lecture slides stats1.13.l18.air
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Lecture slides stats1.13.l18.airTRANSCRIPT
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Statistics One
Lecture 18 Repeated measures ANOVA
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Two segments
• Repeated measures: Pros & Cons • Repeated measures: Example
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Lecture 18 ~ Segment 1
Repeated measures Pros & Cons
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Repeated measures: Pros & cons
• Pros – Less cost (fewer subjects required) – More statistical power
• This is the important new concept
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Repeated measures: Pros & cons
• Working memory training example
• Four independent groups (8, 12, 17, 19) – There were 20 subjects per group – Total N = 80
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Repeated measures: Pros & cons
• Working memory training example
• Repeated measures design – N = 20
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Repeated measures: Pros & cons
• More statistical power – Variance across subjects may be systematic – If so, it will not contribute to the error term
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Between groups design (SS)
Systematic/ Between (A)
Unsystematic/ Within (S/A)
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Repeated measures design (SS)
Between (A)
Subjects(S)
Unsystematic = (AxS)
Error in a repeated measures design is the inconsistency of subjects from one condition to another Therefore: FA = MSA / MSAxS
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MS and F
• MSA = SSA / dfA • MSAxS = SSAxS / dfAxS
• F = MSA / MSAxS
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Repeated measures: Pros & cons
• Cons – Order effects – Counterbalancing – Missing data – Extra assumption
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Counterbalancing • Consider a simple design with just two
conditions, A1 and A2 • One approach is a Blocked Design – Subjects are randomly assigned to one of two
“order” conditions • A1, A2 • A2, A1
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Counterbalancing • Another approach is a Randomized Design – Conditions are presented randomly in a
mixed fashion • A2, A1, A1, A2, A2, A1, A2…..
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Counterbalancing
• Now suppose a = 3 and a blocked design • There are 6 possible orders (3!)
• A1, A2, A3 • A1, A3, A2 • A2, A1, A3 • A2, A3, A1 • A3, A1, A2 • A3, A2, A1
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Counterbalancing
• To completely counterbalance, subjects would be randomly assigned to one of 6 order conditions
• The number of conditions needed to completely counterbalance becomes large with more conditions – 4! = 24 – 5! = 120
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Counterbalancing
• With many levels of the IV a better approach is to use a “Latin Squares” design
• Latin Squares designs aren’t completely counterbalanced but every condition appears at every position at least once
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Counterbalancing
• For example, if a = 3, then – A1, A2, A3 – A2, A3, A1 – A3, A1, A2
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Missing data
• Two issues to consider – Relative amount of missing data – Pattern of missing data
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Missing data ~ Relative amount
• How much is a lot? – No hard and fast rules – A rule of thumb is
• Less than 10% on any one variable, OK • Greater than 10%, not OK
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Missing data ~ Pattern?
• Is the pattern random or lawful? – This can easily be detected – For any variable of interest (X) create a new variable (XM)
• XM = 0 if X is missing • XM = 1 if X is not missing
– Conduct a t-test with XM as the IV – If significant then pattern of missing data may be lawful
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Missing data ~ Remedies
• Drop all cases without a perfect profile – Drastic – Use only if you can afford it
• Keep all cases and estimate the values of the missing data points – There are several options for how to estimate
values
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Sphericity assumption
• Homogeneity of variance • Homogeneity of covariance
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Sphericity assumption
• How to test? – Mauchly’s test
– If significant then report an adjusted p-value • Greenhouse-Geisser • Huyn-Feldt
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Segment summary
• Pros – Less cost (fewer subjects required) – More statistical power
• This is the important new concept
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Segment summary
• Cons – Order effects – Counterbalancing – Missing data – Extra assumption
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END SEGMENT
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Lecture 18 ~ Segment 2
Repeated measures ANOVA Example
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Repeated measures: Pros & cons
• Working memory training example
• Four independent groups (8, 12, 17, 19) – There were 20 subjects per group – Total N = 80
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Repeated measures: Pros & cons
• Working memory training example
• Repeated measures design – N = 20
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Working memory training
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Common dataframe subject A1 (8) A2 (12) A3 (17) A4 (19)
1 2 3 4 5 6 …
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R dataframe subject condition IQ
1 A1 (8)
1 A2 (12)
1 A3 (17)
1 A4 (19)
2 A1 (8)
2 A2 (12)
…
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Results: ANOVA
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Results: Post-hoc tests (Holm)
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Results: Post-hoc tests (Bonferroni)
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Results: Paired t-test 12 vs. 17
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Comparison of procedures Procedure p-value for 12 vs. 17
Paired t-test 0.0065
Holm 0.0192
Bonferroni 0.0391
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Repeated measures ANOVA
• Appropriate when comparing group means – Three or more group means – Same subjects tested in each condition
– F-test – Post-hoc testss
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END SEGMENT
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END LECTURE 18
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