Split-Plot Analysis of Variance Part II: Split-Plot Analysis of Variance Part II:
Exploring the Interaction! Exploring the Interaction!
1) Interaction Overview! " 1) Interaction Overview! "
2) Two types of situations with different error terms" 2) Two types of situations with different error terms**"
! !
3) Situation 1: Comparing effect of the between-subjects factor at a particular level of 3) Situation 1: Comparing effect of the between-subjects factor at a particular level of
the within-subjects factor! the within-subjects factor!
4) Situation 1: Denominator calculations! 4) Situation 1: Denominator calculations!
5) Situation 1: Numerator calculations! 5) Situation 1: Numerator calculations!
6) Situation 1: F-values and significance! 6) Situation 1: F-values and significance!
7) Situation 1: Effect size calculation! 7) Situation 1: Effect size calculation!
8) Situation 1: Sample Results Paragraph! 8) Situation 1: Sample Results Paragraph!
9) Situation 2: Comparing effect of the within-subjects factor at a particular level of the 9) Situation 2: Comparing effect of the within-subjects factor at a particular level of the
between-subjects factor! between-subjects factor!
10) Situation 2: Understanding the SPSS syntax! 10) Situation 2: Understanding the SPSS syntax!
11) Situation 2: Understanding SPSS Output! 11) Situation 2: Understanding SPSS Output!
12) Situation 2: Effect size calculation! 1 12) Situation 2: Effect size calculation! 2
Investigating a significant
2-way interaction To understand an interaction, need to conduct simple main effect
Split-Plot design analyses
2-way interaction
The effect of 1 factor on the DV varies depending on the level 1) check out the interaction cell means
of the other factor
2) which set of simple main effects makes more sense theoretically?
3) do the math
The simple main effects of 1 factor are different
what we do in SPSS depends on which set of simple main effects
at the various levels of the other we want
Simple Main Effect
The effect of a factor (e.g., audience) at a specific level of
another factor (e.g., males)
3 4 Problem: The Error term for SME
Gender by Audience Interaction
depends upon the situation
2 Situations
10
Audience
1) Examining a Between-Subject aud0 aud1 aud2 aud3
independent variable at a
8 Men males 8.0 6.2 4.2 4.0
specific level of a Within-
Women Subjects variable females 9.8 6.4 3.0 2.0
6
c
o
r 4 Audience
m 2) Examining a Within-Subjects aud0 aud1 aud2 aud3
a variable at a specific level
M
2 of a Between-Subjects males 8.0 6.2 4.2 4.0
independent variable
females 9.8 6.4 3.0 2.0
0
0 1 2 3
Audience 5 6
Split-Plot Analysis of Variance Part II: Problem: The Error term for SME
Exploring the Interaction!
depends upon the situation
1) Interaction Overview! "
2 Situations
2) Two types of situations with different error terms"
! Audience
1) Examining a Between-Subject aud0 aud1 aud2 aud3
3) Situation 1: Comparing effect of the between-subjects factor at a particular level of independent variable at a
the within-subjects factor**! males 8.0 6.2 4.2 4.0
specific level of a Within-
4) Situation 1: Denominator calculations! Subjects variable females 9.8 6.4 3.0 2.0
5) Situation 1: Numerator calculations!
6) Situation 1: F-values and significance!
7) Situation 1: Effect size calculation! Audience
8) Situation 1: Sample Results Paragraph! 2) Examining a Within-Subjects
aud0 aud1 aud2 aud3
variable at a specific level
of a Between-Subjects males 8.0 6.2 4.2 4.0
9) Situation 2: Comparing effect of the within-subjects factor at a particular level of the independent variable
between-subjects factor! females 9.8 6.4 3.0 2.0
10) Situation 2: Understanding the SPSS syntax!
11) Situation 2: Understanding SPSS Output!
12) Situation 2: Effect size calculation! 7 8 Gender by Audience Interaction
What about gender
at the different levels of audience? 10
8
aud0 aud1 aud2 aud3 aud0 aud1 aud2 aud3 Men
Women
males 8.0 6.2 4.2 4.0 males 8.0 6.2 4.2 4.0
6
females 9.8 6.4 3.0 2.0 females 9.8 6.4 3.0 2.0 s
e
f
b 4
aud0 aud1 aud2 aud3 aud0 aud1 aud2 aud3 n
M
males 8.0 6.2 4.2 4.0 males 8.0 6.2 4.2 4.0 2
females 9.8 6.4 3.0 2.0 females 9.8 6.4 3.0 2.0
0
0 1 2 3
Audience
9 10
Interested in: Effect of gender at the different levels of audience
aud0 aud1 aud2 aud3 aud0 aud1 aud2 aud3
males 8.0 6.2 4.2 4.0 males 8.0 6.2 4.2 4.0
females 9.8 6.4 3.0 2.0 females 9.8 6.4 3.0 2.0
aud0: F(?, ?) = ?? / ?? aud0: F(?, ?) = ?? / ??
