Factorial Analysis of
Variance
I. Why do a multifactor experiment?
A.
Add precision to predictions
B.
Explain effects
II. Two-way fixed effects ANOVA: SR(GF X DF)
A.
Format
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Drug |
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Group |
1 |
2 |
3 |
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Bipolar |
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Unipolar |
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B.
Sums of Squares
p
_ _
SSGroup= nq S (Yi.-Y..)2
i=1
q _ _
SSDrug= np S (Y.j-Y..)2
j=1
p q _
_ _ _
SSGXD= n S S (Yij-Yi.-Yj. +Y..)2
i=1 j=1
_
SSS(GXD) = S S S (yijk-Yij.)2
i j k
_
SSTotal = S S S (yijk-Y...)2
i j k
Note the meaning of the interaction sum of squares:
the effects on the dependent variable of drugs and group membership that are
not predictable from the overall mean or the main effects of drug or
group.
For example, the overall
mean is 
(what you would
predict if you only knew that the subject was depressed and receiving some
drug).
The effect of being bipolar is B=
- 
The effect of receiving drug 1 is D=
-
So, the effect of being bipolar and receiving drug 1 is: 
- 
-B-D
- 
- (
- 
) - (
-
)
- 
- 
+ 
- 
+
- 
- 
+
Which is one iteration of the SSGXD
summation above.
C. Effect Size
Several measures of effect size are in common
use. All are designed to give the user
a summary statistic describing the spread between the groups relative to some
standard. The most common is
Eta-squared (h2). Unfortunately, two h2 ‘s are in common use. One is the proportion of variance explained
by a predictor (SSA/Sstotal). This h2 is identical to the R2
commonly reported in regression analyses.
Another h2 is the partial h2:
Partial h2 = SSA /(SSA+SSerror)
Thus, partial h2 does not depend on the
variance explained by other factors in a multifactor experiment. The partial h2’s also do not sum to 1.0 as
will R2 (and the identical h2 measure). SPSS reports partial h2 under the label
“Eta-squared”.
D. Expected Mean Square Table
Source E(MS) df Error line
1. Group ndsG2 + ss(G X D)2 g-1 4
2. Drugs ngs D 2+ ss(G X D)2 d-1 4
3. G X D ns G D 2+ ss(G X D)2
(g-1)(d-1) 4
4. S(GXD) ss(G X D)2 (n-1)gd
E. Example
1. Data
Drug
Group 1 2 3
Bipolar 8
4 0 10 8 6
8 6 4
Unipolar 14 10 6 4 2 0 15 12 9
2. Means
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|
Drug
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Group |
1 |
2 |
3 |
|
Bipolar
|
4 |
8 |
6 |
6 |
Unipolar
|
10 |
2 |
12 |
8 |
|
|
7 |
5 |
9 |
7 |
3. Analysis
Source SS df MS F p h2
Group 18 1 18 2.04 ns .145
Drug 48 2 24 2.72 ns .312
GXD 144 2 72 8.15 <.01 .576
S(GXD) 106 12 8.83
Total 316 17
F. What if we had ignored the groups?
Source SS df MS F p
Drug 48 2 24 1.34 ns
S(D) 268 15 17.87
III. Analysis of Interactions
A. Plot the
data.
B.
Conceptual types of interactions: disordinal (cross-over) and ordinal
Note difference in interpretations. In ordinal case, general summary of data
given by overall pattern of means applies to all groups. In the disordinal case it does not.
C.
Interaction numbers.
1. Subtract
effects of independent variables from cell means (i.e., subtract row mean and
column mean and add grand mean).
2. Example yields: -2
4 -2
2 -4 2
D.
Decomposition by contrasts
1. Convert interaction numbers into contrast
weights.
2. Example yields: -1
2 -1
1 -2 1
which can be considered to be the product of a
"linear" contrast [-1 1] on groups and a quadratic [1 -2 1] on drugs:
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Drugs (quadratic) |
||
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Groups |
-1 |
2 |
-1 |
|
1 |
-1 |
2 |
-1 |
|
-1 |
1 |
-2 |
1 |
to obtain the interaction contrast (cell entries)
multiply the top row by the first column.
_
3. Remember
that SScontrast= n (SciYi)2
Sci2
Source SS df MS F p
Group 18 1 18 2.04 ns
Drugs 48 2 24 2.72 ns
Linear 12 1 12 1.36 ns
Quad 36 1 36 4.08 ns
GXD 144 2 72 8.15 <.01
G X L 0 1 0 0 ns
G X Q 144 1 144 16.31 <.01
S(GXD) 106 12 8.83
Total 316 17
E. Simple
Effects
1. Simple
effects are essentially contrasts involving only one level of one or more
factors in multifactor designs.
2. For example, for simple effect of use
contrast
Group
@ Drug 1
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Drugs |
|||
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Groups |
1 |
2 |
3 |
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Unipolar |
+1 |
0 |
0 |
|
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Bipolar |
-1 |
0 |
0 |
|
Group
@ Drug 2
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|
Drugs |
|||
|
Groups |
1 |
2 |
3 |
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Unipolar |
0 |
+1 |
0 |
|
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Bipolar |
0 |
-1 |
0 |
|
Group @
Drug 3
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Drugs |
|||
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Groups |
1 |
2 |
3 |
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Unipolar |
0 |
0 |
+1 |
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Bipolar |
0 |
0 |
-1 |
|
For
the simple effect use
this contrast and this contrast
(and
add the sums of squares)
Drug
@ Group 1
Drug
@ Group 2
3. A simple
effect is a combination of a main effect and an interaction; e.g.:
S SSG @ Di = SSG
+ SSGXD
and S dfG @ Di = dfG
+ dfGXD
So, when analyzing for more than one set of simple
main effects, consider setting new alpha levels for significance tests, just as
one would with any other set of non-orthogonal contrasts.
4. Example
Source SS df
MS F p
Group 162 3
@
Drug 1 54 1 54 6.12 <.05
@
Drug 2 54 1 54 6.12 <.05
@
Drug 3 54 1 54 6.12 <.05
Drugs 192 4
@
Group 1 24 2 12 1.36 ns
@
Group 2 168 2 84 9.51 <.01
S(GXD) 106
12 8.83
By estimating the error variance more precisely, a
simple effects analysis on one factor may be more powerful than a one-way ANOVA
performed at each level of the interacting factor (see above).
III. Multi-way Fixed Effects ANOVA
A. Nothing
new
B. Example:
SR(GFXTFXDF); Group (Bi/Unipolar) X Treatment
(Behavioral/Psychodynamic) X Drug (Lithium, Prozac, Ativan) n=5
Source E(MS) df
Group ntdsg2+ss(gtd)2 (g-1)=1
Drugs ntgsd2+ ss(gtd)2 (d-1)=2
Treatment ngdst2+ ss(gtd)2 (t-1)=1
G X D ntsgd2+ ss(gtd)2 (g-1)(d-1)=2
G X T ndsgt2+ ss(gtd)2 (g-1)(t-1)=1
D X T ngsdt2+ ss(gtd)2 (d-1)(t-1)=2
G X D X T nsgdt2+ ss(gtd)2 (g-1)(d-1)(t-1)=2
S(G X D X T) ss(gtd)2 (n-1)gtd=
48
Total 59