Seth M' Noar, Ph'D'

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Factorial (Multiple Factor) ANOVA

• Seth M. Noar, Ph.D.
• Department of Communication
• University of Kentucky

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Familywise error rate

• When you run multiple tests you get a greater
  error rate than originally proposed (.05 x 5
  .25)
• This is referred to as familywise error rate
• Bonferroni procedure corrects familywise error
  rate back to original chosen rate (.05 / 5
  .01)
• Also, Tukey follow-up test controls for
  familywise error rate

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Factorial ANOVA

• Single factor (one-way) ANOVA only allows for 1
  IV
• Factorial ANOVA allows for multiple IVs
  (factors)
• When one would apply it
• 2 or more IVs 2 or more nominal levels
• 1 DV continuous (must be interval)

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Example

• IVs medication and gender
• DV measure of health
• 2 x 2 factorial ANOVA
• (2-way ANOVA)

Med
No Med
Male
Female
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Factorial ANOVA Effects

• Main effects and interactions
• Examine
• Main effect of each IV
• Interaction between IVs
• Example
• Main effect of gender
• Main effect of medication
• Interaction between gender and medication

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2 x 2 ANOVA
IV 2
Main Effect 1
IV 1
Main Effect 2
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Med
No Med
15
Male
Female
15
10
20
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Med
No Med
20
Male
Female
10
15
15
9
Med
No Med
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Male
Female
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15
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Logic of Factorial ANOVA

• MS effect (between)
• MS error (within)
• Logic is same as single-factor ANOVA
• Calculate between group variance
• Calculate within group variance
• Divide to get an F value
• We always expect within group variability
• However, we would only expect between group
  variability if the groups were different

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Logic of Factorial ANOVA

• Each main effect examines the significance of
  that particular IV (essentially ignoring the
  other potential effects)
• The interaction examines if there are any
  special effects due to the combination of IVs.
• Is there any systematic variability that is not
  due to other sources?

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F tests

• F tests accompany all main effects and all
  interactions
• If interaction is significant, you cannot
  interpret the main effects
• You must look at the interaction for the more
  complex picture
• Interaction effects of one IV depend on the
  level of another IV

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Main Effects and Interactions

• The more IVs one has, the more main effects and
  interactions
• 2 x 2 x 3 (A, B, C)
• 3 way ANOVA
• 3 main effects
• A, B, and C
• 4 Interactions
• A x B, A x C, B x C, A x B x C

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Correlation
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Correlation

• Allows us to examine relationships / associations
  between variables
• It is a co-relation between 2 variables (Karl
  Pearson, 1895)
• There are various types of correlation
  coefficients

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Correlation (contd)

• Pearson product-moment correlation coefficient
  (r)
• Standard measure of the linear relationship
  between two variables
• Most common correlation coefficient
• When one would apply Pearsons r
• 2 continuous variables (must be interval)

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Correlation (contd)

• Correlations vary both in magnitude and direction
• Magnitude strength of relationship
• Direction positive or negative
• Correlations fall between -1 and 1

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Correlation (contd)

• Positive correlation High values of one
  variable are associated with high values of
  another variable
• Attitude-Behavior, r .40
• Negative correlation High values of one
  variable are associated with low values of
  another variable
• Stress-Immune, r -.35

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Correlation (contd)

• Correlation is NOT causation. Why?
• Causation has much more to do with research
  design than with statistics

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Causation

• 3 necessary conditions for demonstrating
  causation
• Association between variables
• Temporal (time) ordering
• Isolation of effect (ruling out of other
  variables)

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Logic of Correlation

• Examines linear relationship between 2 variables
• Formula examines how much score deviation two
  distributions have in common
• We can also examine significance of r using the t
  distribution
• Null hypothesis is r 0
• What are the chances of our correlation being
  significantly different from zero?

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Logic of Correlation (contd)

• A squared correlation is an effect size
• r .40, r2 .16
• 16 of variance is shared between these variables
• Also, correlation itself is not an interval scale
• .20 squared .04
• .40 squared .16

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Partial Correlation

• Standard correlations do not consider influence
  of other variables
• However, we can partial out variance if we wish
• Partial correlation correlation between 2
  variables with other variance partialed out

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Regression Analysis
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Regression Analysis

• A tool for prediction
• If we know variable X, what does it tell us about
  variable Y?
• When one would apply linear regression
• 2 continuous variables (must be interval)
• Look familiar?

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Questions

• If we know X about a person, what are the chances
  that they will be successful
• In graduate school
• At a certain job
• Whats the single best predictor of
• Health behavior
• Political affiliation
• Success in therapy

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Standard Error of the estimate

• A tool for prediction

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Regression Line

• A tool for prediction

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