Calculate Calculate
4 F-ratios aud1: F(?, ?) = ?? / ?? 4 F-ratios aud1: F(?, ?) = ?? / ??
aud2: F(?, ?) = ?? / ?? aud2: F(?, ?) = ?? / ??
aud3: F(?, ?) = ?? / ?? aud3: F(?, ?) = ?? / ??
11 12 aud0 aud1 aud2 aud3 aud0 aud1 aud2 aud3
males 8.0 6.2 4.2 4.0 males 8.0 6.2 4.2 4.0
females 9.8 6.4 3.0 2.0 females 9.8 6.4 3.0 2.0
aud0: F(?, ?) = ?? / ?? aud0: F(?, ?) = ?? / ??
Calculate Calculate
4 F-ratios aud1: F(?, ?) = ?? / ?? 4 F-ratios aud1: F(?, ?) = ?? / ??
aud2: F(?, ?) = ?? / ?? aud2: F(?, ?) = ?? / ??
aud3: F(?, ?) = ?? / ?? aud3: F(?, ?) = ?? / ??
13 14
For each F-ratio For each F-ratio
• There is a Numerator • There is a Numerator
• There is Denominator • There is Denominator
• F = MSeffect Numerator • F = MSeffect Numerator
MSerror MSerror Denominator
• Must determine both, and then calculate F • Must determine both, and then calculate F
15 16 Calculate the Denominator for the F-ratio
First, we'll determine the proper denominator Calculate F-ratio By Hand
1) Calculate a new pooled MS (denominator) to put into each
error
F-ratio.
Next, we'll determine the numerators
2) Calculate d.f. for new pooled MS errordenominator).
3) Use ONEWAY command to get appropriate MS betweenand d.f.
for them.
Finally, we'll calculate the F-ratios
4) Calculate F-ratios by hand
5) Determine critical F-ratio using df MS and MS
between error
and compare to obtained F-values
17 18
Calculate the Numerator for the F-ratio Combine them, calculate F,
and determine significance.
Calculate F-ratio By Hand Calculate F-ratio By Hand
1) Calculate a new pooled MS (denominator) to put into each 1) Calculate a new pooled MS (denominator) to put into each
error error
F-ratio. F-ratio.
2) Calculate d.f. for new pooled MS errordenominator). 2) Calculate d.f. for new pooled MS errordenominator).
3) Use ONEWAY command to get appropriate MS betweenand d.f. 3) Use ONEWAY command to get appropriate MS betweenand d.f.
for them. for them.
4) Calculate F-ratios by hand 4) Calculate F-ratios by hand
5) Determine critical F-ratio using df MS and MS 5) Determine critical F-ratio using df MS and MS
between error between error
and compare to obtained F-values and compare to obtained F-values
19 20 Split-Plot Analysis of Variance Part II:
Exploring the Interaction!
Calculating F-ratios by hand
1) Interaction Overview! "
2) Two types of situations with different error terms"
!
3) Situation 1: Comparing effect of the between-subjects factor at a particular level of
the within-subjects factor!
4) Situation 1: Denominator calculations**! 1) **Calculate a new pooled MS error(denominator) to put into
5) Situation 1: Numerator calculations! each F-ratio.
6) Situation 1: F-values and significance! 2) Calculate d.f. for new pooled MS (denominator).
error
7) Situation 1: Effect size calculation!
3) Use ONEWAY command to get appropriate MS and d.f.
8) Situation 1: Sample Results Paragraph! between
for them.
9) Situation 2: Comparing effect of the within-subjects factor at a particular level of the
4) Calculate F-ratios by hand
between-subjects factor!
5) Determine critical F-ratio using df MS between and MS error
10) Situation 2: Understanding the SPSS syntax!
and compare to obtained F-values
11) Situation 2: Understanding SPSS Output!
12) Situation 2: Effect size calculation!
