Repeated Measures ANOVA - Overview
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Transcript of Repeated Measures ANOVA - Overview
Independent-measures analysis of
variance (Oneway ANOVA)
Repeated measures designs (t-tests)
Individual differences
Oneway ANOVA
WithinTotal
i
Between
i
iiiiWithin
Total
SSor SS N
G
n
TSS
group eachfor n
XXMXSS
N
GX
N
XXMXSS
i
22
2
22
22
2
22
)()(
)()(
Independent Samples
Compare two groups that are unrelated to each other
Numerator is difference between groups
Does not control for the impact of individual differences
Related Samples
Compare two measures from one person or one related pair of people
Numerator is difference within pair
Controls for the impact of individual differences
Null hypothesis: in the population, there are no
mean differences among the treatment groups
Alternate hypothesis states that there are
mean differences among the treatment groups.
...: 3210 H
H1: At least one treatment mean μ
is different from another
removed) sdifference individual(
effect, treatmentno withexpected es)(differenc variance
s treatmentbetween es)(differenc varianceF
F ratio based on variances
• Same structure as independent measures
• Variance due to individual differences is
not present
Participant characteristics that vary from one person to another.
• Not systematically present in any treatment group or by research design
Characteristics may influence measurements on the outcome variable
• Eliminated from the numerator by the research design
• Must be removed from the denominator statistically
Numerator of the F ratio includes
• Systematic differences caused by treatments
• Unsystematic differences caused by random factors (reduced because same individuals in all treatments)
Denominator estimates variance reasonable to expect from unsystematic factors
• Effect of individual differences is removed
• Residual (error) variance remains
Total Variability
Between Treatments
Within Treatments
Between Subjects
Error Variance
Repeated Measures ANOVA
(Equations follow)
First stage
• Identical to independent samples ANOVA
• Compute SSTotal =SSBetween treatments + SSWithin treatments
Second stage
• Removing the individual differences from the denominator
• Compute SSBetween subjects and subtract it from SSWithin treatments to find SSError
There is a computational equation for the Sum of Squared Deviations from the Mean:
With ANOVA, we sum groups of means
• G2 = Sum of all the scores squared
• T1 = sum of scores in Treatment Group 1
• P1 = sum of scores of Person 1 across time
• TA 1 and TB 1 = sums of scores across levels of a factor.
N
XX
22 )(
N
GXSStotal
22
treatment each insidetreatmentswithin SSSS
N
G
n
TSS treatmentsbetween
22
Note that this is the
Computational Formula
for SS
N
G
k
PSSbetween
22
subjects
subjectsbetweentreatments withinerror SSSSSS
dftotal = N – 1
dfwithin treatments = Σdfinside each treatment
dfbetween treatments = k – 1
dfbetween subjects = n – 1
dferror = dfwithin treatments – dfbetween subjects
error
errorerror
df
SSMS
treatments between
treatments betweenntstreatme between
df
SSMS
error
mentstreat between
MS
MSF
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Solve the ANOVA table problem
Try to finish in 10 minutes (1 minute per slot to complete)
SS df MS F
Between Treatments 10
Within Treatments
Between Subjects 26
Error
Total 110
A sample of n=10 individuals participate in a repeated measures study of 4 treatment conditions. Is there evidence of significant differences among the treatments? Hint: begin with df.
Subject Before 6 months 1 year
A 56 64 69B 79 91 89C 68 77 81D 59 69 71E 64 77 75F 74 88 86G 73 85 86H 47 64 69I 78 98 100J 61 77 85K 68 86 93L 64 77 87M 53 67 76N 71 85 95O 61 79 97P 57 77 89Q 49 65 83R 71 93 100S 61 83 94T 58 75 92U 58 74 92
21 subjects were in the study
The dependent variable was their Growth Mindset, the belief in the ability for intelligence to grow.
They received two months of coaching on how to grow problem solving intelligence.
Their Growth Mindset was measured on a 100-point scale at three times: • before the program,
• six months after it started,
• one year after it started
Much more output than you want
Need to ask for some Options to get SPSS to do as much of the work as possible.
• Descriptives
• Plots
• Multiple Comparisons
• Effect Size
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Follow steps on diagram at right
Percentage of variance explained by the
treatment differences
Partial η2 is percentage of variability that has
not already been explained by other factors
error
treatments between
subjectsbetweentotal
treatments between
SS
SS
SSSS
SS
2
Determine exactly where significant
differences exist among more than two
treatment means
• Tukey’s HSD can be used (almost always
same number of subjects) or Scheffé if
dropouts mean unequal measures.
• Substitute SSerror and dferror in the formulas
Can people increase their actual problem-solving intelligence by
learning that their brains are able to change? Researchers found
the Growth Mindset of 21 people increased from just before they
began a Growth Mindset Coaching Program (M=63.3, s=8.9),
when measured six months (M=78.6, s=9.6) and one year after
starting the program (M=86.1, s=9.6). The differences were
significant (F(2,40)=121.89, p<.001). Post-hoc tests with
Bonferroni correction showed that self-efficacy at each time was
significantly improved from the one before. In fact, the impact of
the training program was large, with about 86% of the variability
in Problem Solving Growth Mindset related to the training
(2=.859). The coaching program seems to be an effective way
to help people tap into the potential of the Growth Mindset.
The observations within each treatment
condition must be independent.
The population distribution within each
treatment must be normal.
The variances of the population
distribution for each treatment should be
equivalent.
Decide if each of the following statements
is True or False.
• For the repeated-measures ANOVA, degrees of freedom for SSerror is (N–k) – (n–1).
T/F
• N is the number of scores and nis the number of participantsFalse