA
Between Subjects: Practice!
Laura McAvinue School of PsychologyTrinity 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?
StepsGroup A Group B Group C
Placebo CBT Gestalt
0 4 1
1 3 2
3 6 2
1 3 0
0 4 0
Mean 1 4 1
1. Sum of Squares
2. Degrees of Freedom
3. Mean Square
4. F Ratio
5. P ValueWhat is the grand mean of these observations?
2
SSTotal
(0-2)2 + (1-2) 2 + (3-2) 2 + (1-2) 2 + (0-2) 2
4 1 1 1 4+ (4-2) 2 + (3-2) 2 + (6-2) 2 + (3-2) 2 + (4-2) 2
4 1 16 1 4+ (1-2) 2 + (2-2) 2 + (2-2) 2 + (0-2) 2 + (0-2) 2
1 0 0 4 4
= 46
∑ (Xij – Grand mean)2
SSBetween
= 30
n∑ (Group mean – Grand mean)2
5 ∑ (1-2) 2 + (4-2) 2 + (1-2) 2
5 ( 1 + 4 + 1)5 (6)
SSwithin
= 16
∑ (Xij – Group meanj)2
SSplacebo (0-1) 2 + (1-1) 2 + (3-1) 2 + (1-1) 2 +(0-1) 2
1 0 4 0 1 = 6SSCBT (4-4) 2 + (3-4) 2 + (6-4) 2 + (3-4) 2 + (4-4) 2
0 1 4 1 0 = 6SSgestalt (1-1) 2 + (2-1) 2 + (2-1) 2 + (0-1) 2 + (0-1) 2
0 1 1 1 1 = 4
Degrees of Freedom
Dftotal N – 1
15 – 114
Dfbetween K – 1
3 – 12
Dfwithin K (n – 1)
3 (5 – 1)12
Mean Square
MSbetween SSbetween / dfbetween
30 / 215
MSwithin SSwithin / dfwithin
16 / 121.33
F Ratio
F Ratio MSbetween / MSwithin
15 / 1.3311.278
Is F > 1? Yes!
Is F big enough to reject Ho?Compare your F value to the F distribution
Dfnumerator = Dfbetween Dfdenominator = 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
Group Social Anxiety
1 (Placebo)... 0
1 1
1 3
1 1
1 0
2 (CBT)... 4
2 3
2 6
2 3
Enter the data into SPSS...
Run the ANOVA
• Analyse, Compare Means, One Way ANOVA
– Dependent List: Social Anxiety– Factor: Group– Options: Descriptives
Homogeneity of variance testMeans plot
Examine the means plot
GestaltCBTplacebo
Therapy
4
3.5
3
2.5
2
1.5
1
Me
an
of
so
cia
lan
x
Are the means more or less the same or does one seem a little different?
Examine the test for Homogeneity of Variance
Is Levine’s statistic significant?
Can we assume homogeneity of variance among groups?
Test of Homogeneity of Variances
socialanx
.000 2 12 1.000
LeveneStatistic df1 df2 Sig.
Examine the ANOVA table
•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
ANOVA
socialanx
30.000 2 15.000 11.250 .002
16.000 12 1.333
46.000 14
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
Multiple Comparisons
•According to the Bonferroni & Tukey posthoc tests, which means are significantly different from the others?
Multiple Comparisons
Dependent Variable: socialanx
-3.000* .730 .004 -4.95 -1.05
.000 .730 1.000 -1.95 1.95
3.000* .730 .004 1.05 4.95
3.000* .730 .004 1.05 4.95
.000 .730 1.000 -1.95 1.95
-3.000* .730 .004 -4.95 -1.05
-3.000* .730 .004 -5.03 -.97
.000 .730 1.000 -2.03 2.03
3.000* .730 .004 .97 5.03
3.000* .730 .004 .97 5.03
.000 .730 1.000 -2.03 2.03
-3.000* .730 .004 -5.03 -.97
(J) TherapyCBT
Gestalt
placebo
Gestalt
placebo
CBT
CBT
Gestalt
placebo
Gestalt
placebo
CBT
(I) Therapyplacebo
CBT
Gestalt
placebo
CBT
Gestalt
Tukey HSD
Bonferroni
MeanDifference
(I-J) Std. Error Sig. Lower Bound Upper Bound
95% Confidence Interval
The mean difference is significant at the .05 level.*.
Example 3: T tests v ANOVAs
Software/ Kevin Thomas/ ANOVA data setAnalyse the ‘age’ & ‘adherence’ variables using an independent samples t test & an ANOVA
Independent Samples Test
.000 1.000 -3.873 8 .005 -3.00000 .77460 -4.78622 -1.21378
-3.873 8.000 .005 -3.00000 .77460 -4.78622 -1.21378
Equal variancesassumed
Equal variancesnot assumed
adherenceF Sig.
Levene's Test forEquality of Variances
t df Sig. (2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the
Difference
t-test for Equality of Means
ANOVA
adherence
22.500 1 22.500 15.000 .005
12.000 8 1.500
34.500 9
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
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?
Ho: There is no difference between the groupsLevel of processing has no effect on recall
H1 : At least one group is significantly differentAs level of processing increases, incidental memory increases
Run the ANOVA
ANOVA dataset: Group & Recall variables
ANOVA
recall
209.067 2 104.533 10.516 .000
268.400 27 9.941
477.467 29
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
•What is the F value?•Is F > 1?•Is it statistically significant?•What can we conclude?
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?
What does the means plot suggest?
imageryadjectivecounting
group
14.00
13.00
12.00
11.00
10.00
9.00
8.00
7.00
Mea
n o
f re
call
Which means are different?
What do the posthoc tests suggest?
Multiple Comparisons
Dependent Variable: recall
-4.00000* 1.41002 .022 -7.4960 -.5040
-6.40000* 1.41002 .000 -9.8960 -2.9040
4.00000* 1.41002 .022 .5040 7.4960
-2.40000 1.41002 .223 -5.8960 1.0960
6.40000* 1.41002 .000 2.9040 9.8960
2.40000 1.41002 .223 -1.0960 5.8960
-4.00000* 1.41002 .026 -7.5990 -.4010
-6.40000* 1.41002 .000 -9.9990 -2.8010
4.00000* 1.41002 .026 .4010 7.5990
-2.40000 1.41002 .301 -5.9990 1.1990
6.40000* 1.41002 .000 2.8010 9.9990
2.40000 1.41002 .301 -1.1990 5.9990
(J) groupadjective
imagery
counting
imagery
counting
adjective
adjective
imagery
counting
imagery
counting
adjective
(I) groupcounting
adjective
imagery
counting
adjective
imagery
Tukey HSD
Bonferroni
MeanDifference
(I-J) Std. Error Sig. Lower Bound Upper Bound
95% Confidence Interval
The mean difference is significant at the .05 level.*.
Effect Size
Calculate Eta squared Calculate Omega squared
2 SSbetweenSStotal
2 SSbetween (k 1)MSwithin
SStotal MSwithin209.09 (3 1)9.94
477.46 9.94189.18
487.4.39209.06 /
477.46
= .44
• 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).