Chapters 11&12

1
Chapters 1112

• Factorial and Mixed Factor ANOVA and ANCOVA

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ANOVA Review

• Compare 2 mean scores
• One way (1 factor or IV)
• Repeated measures (multiple factors)
• Main effects
• Interactions
• F-ratio
• P-value
• Post hoc tests and corrections
• Within and between

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Multiple Factor ANOVA

• aka Factorial ANOVA incorporates more than one
  IV (factor).
• Only one DV
• Factor IV
• Levels are the groups within each factor.
• In the reaction time example, there was one
  factor (drug) with three levels (beta blocker,
  caffeine, and placebo).
• Mixed factor is both within and between in the
  same analysis.

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Factorial ANOVA Example

• Studies are explained by their levels
• 2 x 3 or 3 x 3 x 4
• The effect of three conditions of muscle glycogen
  at two different exercise intensities on blood
  lactate. There are 2 IV (factors glycogen and
  exercise intensity) and 1 DV (blood lactate).
• 3 levels of muscle glycogen depleted, loaded,
  normal.
• 2 levels of exercise intensity 40 and 70
  VO2max.
• 2 x 3 ANOVA, two-way ANOVA.
• 60 subjects randomized to the 6 cells (n 10 per
  cell). Between subjects.

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Factorial ANOVA Example

• Each subject, after appropriate glycogen
  manipulation, performs 30 minute cycle ergometer
  ride at either low intensity (40) or high
  intensity (70).
• Blood is sampled following ride for lactate level.

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3 F ratios in 2-way ANOVA

• 2 Main Effects a main effect looks at the
  effect of one IV while ignoring the other IV(s),
  i.e., collapsed across the other IV(s). Based
  on the marginal means (collapsed).
• Main effect for Intensity
• based on row marginal means (collapsed across
  glycogen state).
• If significant, look at mean values to see which
  one is larger (since there are only 2 means).

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3 F ratios in 2-way ANOVA

• Main effect for glycogen state
• Compare column marginal means.
• If significant, perform follow-up procedures on
  the 3 means (collapsed across intensity).
• Main effects are easily followed up if the
  interaction (see below) is not significant.
• Each main effect is treated as a single factor
  ANOVA while ignoring the other factor.
• If the interaction is significant, focus on the
  interaction even if the main effects are
  significant. Ignore the main effects

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Exercise Intensity Marginal Means
Glycogen Condition
Exercise Intensity
Glycogen Marginal Means
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Main Effect for Intensity
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Main Effect for Glycogen
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3 F ratios in 2-way ANOVA

• Interaction does the effect of one IV (factor)
  change across levels of the other factor(s).
• Significant interaction indicates that the
  effects of muscle glycogen on blood lactate
  differs across levels of exercise intensity.
• Or equivalently, a significant interaction
  indicates that the effects of exercise intensity
  on blood lactate differs across different
  levels of muscle glycogen.
• Interactions tell you that the slopes of lines of
  the plotted data are not parallel.
• In other words the groups did not react the same
  way.

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Interactions

• The first F ratio to consider is the highest
  order (most complicated) interaction. In this
  example, there is only one interaction.
• If the interaction is significant, then ignore
  the main effects and analyze the interaction.
• When a significant interaction occurs, the main
  effects can be misleading.

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Interaction of Intensity and Glycogen
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Interactions

• Options for Follow Up Procedures
• Perform multiple pairwise comparisons need to
  control familywise Type I error rate.
  (Bonferroni)
• Tests of Simple Main Effects
• Compare cell means within the levels of each
  factor.
• Examples
• 1. Perform two 1x3 ANOVAs one for each level of
  exercise intensity.
• 2. Perform three 1x2 ANOVAs (single df
  comparisons, t tests) for each level of glycogen
  state.
• 3. Perform both 1 and 2 above.

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Or
and
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Interactions

• Options for Follow Up Procedures (continued)
• Analysis of interaction comparisons transform
  the factorial into a set of smaller factorials.
• Plot interaction and describe.
• The choice of follow-up procedure depends on the
  research question(s) one may be better in one
  situation vs. another.

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Main Effect for Intensity
Main Effect for Glycogen
Interaction of Intensity and Glycogen
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ANCOVA

• Analysis of Covariance
• Combined use of ANOVA and Regression
• Adjust for covariate by regressing covariate on
  the DV, then doing an ANOVA on the adjusted DV.
• Can remove pre-treatment variations (as measured
  by the covariate) from the post-treatment means
  prior to testing groups for differences in the
  DV.
• Example compare strength in subjects who did
  Swiss Ball exercise vs. controls.
• Randomization may not equate groups on body
  weight.
• Covary for body weight prior to comparing groups.

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ANCOVA

• Issues with ANCOVA
• Covariate should be highly correlated with DV.
• Covariate should not be correlated with IV.
• Homogeneity of Regression
• Slopes of regression lines between covariate and
  DV must be equal across levels of the IV.
• Violation implies an interaction between the
  covariate and IV.
• Groups may differ on other variables that are not
  adjusted.
• Abuse arguably inappropriate to correct for
  pre-existing group differences if those groups
  were not formed by randomization.

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ANCOVA

• Advantages
• More Power due to decreased variance that must
  be explained by the IV (smaller error term in the
  F ratio).
• Covariate accounts for some of the variance in
  the DV ? ? variance that must be explained by IV
  to reach significance.
• Some suggest use of covariate solely to increase
  power.
• Adjusts for pre-treatment differences between
  groups.
• If pre-treatment differences exist because groups
  were not randomly formed, then ANCOVA will not
  magically eliminate the bias that may exist with
  non-random assignment.

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Next Class

• Tonight factorial ANOVA and ANCOVA in lab and
  stat practice
• Research paper due and stat practice
• Final exam next week