Factorial ANOVA
One-way ANOVA has one independent variable, factorial ANOVA has two or more independent
variables making it similar to multiple regression.
ANOVA has categorical independent variables, multiple regression has either categorical or
numeric independent variables.
ANOVA needs similar numbers of observations in each cell, while multiple regression does not.
ANOVA is hard to interpret with more than 3 independent variables, while multiple regression
is better with lots of independent variables.
ANOVA is better with controlled experiments and multiple regression for naturalistically
obtained data.
Doing multiple t-tests for each variable doesn't let you see interactions.
Interaction When one independent variable affects another independent variable.
10 0
90
80
70
60
50
M en
Wome n
40
30
20
10
0
M etho d 1
What does this show?
M e tho d 2
12 0
10 0
80
60
M en
Wome n
40
20
0
M etho d 1
M e tho d 2
What does this show?
10 0
90
80
70
60
50
M en
Wome n
40
30
20
10
0
M etho d 1
What does this show?
M e tho d 2
70
60
50
40
M en
Wome n
30
20
10
0
M etho d 1
M e tho d 2
What does this show?
Between groups variables Different people are in each group. (Similar to an independent samples ttest).
Example: One group gets method one and the other method 2.
Within groups variables The same people are in different groups. (Similar to a paired t-test).
Example: People respond to test items that are verbs and nouns (or high, medium, low
frequency collocates).
Example taken from http://www.ling.ohio-state.edu/~ehume/papers/Hall-Hume-Johnson-Reiter.pdf
Subjects rated pairs of words differing (or not) in the consonants [d ð], [ð ɾ] [d ɾ].
Dependent variable: They judged them on a 5 point scale (1=very similar, 5= not similar at all).
Independent variables:
The phone pair ([d ð], [ð ɾ], [d ɾ], [ð ð], [ɾ ɾ], [d d])
Listener group: Native Spanish, Intermediate Spanish, Beginning Spanish, No Spanish
Results
Average standardized rating scores were analyzed in a repeated measures analysis of variance
with factors: consonant PAIR ([d ð], [ð ɾ], [d ɾ], [ð ð], [ɾ ɾ], [d d]), and listener GROUP
(Native, Intermediate, Beginner, no Spanish).
PAIR main effect (F[5,280] = 547, p < 0.01) - ratings to same pairs (e.g.d/d) were lower
than ratings to "different pairs" (e.g. d/ð).
PAIR * GROUP interaction (F[15,280] = 6.18, p < 0.01) - the pattern of PAIR ratings
differed depending on which group the listener is in. Linguistic knowledge
affected the rating patterns. This is shown in figure 2.
Since Spanish speakers don't distinguish [d ð] as belonging to separate phonemes, but English speakers
do, there is a difference on this pair.
Another example Speakers of different languages are taught under different conditions.
In this ANOVA the main effects are L1 and condition. The interaction effect is L1 by condition.
Important note: If an interaction is significant that overrides any main effect of the variables in that
interaction.
How do you deal with interactions? Make graph to see interactions, inspect post hoc tests to see which
ones are significant.
Reporting Factorial ANOVA results
Main Effect of L1: F (3, 566) = 27.19, p < 0.0005
Main Effect of Condition: F (2, 566) = 552.29, p < 0.0005
L1 by Condition: F (6, 566) = 7.31, p < 0.0005
Practice
How much vocabulary children learned (GNSC1.1) was measured. In some cases children heard music
(MusicT1) while learning, in others they were shown pictures (Pictures T1). Sometimes both were
experienced at the same time. It is also possible that students of different genders (Gender) may learn
differently with these methods.
Open this file.
Click on Analyze > General linear model > Univariate
Put three independent variables in Fixed factors and GNSC!.1 in Dependent variable box
Click on Post hoc and move independent variables into box.
Mark Tukey and LSD > Continue > OK
Click on Options > mark Descriptive statistics and move variable into Display means box >
Mark descriptive statistics
Main Effects
Interaction Effects
Tests of Between-Subjects Effects
Dependent Variable:gnsc1.1
Source
Type III Sum of
Squares
Corrected Model
df
Mean Square
F
Sig.
22.399a
7
3.200
1.393
.226
Intercept
79.125
1
79.125
34.456
.000
gender
10.302
1
10.302
4.486
.039
1.668
1
1.668
.726
.398
MusicT1
.560
1
.560
.244
.623
gender * PicturesT1
.253
1
.253
.110
.741
gender * MusicT1
5.788
1
5.788
2.521
.118
PicturesT1 * MusicT1
1.534
1
1.534
.668
.417
gender * PicturesT1 *
1.369
1
1.369
.596
.443
Error
128.601
56
2.296
Total
232.000
64
Corrected Total
151.000
63
PicturesT1
MusicT1
a. R Squared = .148 (Adjusted R Squared = .042)
How much of the variance is accounted for?
Hands on practice 1
People were judged on how good their accent was (SentenceAccent). What is the effect of gender (sex)
and whether they studied in an immersion setting (Status: early, non, late) on their accent?
Open this file.
Click on Analyze > General linear model > Univariate
Put independent variables in Fixed factors and dependent in Dependent variable box
Click on Post hoc and move independent variables into box.
Mark LSD > Continue > OK
Click on Model > Sum of squares > Type II
Click on Options > mark Descriptive statistics and move variable into Display means box >
Mark descriptive statistics
1 How would you report the statistical results?
2 What variables are significant?
3 Is the interaction significant?
4 How much of the variance is accounted for?
5 What are the average scores for men and women?
6 What are the average scores for the three statuses?
7 Which of the statuses are significantly different from each other?