More sophisticated ANOVA applications

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More sophisticated ANOVA applications

• Repeated measures and factorial
• PSY295-001 SP2003

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Major Topics

• What are repeated-measures?
• An example
• Assumptions
• Advantages and disadvantages
• Review questions

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Effects of Counseling For Post-Traumatic Stress
Disorder

• Foa, et al. (1991)
• Provided supportive counseling (and other
  therapies) to victims of rape
• Do number of symptoms change with time?
• Point out lack of control group
• Not a test of effectiveness of supportive
  counseling
• Foa actually had controls.

Cont.
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Effect of Counseling--cont.

• 9 subjects measured before therapy, after
  therapy, and 3 months later
• We are ignoring Foas other treatment conditions.

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Therapy for PTSD

• Dependent variable number of reported symptoms.
• Question--Do number of symptoms decrease over
  therapy and remain low?
• Data on next slide

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The Data
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Plot of the Data
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Preliminary Observations

• Notice that subjects differ from each other.
• Between-subjects variability
• Notice that means decrease over time
• Faster at first, and then slower
• Within-subjects variability

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Partitioning Variability
Total Variability
Between-subj. variability
Within-subj. variability
Time
Error
This partitioning is reflected in the summary
table.
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Summary Table
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Interpretation

• Note parallel with diagram
• Note subject differences not in error term
• Note MSerror is denominator for F on Time
• Note SStime measures what we are interested in
  studying

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Assumptions

• Correlations between trials are all equal
• Actually more than necessary, but close
• Matrix shown below

Cont.
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Assumptions--cont.

• Previous matrix might look like we violated
  assumptions
• Only 9 subjects
• Minor violations are not too serious.
• Greenhouse and Geisser (1959) correction
• Adjusts degrees of freedom

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Multiple Comparisons

• With few means
• t test with Bonferroni corrections
• Limit to important comparisons
• With more means
• Require specialized techniques
• Trend analysis

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Advantages of Repeated-Measures Designs

• Eliminate subject differences from error term
• Greater power
• Fewer subjects needed
• Often only way to address the problem
• This example illustrates that case.

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Disadvantages

• Carry-over effects
• Counter-balancing
• May tip off subjects

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Major Points

• What is a factorial design?
• An example
• Main effects
• Interactions
• Simple effects

Cont.
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Major Points-cont.

• Unequal sample sizes
• Magnitude of effect
• Review questions

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What is a Factorial

• At least two independent variables
• All combinations of each variable
• R X C factorial
• Cells

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Video Violence

• Bushman study
• Two independent variables
• Two kinds of videos
• Male and female subjects
• See following diagram

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2 X 2 Factorial
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Bushmans Study-cont.

• Dependent variable number of aggessive
  associates
• 50 observations in each cell
• We will work with means and st. dev., instead of
  raw data.
• This illustrates important concepts.

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The Data (cell means and standard deviations)
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Plotting Results
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Effects to be estimated

• Differences due to videos
• Violent appear greater than nonviolent
• Differences due to gender
• Males appear higher than females
• Interaction of video and gender
• What is an interaction?
• Does violence affect males and females equally?

Cont.
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Estimated Effects--cont.

• Error
• average within-cell variance
• Sum of squares and mean squares
• Extension of the same concepts in the one-way

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Summary Table
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Conclusions

• Main effects
• Significant difference due to video
• More aggressive associates following violent
  video
• Significant difference due to gender
• Males have more aggressive associates than
  females.

Cont.
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Conclusions--cont.

• Interaction
• No interaction between video and gender
• Difference between violent and nonviolent video
  is the same for males (1.5) as it is for females
  (1.4)
• We could see this in the graph of the data.

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Elaborate on Interactions

• Diagrammed on next slide as line graph
• Note parallelism of lines
• Means video differences did not depend on gender
• A significant interaction would have nonparallel
  lines
• Ordinal and disordinal interactions

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Line Graph of Interaction
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Simple Effects

• Effect of one independent variable at one level
  of the other.
• e.g. Difference between males and females for
  only violent video
• Difference between males and females for only
  nonviolent video

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Unequal Sample Sizes

• A serious problem for hand calculations
• Can be computed easily using computer software
• Can make the interpretation difficult
• Depends, in part, on why the data are missing.

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Minitab Example

• Analysis of Variance for AGGASSOC
• Source DF SS MS F
  P
• GENDER 1 66.1 66.1 4.49
  0.035
• VIDEO 1 105.1 105.1 7.14
  0.008
• Interaction 1 0.1 0.1 0.01
  0.927
• Error 196 2885.6 14.7
• Total 199 3057.0

Cont.
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Minitab--cont.
Individual 95 CI GENDER Mean
-------------------------------------- 1
6.95 (------------------
--) 2 5.80 (--------------------)
---------------------------
----------- 5.60
6.30 7.00 7.70 Individual 95 CI VIDEO
Mean --------------------------------
------ 1 7.10
(-----------------) 2 5.65
(-----------------)
--------------------------------------
5.60 6.40 7.20
8.00