Using ANOVA to test two IVs at once.

1
Using ANOVA to test two IVs at once.
2
Using ANOVA to testing two IVs at once.

• Why test two IVs at once?
• An example experiment
• Hypotheses that are tested
• Possible outcomes eight graphs that say it all
• What the ANOVA table looks like

3
Why test two IVs at once?

• Testing two or more IVs is called a factorial
  design
• When two factors (IVs) are addressed in a
  single experiment it is called a two-factor
  design
• Several sets of hypotheses are tested with a
  two-factor ANOVA
• Why design a two-factor experiment?
• Efficiency of time and money test 3 hypotheses
  all at once
• Test if IV1 influences the DV
• Test if IV2 influences the DV
• Test if the effects of one IV on the DV depend
  upon the other.
• Power Enhance the sensitivity of your test
  relative to two separate one-factor designs

4
An example experiment simplified from page 421

• A sleep researcher is interested in whether a
  drug marketed as a remedy for insomnia really
  does induce sleep. She is also interested in
  whether tolerance to the drugs putative
  sleep-inducing effects develops with repeated use
  of the drug. She knows that she could test these
  ideas using separate one-factor studies, but only
  has time to do one study. She decides to give
  either placebo or active drug (IV1 2 levels) to
  people with different experience with this drug,
  either people who have never used it or people
  who use it every night (IV2 2 levels). 32
  people are recruited for a one night study 16
  people who have never used the drug and 16 people
  who use it every night. These two groups are
  split 8 in each group get placebo and 8 get the
  active drug. The DV is the number of minutes it
  takes for these folks to get to sleep.

5
Diagram of sleep study 2 by 2 matrix
6
Diagram of sleep study 2 by 2 matrix
Drug (columns)
Placebo
Active
8 people who have never used are given placebo
medication.
8 people who have never used are given active
medication.
Never used
Experience (rows)
8 people who use nightly are given Placebo
medication.
8 people who use nightly are given active
medication.
Use nightly
7
Diagram of sleep study 2 by 2 matrix
Drug (columns)
Placebo
Active
Row mean (16 scores)
Cell mean (mean of 8 scores)
Cell mean (mean of 8 scores)
Never used
Experience (rows)
Row mean ( 16 scores)
Cell mean (mean of 8 scores)
Cell mean (mean of 8 scores)
Use nightly
Column mean (16 scores)
Column mean (16 scores)
8
Diagram of sleep study 2 by 2 matrix
Used to test effect of row IV.
Drug (columns)
Placebo
Active
Row mean (16 scores)
Cell mean (mean of 8 scores)
Cell mean (mean of 8 scores)
Never used
Used to test interaction of 2 IVs
Experience (rows)
Row mean ( 16 scores)
Cell mean (mean of 8 scores)
Cell mean (mean of 8 scores)
Use nightly
Used to test effect of column IV.
Column mean (16 scores)
Column mean (16 scores)
9
Questions to be answered in sleep study

• Does the drug influence the time it takes to get
  to sleep?
• (main effect of drug)
• Does experience with the drug influence time it
  takes to get to sleep?
• (main effect of experience)
• Does the drugs effect on time it takes to get to
  sleep depend upon the users experience with it?
• (interaction of effect of drug and experience)

10
Null and alternative hypotheses 3 sets!
Main effect of column IV1 (Drug) H0 The column
means are equal HA The column means are not
equal Main effect of row IV2 (Experience) H0
The row means are equal HA The row means are
not equal Main effect of interaction of the two
IVs H0 The cell means are equal HA The cell
means are not equal
The drug produced an effect!
Experience produced an effect!
The effect of the drug depends On how much
experience people have with it!
11
There are eight possible outcomes

• For each possible outcome, Ill show you
• Some sample means that are consistent with that
  outcome.
• A graph of those means.
• A partially completed ANOVA table.
• On the test you will need to
• Know how to produce graphs from means and/or
  sample data.
• Know how to interpret graphs.
• Understand two-factor ANOVA tables.
• You will NOT have to understand two-factor df.

