Factorial ANOVA
Basic Concepts
Two-Way ANOVA
• We have two grouping variables,
commonly referred to as:
– Factors
– Independent Variables
• best term if manipulated experimentally
– Predictors
– Grouping Variables
– Classification Variables
• We have one continuous variable,
commonly referred to as the
– Dependent variable
• best term if data collected experimentally
– Criterion variable
– Outcome variable
– Response variable
A 2 x 2 Design
• Has two levels of Factor A and two levels
of Factor B.
• This results in four combinations of level of
A and level of B.
• Each such combination is referred to as a
cell.
Your Party is Going Strong
• And then green aliens crash it
Data Collection
• You and colleagues monitor the aliens’
behavior.
• Half of them consume your ethanol-based
punch.
• Half of them sample your room-mate’s
barbiturate tablets.
• Post consumption, you blindly rate each
alien’s level of intoxication, and you
compute cell means.
Green Alien Party Crashers
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
30
10
20
marginal
05
25
15
30
no barb.
20
one barb.
10
0
no alch.
one alch
Main Effects & Marginal Means
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
30
10
20
marginal
05
25
15
• Effect of Alcohol, ignoring Barbiturates
– 20 > 10
– Drinking alcohol increased intoxication.
• Effect of Barbiturates, ignoring Alcohol
– 25 > 5
– Taking Barbiturates increased intoxication.
Simple Main Effects
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
30
10
20
marginal
05
25
15
• The effect of one factor at specified level
of another factor.
• Effect of Alcohol when Barb = none
• 10 – 0 = 10
• Effect of Alcohol when Barb = one
• 30 – 20 = 10
• 10 = 10, no interaction
Simple Main Effects
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
30
10
20
marginal
05
25
15
• Effect of Barb when Alch = none
• 20 – 0 = 20
• Effect of Barb when Alch = one
• 30 – 10 = 20
• 20 = 20, no interaction
Additive Model
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
30
10
20
marginal
05
25
15
• The effect of adding Alch and Barb is the
simple sum of their separate effects.
• Ingest both A and B, intoxication = A + B =
10 + 20 = 30.
• It does not work this way in humans!
Green Aliens
• Lines are parallel, no
interaction
• Barb line higher than
no-barb line, main
effect of barbiturates
• Slope of both lines is
positive, main effect
of alcohol
30
no barb.
20
one barb.
10
0
no alch.
one alch
Comparative Psychology
• For comparative purposes, you also
observe human guests, including Pee Dee
the Pirate.
Humans
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
40
10
25
marginal
05
30
17.5
40
30
no barb.
20
one barb.
10
0
no alch.
one alch.
Main Effects & Marginal Means
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
40
10
25
marginal
05
30
17.5
• Effect of Alcohol, ignoring Barbiturates
– 25 > 10
– Drinking alcohol increased intoxication.
• Effect of Barbiturates, ignoring Alcohol
– 30 > 5
– Taking Barbiturates increased intoxication.
Simple Main Effects
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
40
10
25
marginal
05
30
17.5
• Effect of Alcohol when Barb = none
• 10 – 0 = 10
• Effect of Alcohol when Barb = one
• 40 – 20 = 20
• 10  20, there is an interaction.
Simple Main Effects
Barbiturate
none
one
marginal
Alcohol
none
one
00
10
20
40
10
25
marginal
05
30
17.5
• Effect of Barb when Alch = none
• 20 – 0 = 20
• Effect of Barb when Alch = one
• 40 – 10 = 30
• 20  30, there is an interaction.
Nonadditive Combination
•
•
•
•
One drink makes you 10 units intoxicated.
One barb makes you 20 units intoxicated
Drink & Barb  Drink + Barb, that is,
10 + 20 (+ Interaction) = 40, not 30
Monotonic Interaction
40
30
no barb.
20
one barb.
10
0
no alch.
one alch.
• The lines are not parallel, there is an
interaction.
• Both lines have positive slope, the direction of
effect of alcohol is the same at both levels of
barbiturate (but effect stronger at barb = one).
• The interaction is monotonic.
Interpreting a Monotonic Interaction
• You can still interpret the main effects.
• Drinking alcohol will make you more
intoxicated whether you took a barbiturate
or not.
• --- but if you take a barbiturate, the alcohol
will have a greater effect than it would if
you had not taken a barbiturate.
Purple Aliens Arrive Later
• You observe them too.
Purple Alien Party Crashers
Barbiturate
none
one
marginal
Alcohol
none
one
00
20
30
10
15
15
marginal
10
20
15
30
no barb.
20
one barb.
10
0
no alch.
one alch
Main Effects & Marginal Means
Barbiturate
none
one
marginal
Alcohol
none
one
00
20
30
10
15
15
marginal
10
20
15
• Effect of Alcohol, ignoring Barbiturates
– 15 = 15
– Drinking alcohol did not affect intoxication.
• Effect of Barbiturates, ignoring Alcohol
– 20 > 10
– Taking Barbiturates increased intoxication.
Simple Main Effects
Barbiturate
none
one
marginal
Alcohol
none
one
00
20
30
10
15
15
marginal
10
20
15
• Effect of Alch when Barb = none
• 20 – 0 = 20
• Effect of Alch when Barb = one
• 10 – 30 = -20
• +20  -20, there is an interaction.
Simple Main Effects
Barbiturate
none
one
marginal
Alcohol
none
one
00
20
30
10
15
15
marginal
10
20
15
• Effect of Barb when Alch = none
• 30 – 0 = 30
• Effect of Barb when Alch = one
• 10 – 20 = -10
• 30  -10, there is an interaction.
Nonmonotonic Interaction
30
no barb.
20
one barb.
10
0
no alch.
one alch
• The lines are not parallel, there is an
interaction.
• One line has positive slope, the other
negative, the direction of effect of alcohol
depends on whether a barbiturate was taken.
• The interaction is nonmonotonic.
Interpreting a Nonmonotonic
Interaction
• Probably not a good idea to interpret the
main effects.
• What is the effect of alcohol on purple
aliens?
• Main effect is zero, but alcohol does have
an effect.
• The effect depends on whether a
barbiturate was taken or not.
Interpreting a Nonmonotonic
Interaction
• How do barbiturates affect purple aliens?
• Main effect is it makes them more
intoxicated -• but if they have been drinking alcohol, the
barbiturate has the effect of reducing
intoxication.
• Again, it does not work like this with
humans!
Three-Way Factorial ANOVA
• Three Factors
• Three Main Effects – A, B, and C
• Three Two-Way Interactions
– A x B, A x C, and B x C
• One Three-Way Interaction – A x B x C
Factor C = Species of Party Dude
• We have a Alcohol x Barbiturate x Species
ANOVA
• The A x B interaction differs across levels
of Species.
• Accordingly, we have a triple interaction.
Triple Interaction
Green
Human
40
30
30
no barb.
20
no barb.
20
one barb.
10
one barb.
10
0
0
no alch.
one alch
no alch.
Purple
30
no barb.
20
one barb.
10
0
no alch.
one alch
one alch.
Hypotheses Tested in Two-Way
ANOVA
• 1 = 2 = . . . = a
– That is, the mean of the criterion variable is
constant across the a levels of factor A.
• 1 = 2 = . . . = b
– That is, Factor B does not affect the mean of
the criterion variable.
• Factors A and B do not interact with one
another, A and B combine additively to
influence the criterion variable.