21 22
Tests of Between-Subjects Effects
Measure: MEASURE_1
Need to pool error terms when doing simple main effects of the Transformed Variable: Average
independent groups variable at the different levels of the repeated
Type III Sum Partial Eta
measures variable Source of Squares df Mean Square F Sig. Squared
Intercept 1188.100 1 1188.100 760.384 .000 .990
gender .900 1 .900 .576 .470 .067
effects of Gender at different levels of Audience: Error 12.500 8 1.563
Gender & interaction have different error terms, so pool them
(use sphericity assumed line)
Step 1: Calculate a new pooled MS error(denominator)
Tests of Within-Subjects Effects
new MS error = (SS e1 + SS )e2 (df e1 + df e2
Measure: MEASURE_1
= (31.10 + 12.50) / (24 + 8) Type III Sum Partial Eta
= 43.60 / 32 Source of Squares df Mean Square F Sig. Squared
audience Sphericity Assumed 220.500 3 73.500 56.720 .000 .876
= 1.36
Greenhouse-Geisser 220.500 2.171 101.588 56.720 .000 .876
Huynh-Feldt 220.500 3.000 73.500 56.720 .000 .876
Lower-bound 220.500 1.000 220.500 56.720 .000 .876
Step 2: Calculate df for new MS (denominator)
error audience * genderSphericity Assumed 20.900 3 6.967 5.376 .006 .402
new df = (SS + SS ) / (SS 2/df + SS 2/df ) Greenhouse-Geisser 20.900 2.171 9.629 5.376 .014 .402
error e1 e2 e1 e1 e2 e2 Huynh-Feldt 20.900 3.000 6.967 5.376 .006 .402
= (31.10+12.50) 2 / (31.102/24 + 12.502/8) Lower-bound
= 1900.96 / 59.83 20.900 1.000 20.900 5.376 .049 .402
Error(audience) Sphericity Assumed 31.100 24 1.296
= 31.77 Greenhouse-Geisser 31.100 17.364 1.791
= 31 in this case, round down Huynh-Feldt 31.100 24.000 1.296
Lower-bound 31.100 8.000 3.888
23 24 aud0 aud1 aud2 aud3
Need to pool error terms when doing simple main effects of the
males 8.0 6.2 4.2 4.0
independent groups variable at the different levels of the repeated
measures variable
females 9.8 6.4 3.0 2.0
effects of Gender at different levels of Audience:
Gender & interaction have different error terms, so pool them
Step 1: Calculate a new pooled MS errordenominator)
aud0: F(?, ?) = ?? / 1.36
new MS error = (SS e1 + SS e2/ (df e1+ dfe2
= (31.10 + 12.50) / (24 + 8) Calculate
= 43.60 / 32
4 F-ratios aud1: F(?, ?) = ?? / 1.36
= 1.36
aud2: F(?, ?) = ?? / 1.36
Step 2: Calculate df for new MS errordenominator)
2 2 2
new df error = (SS e1 + SS e2/ (SS e1 /dfe1+ SS e2/dfe2 aud3: F(?, ?) = ?? / 1.36
= (31.10+12.50)/ (31.102/24 + 12.502/8)
= 1900.96 / 59.83
= 31.77
= 31 in this case, round down
25 26
Need to pool error terms when doing simple main effects of the
independent groups variable at the different levels of the repeated
measures variable
If they have different Error Terms
effects of Gender at different levels of Audience:
Calculate F-ratio By Hand Gender & interaction have different error terms, so pool them
1) Calculate a new pooled MS errordenominator) to put into each
F-ratio. Step 1: Calculate a new pooled MS errordenominator)
new MS = (SS + SS ) / (df + df )
error e1 e2 e1 e2
2) **Calculate d.f. for new pooled MS errordenominator). = (31.10 + 12.50) / (24 + 8)
= 43.60 / 32
3) Use ONEWAY command to get appropriate MS and d.f. = 1.36
between
for them.
