Dr. Lakey – Diekhoff (1996) – Chapter 9
FACTORIAL ANALYSIS OF VARIANCE
2+ Independent Variables – One Dependent Variable
Examines Separate (Main) Effects and Combined (Interaction) Effects
9.1 Types of Factorial ANOVA
Factors & Levels
 Two-Way or Two-Factor ANOVA: #levels x #levels
 Three-Way or Three-Factor ANOVA: #levels x #levels x #levels
 Four-Way or Four-Factor ANOVA? Avoid!
Independent, Related, and Mixed Groups Designs
(1) Independent Samples/Groups: Between-Groups
“Completely Randomized Factorial ANOVA”
Or “Two- or Three-Factor Randomized Groups Design ANOVA.”
(2) Related Samples/Groups: Within-Groups (Paired Scores)
“Randomized-Blocks Factorial ANOVA”
Or “Two- or Three-Factor Repeated Measures ANOVA”
Or “Treatments-by-Treatments-by-Subjects ANOVA”
Or “Treatments-by-Treatments-by-Treatments-by-Subjects ANOVA.”
(3) Both Independent and Related Samples/Groups:
“Split-Plot Factorial ANOVA”
Or “Mixed Design Factorial ANOVA.”
Most Common Design: “Three-Factor Mixed Design ANOVA with
Repeated-Measures on One Factor”
9.2 Main Effects and Interaction Effects
Visualization with Line Graphs or Results Figures
(1) Significant Factor A Effect
Average the two A1 means and average the two A2 means: Are these two
new means significantly different?
(2) Significant Factor B Effect
Average the two B1 means and average the two B2 means: Are these two
new means significantly different?
(3) Significant AxB Interaction Effect
Do the lines cross or will they cross if extended out?
If the lines are not parallel, they have Significantly Different Slopes.
9.3 Partitioning the Variance in a Factorial Design
 F = MSBG / MSWG = (treatment + error) / (error)
 Total Variance (SS) broken down (partitioned) into
subgroup SS components.
 Total SS = [Between-Group SS] + Within-Group SS
Total SS = SSBG[SSA + SSB + SSAxB] + SSWG
RECALL: SS = (X – MEAN)2
SSWG = (X – MEANCELL)2 + (X – MEANCELL)2 + …
for all cells
SSBG = (MEANCELL – MEANTOTAL)2 + (MEANCELL
– MEANTOTAL)2 + … for all cells
SSA = (MEANROW – MEANTOTAL)2 + (MEANROW –
MEANTOTAL)2 + … for all ROWS
SSB = (MEANCOLUMN – MEANTOTAL)2 +
(MEANCOLUMN – MEANTOTAL)2 +
… for all COLUMNS
AND: SSAxB = SSBG – SSA + SSB
TABLE 9.2 SUMMARY TABLE (p282)
SOURCE
SS
DF
MS
F
p
Omega
(est)
(if sign.)
BETWEEN
GROUPS
A
B
AxB
WITHIN
GROUPS
TOTAL
9.6 Interpreting Factorial ANOVA
Significant Main Effects:
Significant Differences Between Marginal Means
Factor A Main Effect: Two or More Row Means
Factor B Main Effect: Two or More Column Means
Significant Interaction Effects:
Significant Differences Between Slopes.
(Usually the most important result!)
2
Post-Hoc Data Snooping
Tukey’s HSD (use .05 level)
Modified for Marginal Means (just adjust HM)
Scheffe’ Test
Less Powerful with a Very Awkward Computation
(let SPSS do it if you want it…)
Strength of Association
Omega-Square… best
Eta-Square… OK
(Believe that SPSS produces incorrect eta2 results for each of the A and B factors as
well as the AxB interaction; SPSS produces correct eta2 result for BG factor)
For Single Factors:
 Omega2A = [SSA – (dfA) MSWG ] / [SSTotal + MSWG],
etc.
