Transcript Lecture 11: One Way ANOVA Repeated Measures

Lecture 11:
One Way ANOVA
Repeated Measures
Laura McAvinue
School of Psychology
Trinity College Dublin
Analysis of Variance
One way ANOVA
Factorial ANOVA
One Independent
Variable
More than One
Independent Variable
Between
Repeated
subjects
measures /
Within
subjects
Different
participants
Same
participants
Two
way
Three
way
Four
way
One Way Repeated Measures ANOVA
• A statistical technique for testing for differences
between the means of several groups
– Groups are related in some way
– Usually same participants in each group
• Similar to the paired samples T-Test
• But no restriction on the number of groups
A few examples…
• What are the independent & dependent variables
in each of the following studies?
– Longitudinal study of annual growth of children over
the first five years of life
– The effects of three levels of methylphenidate on
reaction time performance of the same participants
– The measurement of depression before therapy, after
therapy and at a nine month follow up
• Same group of people is tested under different
conditions or at different times
Between Subjects ANOVA
Repeated Measures ANOVA
Data points in each group are
unrelated
Data points in each group are
related
Group
A
5
Group
B
4
Group
C
3
Time
A
1
Time
B
2
Time
C
4
6
2
7
5
5
9
9
4
3
7
8
6
2
5
4
3
5
8
9
3
2
2
2
4
Population
draw one sample
Cond 1
Cond 2 Cond 3
Number
of trials
required to
pass test
µ1
µ2
µ3
measure effect of IV on a DV
Did the
manipulation
alter the
sample to
Manipulate such an
extent that
the
the sample
sample
represents
another
population
during at
least one
condition?
Null Hypothesis Significance Testing
• Step 1:
Halt – At least one condition mean is
significantly different from the others
• Step 2:
Ho – All of the condition means are equal
• Step 3:
Collect your data
• Step 4:
Run the ANOVA
• Step 5:
Obtain the F statistic and associated p value
• Step 6:
Decide whether to reject or fail to reject Ho on
the basis of the p value
Total Variance, Between
Subjects ANOVA
Between Groups
Within Groups
Variance
Variance
between
within
the means
each group
Captures the effect of the
manipulation / treatment
The bigger the
effect of your
manipulation,
the bigger the
Between
Groups
Variance
Variance due to random
error / chance
F Ratio
Compares the variance due to the treatment /
manipulation to the variance due to random error /
chance
MS between groups
MS within groups
Between Groups Variance
Within Groups Variance
Treatment Effect + Differences due to chance / error
Differences due to chance / error
If the treatment / manipulation has an effect, then…
MS between groups > MS within groups
F>1
Total Variance, Between
Subjects ANOVA
Between Groups
Within Groups
Variance due to random
error / chance
Variance
due to
individual
differences
between
subjects
Variance due to
random error
Repeated Measures Design
• Powerful
• Error Variance
– Variance between participants
– Unexplained Random Error
• Because we test participants repeatedly with this design,
we can quantify this variance between participants
• We can remove this variance from the error term
– Making the error term smaller,
– Making the F ratio bigger,
– Making it easier to obtain a statistically significant F
Four Kinds of Variance
• Total Variance
– SStotal
– ∑ (xij - Grand Mean )2
• Variance due to the manipulation
– SStreatment
– n∑ (Group meanj - Grand Mean )2
Four Kinds of Variance
• Variance due to individual differences between
participants
– SSparticipants
– No. of conditions ∑ (Participant mean - Grand Mean )2
• Variance due to random error
– SSrandom error = SStotal - SStreatment - SSparticipants
F
MS Treatment
MS Random error
Condition / Treatment
Person 1
2
3
1
2
4
7
2
10
12
13
3
22
29
30
4
30
31
34
Condition / Treatment
Person 1
2
3
1
2
4
7
2
10
12
13
3
22
29
30
4
30
31
34
18.67
SStotal : ∑ (xij - Grand Mean )2
∑(2 – 18.67) 2 + (10 – 18.67)2 +…+ (34 – 18.67) 2
Condition / Treatment
Person 1
2
3
1
2
4
7
2
10
12
13
3
22
29
30
4
30
31
34
Mean
16
19
21
18.67
SStreatment : n ∑ (Treatment Mean - Grand Mean )2
4 ∑(16 – 18.67) 2 + (19 – 18.67)2 + (21 – 18.67) 2
Condition / Treatment
Person
1
2
3
1
2
4
7
Mean
per
person
4.33
2
10
12
13
11.67
3
22
29
30
27
4
30
31
34
31.67
Mean
16
19
21
18.67
SSparticipants : no. of conditions ∑ (Participant Mean - Grand Mean )2
3 ∑(4.33 – 18.67) 2 + (11.67 – 18.67)2 +…+ (31.67 – 18.67) 2
Condition / Treatment
Person
1
2
3
1
2
4
7
Mean
per
person
4.33
2
10
12
13
11.67
3
22
29
30
27
4
30
31
34
31.67
Mean
16
19
21
18.67
SSrandom error : SStotal - SStreatment - SSparticipants
ANOVA table
Source of
variation
SS
Between
participants
(individual
differences)
K ∑ (Participant
Mean - Grand
Mean )2
df
MS
F
p
n-1
Treatment
n ∑ (Treatment
meanj - Grand
Mean )2
Random
Error
SStotal - SStreatment (n -1)x SSe / dfe
- SSparticipants
(k -1)
Total
∑ (xij - Grand
Mean )2
K-1
SST / dfT
MSTreatment
MSerror
Prob. of
observing
F-value
when Ho is
true
N-1
N = Total no. of observations, n = no. of people in each condition, K = no. of conditions
Post hoc testing
• Significant F value
– At least one condition mean is significantly different
from the others
• But which one?
• Posthoc tests
– Bonferroni
– Tukey
– Sidak
A few assumptions…
Data in each group should be…
•
Interval scale
•
Normally distributed
•
•
Histograms, box plots
Homogeneity of variance
•
Variance within each condition should be roughly
equal
Extra Assumption
•
Sphericity
•
Variance of the differences between conditions is
the same
•
•
Variance t1-t2  Variance t1-t3  Variance t2-t3
Correlation between pairs of groups is the same
•
corr t1-t2  corr t1-t3  corr t2-t3
Testing Sphericity
•
Mauchly’s Test
•
If p > .05, assume equality of variances
•
If p < .05, then the data fails to meet the assumption
of sphericity
•
•
•
Need to use one of the correction factors
E.g. Greenhouse-Geisser
A more conservative test, using different dfs
Advantages & Disadvantages of
Repeated Measures Designs
•
Major Advantage
•
•
•
Individual differences between participants can
be removed from the analysis
More power
Disadvantage
•
•
•
Risk of carry-over effects from one condition to
the next
Practice effects
Reduce by counterbalancing