4) Calculate F-ratios by hand Step 2: Calculate df for new MS error(denominator)
new df error = (SS e1+ SS )e2 (SS e12/dfe1+ SS e22/dfe2
= (31.10+12.50)/ (31.102/24 + 12.502/8)
5) Determine critical F-ratio using df MS between and MS error
= 1900.96 / 59.83
and compare to obtained F-values = 31.77
= 31 in this case, round down
27 28 Tests of Between-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average Need to pool error terms when doing simple main effects of the
independent groups variable at the different levels of the repeated
Type III Sum Partial Eta
Source of Squares df Mean Square F Sig. Squared measures variable
Intercept
1188.100 1 1188.100 760.384 .000 .990
gender .900 1 .900 .576 .470 .067
Error 12.500 8 1.563 effects of Gender at different levels of Audience:
Gender & interaction have different error terms, so pool them
Step 1: Calculate a new pooled MS error(denominator)
Tests of Within-Subjects Effects
Measure: MEASURE_1 new MS error = (SS e1+ SS ) e2(df e1 + df e2
= (31.10 + 12.50) / (24 + 8)
Type III Sum Partial Eta
Source of Squares df Mean Square F Sig. Squared = 43.60 / 32
audience Sphericity Assumed 220.500 3 73.500 56.720 .000 .876 =1.36
Greenhouse-Geisser 220.500 2.171 101.588 56.720 .000 .876
Huynh-Feldt 220.500 3.000 73.500 56.720 .000 .876
Lower-bound 220.500 1.000 220.500 56.720 .000 .876
audience * gendeSphericity Assumed 20.900 3 6.967 5.376 .006 .402 Step 2: Calculate df for new MS error(denominator)
Greenhouse-Geisser 20.900 2.171 9.629 5.376 .014 .402 2 2 2
new df error = (SS e1+ SS ) e2(SS e1 /df e1 + SS e2 /dfe2
Huynh-Feldt 20.900 3.000 6.967 5.376 .006 .402 = (31.10+12.50) 2/ (31.10 /24 + 12.50 /8)
Lower-bound 20.900 1.000 20.900 5.376 .049 .402
Error(audience) Sphericity Assumed 31.100 24 1.296 = 1900.96 / 59.83
= 31.77
Greenhouse-Geisser 31.100 17.364 1.791
Huynh-Feldt 31.100 24.000 1.296 =31 in this case, round down
Lower-bound 31.100 8.000 3.888
29 30
aud0 aud1 aud2 aud3 aud0 aud1 aud2 aud3
males 8.0 6.2 4.2 4.0 males 8.0 6.2 4.2 4.0
females 9.8 6.4 3.0 2.0 females 9.8 6.4 3.0 2.0
aud0: F(?, 31) = ?? / 1.36 aud0: F(?, 31) = ?? / 1.36
Calculate Now find a
4 F-ratios numerator
aud1: F(?, 31) = ?? / 1.36 aud1: F(?, 31) = ?? / 1.36
for each SME
aud2: F(?, 31) = ?? / 1.36 aud2: F(?, 31) = ?? / 1.36
aud3: F(?, 31) = ?? / 1.36 aud3: F(?, 31) = ?? / 1.36
31 32 Split-Plot Analysis of Variance Part II:
Exploring the Interaction!
1) Interaction Overview! " If they have different Error Terms
2) Two types of situations with different error terms"
!
3) Situation 1: Comparing effect of the between-subjects factor at a particular level of Calculate F-ratio By Hand
the within-subjects factor!
4) Situation 1: Denominator calculations! 1) Calculate a new pooled MS errordenominator) to put into each
5) Situation 1: Numerator calculations**! F-ratio.
6) Situation 1: F-values and significance! 2) Calculate d.f. for new pooled MS (denominator).
error
7) Situation 1: Effect size calculation!
8) Situation 1: Sample Results Paragraph! 3) **Use ONEWAY command to get appropriate MS between and
d.f. for them.
9) Situation 2: Comparing effect of the within-subjects factor at a particular level of the 4) Calculate F-ratios by hand
between-subjects factor!
10) Situation 2: Understanding the SPSS syntax! 5) Determine critical F-ratio using df MS between and MS error
and compare to obtained F-values
11) Situation 2: Understanding SPSS Output!
12) Situation 2: Effect size calculation!
33 34
Step 3: Use ONEWAY command to get appropriate MS
between
ONEWAY
aud0 aud1 aud2 aud3 BY gender
/MISSING ANALYSIS.
Gender at: Oneway
From ONEWAY command New Pooled MS
error
aud0: F(?, 31) = ?? / 1.36
aud1: F(?, 31) = ?? / 1.36
aud2: F(?, 31) = ?? / 1.36
aud3: F(?, 31) = ?? / 1.36
35 36 aud0 aud1 aud2 aud3 Split-Plot Analysis of Variance Part II:
Exploring the Interaction!
males 8.0 6.2 4.2 4.0
females 9.8 6.4 3.0 2.0 1) Interaction Overview! "
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