12
No main effects and no interaction
Drug (columns)
Placebo
Active
35.5
34.5
35.0
Never
Experience (rows)
35.5
36.5
36.0
Nightly
35.5
35.5
13
No main effects and no interaction
Drug (columns)
Placebo
Active
35.5
34.5
35.0
Never
Experience (rows)
35.5
36.5
36.0
Nightly
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
14
No main effects and no interaction
Placebo
Drug (columns)
70
Placebo
Active
Active
35.5
34.5
35.0
35
Never
Time to sleep
Experience (rows)
0
35.5
36.5
36.0
Nightly
Never
Nightly
Experience
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
15
No main effects and no interaction
Placebo
Drug (columns)
70
Placebo
Active
Active
35.5
34.5
35.0
35
Never
Time to sleep
Experience (rows)
0
35.5
36.5
36.0
Nightly
Never
Nightly
Experience
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
16
No main effects and no interaction
Placebo
Drug (columns)
70
Placebo
Active
Active
35.5
34.5
35.0
35
Never
Time to sleep
Experience (rows)
0
35.5
36.5
36.0
Nightly
Never
Nightly
Experience
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
17
No main effects and no interaction
Placebo
Drug (columns)
70
Placebo
Active
Active
35.5
34.5
35.0
35
Never
Time to sleep
Experience (rows)
0
35.5
36.5
36.0
Nightly
Never
Nightly
Experience
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
18
No main effects and no interaction
Drug (columns)
Placebo
70
Placebo
Active
Active
35.5
34.5
35.0
35
Never
Time to sleep
Experience (rows)
0
35.5
36.5
36.0
Nightly
Never
Nightly
Experience
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
19
No main effects and no interaction
Drug (columns)
Placebo
70
Placebo
Active
Active
35.5
34.5
35.0
35
Never
Time to sleep
Experience (rows)
0
35.5
36.5
36.0
Nightly
Never
Nightly
Experience
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
20
No main effects and no interaction
Drug (columns)
Placebo
70
Placebo
Active
Active
35.5
34.5
35.0
35
Never
Time to sleep
Experience (rows)
0
35.5
36.5
36.0
Nightly
Never
Nightly
Experience
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
21
No main effects and no interaction
Drug (columns)
Placebo
70
Placebo
Active
Active
35.5
34.5
35.0
35
Never
Time to sleep
Experience (rows)
0
35.5
36.0
36.5
Nightly
Never
Nightly
Experience
35.5
35.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X Tota
l XX.X 31
22
Main effect of Drug, no effect of Exp. or
interaction
Placebo
Drug (columns)
70
Active
Placebo
Active
35.5
25.5
15.5
Never
35
Time to sleep
Experience (rows)
0
35.5
15.5
25.5
Nightly
Never
Nightly
Experience
35.5
15.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X lt.05 Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
23
Main effect of Exp., no effect of Drug or
interaction
Placebo
Drug (columns)
70
Active
Placebo
Active
35.5
35.0
34.5
35
Never
Time to sleep
Experience (rows)
0
55.5
54.5
55.0
Nightly
Never
Nightly
Experience
45.5
44.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X lt.05 Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X Tota
l XX.X 31
24
Interaction and no main effects
Drug (columns)
Placebo
70
Placebo
Active
Active
40.0
20.0
30.0
35
Never
Time to sleep
Experience (rows)
0
20.0
30.0
40.0
Nightly
Never
Nightly
Experience
30.0
30.0
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X lt.05 Within (Error) XX.X 28 XX.X Tota
l XX.X 31
25
Main effects of Drug and Exp., no interaction
Placebo
Drug (columns)
70
Active
Placebo
Active
35.5
15.5
25.5
Never
35
Time to sleep
Experience (rows)
55.5
0
35.5
45.5
Nightly
Never
Nightly
Experience
45.5
25.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X lt.05 Columns (Drug) XX.X
1 XX.X XX.X lt.05 Interaction (EC) XX.X
1 XX.X XX.X n.s. Within (Error) XX.X 28 XX.X Total
XX.X 31
26
Main effect of Drug and Interaction, no Exp. Drug
Placebo
Drug (columns)
70
Active
Placebo
Active
37.5
31.5
25.5
Never
35
Time to sleep
Experience (rows)
0
35.5
30.5
33.0
Nightly
Never
Nightly
Experience
36.5
28.0
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X n.s. Columns (Drug) XX.X
1 XX.X XX.X Interaction (EC) XX.X
1 XX.X XX.X lt.05 Within (Error) XX.X 28 XX.X Tota
l XX.X 31
27
Main effect of Exp. and Interaction, no Drug
effect.
Drug (columns)
Placebo
70
Placebo
Active
Active
28.5
29.0
29.5
Never
35
Time to sleep
Experience (rows)
40.5
35.5
38.0
0
Nightly
Never
Nightly
Experience
34.5
32.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X lt.05 Columns (Drug) XX.X
1 XX.X XX.X n.s. Interaction (EC) XX.X
1 XX.X XX.X lt.05 Within (Error) XX.X 28 XX.X Tota
l XX.X 31
28
Both main effects and interaction are significant
Drug (columns)
Placebo
70
Placebo
Active
Active
35.5
23.0
10.5
Never
35
Time to sleep
Experience (rows)
40.5
30.5
35.5
0
Nightly
Never
Nightly
Experience
38.5
20.5
Source SS df S2 Fobt p Rows (Exp.) XX.X
1 XX.X XX.X lt.05 Columns (Drug) XX.X
1 XX.X XX.X lt.05 Interaction (EC) XX.X
1 XX.X XX.X lt.05 Within (Error) XX.X 28 XX.X XX.X
Total XX.X 31
29
An example test problem.

