2 factor ANOVA

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2 factor ANOVA
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2 X 3 Dose by drug (row by column)
Drug A Drug B Drug C
High dose Sample 1 Sample 2 Sample 3
Low dose Sample 4 Sample 5 Sample 6
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We would test for

• Mean differences between doses (factor A)
• 2) Mean differences between drugs (factor B)
• 3) Mean differences produced by the two factors
  acting together - interactions

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• All three tests combined into one
• reduces chance of type I error.
• Get three F ratios -one for each test

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Total variability
Between treatments variability
Within treatments variability
Factor A
Interaction
Factor B
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• F variance between the means for Factor A (row
  means)
• variance expected from error

F variance between the means for Factor B
(column means) variance expected from
error
F variance not explained by main effects
variance expected from error
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What is a main effect?

• We collapse across one factor and compare the
  other factor.

18 student class 24 student class 30 student class
Program 1 85 80 75 80
Program 2 75 70 65 70
80 75 70
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18 student class 24 student class 30 student class
Program 1 85 80 75 80
Program 2 75 70 65 70
80 75 70
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Interactions

• There is an interaction if the effect of one
  factor depends on the level of another factor.
• The influence of one variable changes according
  to the level of another variable.
• A X B interaction

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Degrees of freedom 2 factor Factor A has a
levels and df a-1 Factor B has b levels and df
b-1 Interaction df (a-1)(b-1) Error df N - ab
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All possible combinations of a 2 factor analysis
Yes A Yes B Yes AXB
Yes A No B Yes AXB
No A Yes B Yes AXB
No A No B Yes AXB
Yes A Yes B No AXB
Yes A No B No AXB
No A Yes B No AXB
No A No B No AXB
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Example College DrinkingThere is a line
connecting each level of the means for A1, A2,
and A3. If we did not consider Gender, then this
line would be the main effect of A. In Panel 1,
the means differ so there would be a main effect
of A. This is not the case in Panel 2.In Panel
3, all the B1s are higher than B2. Thus, there is
a main effect of B. Note that A has no effect for
B1 or B2. So there is no effect of A or an AB
interaction. The latter is because A has no
differential effect dependent on B.In Panel 4,
A has an effect such that as Year increases
drinking increases. Gender (B) has an effect as
the B1 line is higher than the B2 line.Big Point
Alert - The B1 line is parallel to B2 line. Year
effects each Gender in the same fashion. There is
No Interaction!In Panel 5, A has an effect such
that as drinking increases with age for all
groups. B has an effect, as the B1 line is higher
than the B2 line.Big Point Alert - The B1 line
is Steeper than the B2 line. Thus, the year can
have a different effect dependent on your gender.
The lines are not parallel. This is an
Interaction.In Panel 6 the slopes are opposite.
This is a cute one. If you compute the average
for B1 vs. B2, forgetting about Year, then there
is no effect of gender. If you average the Years
forgetting about Gender, the years don't differ.
You would get No main effects. There is only an
Interaction. That's why we check it first.
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In testing food products for palatability,
General Foods employed a 7-point scale from -3
(terrible to 3 (excellent) with 0 representing
"average". The experiment reported here
involved the effects on palatability of a course
versus fine screen and of a low versus high
concentration of a liquid component.
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Mean food palatability determined by screen size
and liquid composition
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• There was a significant effect of liquid
  concentration on palatability. F(1,12) 27.27
  plt.05 MS 10609. The higher concentration of the
  liquid component was reported to be significantly
  more palatable than the lower concentration.
• There was no significant effect of screen size,
  F(1,12) 2.632 pgt.05 MS 1024.
• There was no significant interaction, F(1,12)
  1.08,
• pgt.05, MS 420.25.

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Reporting results of complex design

• What kind of test
• description of variables and definitions of
  levels (conditions) of each
• summary statistics for cells in design matrix
  (figure)
• report F tests for main effects and interactions
• effect size
• statement of power for nonsignificant results
• description of statistically significant main
  effect
• analytical comparisons post hoc where appropriate
  to clarify sources of systematic variation
• simple post hoc effects analysis when interaction
  is statistically significant
• description of statistically significant
  interactions looking at cell means
• conclusion from analysis

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Total variability
Between treatments variability
Within treatments variability
Factor A
Interaction
Factor B
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The analysis is based on the following linear
model
We will calculate F-ratios for each of these FA
FB FAB
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• MS A, MS B, MS AB, MS Within Groups
• MS A estimates s2a
• MS B estimates s2 ß
• MS AB estimates s2aß
• MS Within-Groups estimates s2e

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design