 Eta2A = SSA / SSTotal ,
etc.
For the Between Group Factor Before Partition:
Omega2BG = Omega2A + Omega2B + Omega2AxB
Eta2BG = Eta2A + Eta2B + Eta2AxB
Example 9.1 (p289f)
Dr. Jerry Atric’s Expanded Study
FACTORS
B1
A1
0 MIN FINE
A2
10 MIN FINE
A3
20 MIN FINE
Marginal
Column Means
B2 LONG-TERM
SHORT-TERM
TENURE
13 14 12 17 15 13
Cell Mean = 14.0
7 8 6 11 9 7
Cell Mean = 8.0
2 3 1 6 4 2
Cell Mean = 3.0
TENURE
14 15 13 18 16 14
Cell Mean = 15.0
17 18 16 21 19 17
Cell Mean = 18.0
20 21 19 24 22 20
Cell Mean = 21.0
8.33
18.0
Marginal
Row Means
14.5
13.0
12.0
MEANTotal =
13.17
Results
SOURCE
SS
DF
MS
F
p
Omega
2
Eta
2
(if sign.)
(if sign)
1313
5
262.60
82.06
<.001
.91
.93
A
38
2
19.00
5.94
.007
.02
.03
B
841
1
841.00
262.81
<.001
.59
.60
AxB
434
2
217.00
67.81
<.001
.30
.31
WITHIN
96
30
3.20
1409
35
40.26
BETWEEN
GROUPS
GROUPS
(ERROR)
TOTAL
Tukey HSD Test of Marginal Means
 Use 5% level for all post-hoc tests
 MSERROR is the Denominator of the Sign. F
 Use the Harmonic Mean (the Equal N)
 Adjust HM to Marginal Mean’s N for Single
Factor Post-Hoc Tests
HSD = q.05 x SQRT( MSERROR / HM )
= 4.31 x SQRT( 3.20 / 12 )
= 2.23
1v2: 1.50 n.s.
1v3:
2.50*
2v3: 1.00 n.s.
1v2: 1.50 n.s. (Tukey Test p =.117
vs. Scheffe’ Test p = .139)
1v3: 2.50* (Tukey Test p =.005*
vs. Scheffe’ Test p = .007*)
2v3: 1.00 n.s. (Tukey Test p =.369
vs. Scheffe’ Test p = .403)
Post-Hoc Tests for Significant Interaction Effects
2 x 3 ANOVA: pAxB<.001*
Only two B1 and B2 functions: Obviously these two functions have
significantly different slopes.
Now switch the axes… Same data but a different perspective …
Post-Hoc Tests for Significant Interaction Effects
3 x 2 ANOVA: pAxB<.001*
Now three A1, A2 and A3 functions: Which have significantly different
slopes ???
F-Test for Simple Effects
1. For each pair of line functions (successively drop/cut out one
of the A-level data rows) use SPSS (or other better program) to
re-analyze these data as a 2x2 ANOVA to obtain only the new
Interaction MSAxB value,
2. For each pair of line functions, Compute a new F = new 2x2
Interaction MSAxB/ old 3x2 Error MSWG ,
3. Use the F Table to assess p for significance (new dfnumerator /old
dfdenominator).
4. If p  .05, then the two line functions have
significantly different slopes.
Results For Example 9.1
Using SPSS…
Old 3x2 Analysis: MSError = 3.200 (dfDenominator = 30),
1st New 2x2 (A1vs A2) Analysis: MSAxB = 212.500 (dfNumerator = 1):
F(1,30) = 66.406, p<.001*.
2nd New 2x2 (A1vs A3) Analysis: MSAxB = 433.500 (dfNumerator = 1):
F(1,30) = 135.469, p<.001*.
3rd New 2x2 (A2 vs A3) Analysis: MSAxB = 96.000 (dfNumerator = 1):
F(1,30) = 30.000, p<.001*.
Conclusion: All three function slopes are significantly different
from one another.