• Most people believe that repetition aids recall
  and some people feel that the learning of new
  material interferes with the recall of previously
  learned material, especially if the new material
  is very much like the previously learned
  material. A cognitive scientist tests the
  separate and combined influence of these two
  variables by asking 36 subjects to recall 16
  nonsense syllable (e.g., RET). The effects of
  repetition are assessed by presenting the
  nonsense syllable pairs to groups of subjects (12
  each) 4, 8, or 12 times. Before testing each
  subject for recall all subjects are required to
  learn some intervening material, either
  dissimilar to the syllables (i.e., number n
  18) or similar to syllables (i.e., more
  syllables n 18). There are 6 cells in this 2
  factor experiment, and 6 subjects are in each
  cell. Diagram the experiment, identify all
  hypotheses, and, using the data (number of
  syllables recalled) and ANOVA table on the next
  slide, graph the results and identify any
  significant effects.

30
Diagram of recall study 2 by 3 matrix
Number of repetitions
4
8
12
16 14 16 13 15 16
16 12 11 15 13 14
10 11 12 15 14 10
Numbers
Similarity of material
8 7 4 5 5 6
11 13 9 10 8 9
14 12 16 15 12 13
Syllables
Source SS df S2 Fobt p Rows 121
1 Columns 176 2 Interaction (RC) 35
2 Within (Error) 88 30 Total 420 35
31
Diagram of recall study 2 by 3 matrix
Number of repetitions
4
8
12

• Need to
• Identify hypotheses
• Calculate cell means
• Calculate row and
• column means
• Complete ANOVA
• table
• 4) Make graph

16 12 11 15 13 14
16 14 16 13 15 16
10 11 12 15 14 10
Numbers
Similarity of material
8 7 4 5 5 6
11 13 9 10 8 9
14 12 16 15 12 13
Syllables
Source SS df S2 Fobt p Rows 121
1 Columns 176 2 Interaction (RC) 35
2 Within (Error) 88 30 Total 420 35
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Hypotheses for recall study
Main effect of column IV1 (repetitions) H0 The
column means are equal HA The column means are
not equal Main effect of row IV2 (similarity of
other material) H0 The row means are equal HA
The row means are not equal Interaction of the
two IVs H0 The cell means are equal HA The
cell means are not equal
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Diagram of recall study 2 by 3 matrix
Number of repetitions
4
8
12
Numbers
12.0
13.5
15.0
13.5
Similarity of material
5.8
9.8
10.0
13.7
Syllables
8.9
14.4
11.8
Source SS df S2 Fobt p Rows 121
1 Columns 176 2 Interaction (RC) 35
2 Within (Error) 88 30 Total 420 35
34
Diagram of recall study 2 by 3 matrix
Number of repetitions
4
8
12
Numbers
12.0
13.5
15.0
13.5
Similarity of material
5.8
9.8
10.0
13.7
Syllables
8.9
14.4
11.8
Source SS df S2 Fobt p Rows 121
1 121.0 Columns 176 2 88.0 Interaction
(RC) 35 2 17.5 Within (Error) 88 30
2.9 Total 420 35
35
Diagram of recall study 2 by 3 matrix
Number of repetitions
4
8
12
Numbers
12.0
13.5
15.0
13.5
Similarity to syllables
5.8
9.8
10.0
13.7
Syllables
8.9
14.4
11.8
Source SS df S2 Fobt Fcrit p Rows 121
1 121.0 41.7 4.17 lt.05 Columns 176 2
88.0 30.3 3.32 lt.05 Interaction (RC) 35 2
17.5 6.0 3.32 lt.05 Within (Error) 88 30
2.9 Total 420 35
36
Both main effects and the interaction are
significant.
Numbers
Syllables
16
Both the number of repetitions and the
similarity of learned material influence
recall, and these two factors also interact to
influence recall.
8
Syllables recalled
0
4
12
8
Repetitions