Psych 100A Week 10 Discussion Notes
Dawn Chen
December 5, 2009
TwoFactor BetweenSubjects ANOVA
The twofactor betweensubjects ANOVA is used when there are two independent
variables, and each combination of the levels of the two IVs is given to (or possessed by) a
different group of subjects.
For example, if the first IV has 3 levels and the second IV has 2
levels (a
3
2
×
design), then there are 6 separate groups of subjects (cells).
Summary of Notation Used
If factor
A
has 3 levels, factor
B
has 2 levels, and there are 3 scores in each combination
of the levels of factors
A
and
B
, then the notation we use looks like the following:
Factor
A
1
A
2
A
3
A
Means for
Factor
B
1
B
111
X
211
X
311
X
Cell
mean:
1
1
A B
X
121
X
221
X
321
X
Cell
mean:
2
1
A B
X
131
X
231
X
331
X
Cell
mean:
3
1
A B
X
Mean for
group
1
:
B
1
B
X
Factor
B
2
B
112
X
212
X
312
X
Cell
mean:
1
2
A B
X
122
X
222
X
322
X
Cell
mean:
2
2
A B
X
132
X
232
X
332
X
Cell
mean:
3
2
A B
X
Mean for
group
2
:
B
2
B
X
Means for Factor
A
Mean for group
1
:
A
1
A
X
Mean for group
2
:
A
2
A
X
Mean for group
3
:
A
3
A
X
Grand mean:
G
X
Partitioning Variation
There are now four different sources of variation in a score:
The variation due to factor
A
,
the variation due to factor
B
, the variation due to the interaction of factors
A
and
B
, and the
variation due to random error.
Therefore, the total variation is now partitioned into these four
kinds of variation:
total
A
B
A B
error
SS
SS
SS
SS
SS
×
=
+
+
+
The following equation shows how to partition a score
X
.
A
X
represents the mean for
the
A
group that
X
belongs to,
B
X
represents the mean for the
B
group that
X
belongs to, and
AB
X
represents the mean for the
AB
cell that
X
belongs to.
From each term in this equation, we
obtain the corresponding sum of squares by calculating the term for each score, squaring it, and
summing them up for all scores.
(
(
(
(
total
A
B
A B
error
G
A
G
B
G
AB
A
B
G
AB
SS
SS
SS
SS
SS
X
X
X
X
X
X
X
X
X
X
X
X
×

=

+

+


+
+

dncurlybracketlefthorizcurlybracketextdncurlybracketmidhorizcurlybracketextdncurlybracketright
dncurlybracketlefthorizcurlybracketextdncurlybracketmidhorizcurlybracketextdncurlybracketright dncurlybracketlefthorizcurlybracketextdncurlybracketmidhorizcurlybracketextdncurlybracketright dncurlybracketlefthorizcurlybracketexthorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketmidhorizcurlybracketexthorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketright dncurlybracketlefthorizcurlybracketextdncurlybracketmidhorizcurlybracketextdncurlybracketright
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Hypothesis Testing
In the twofactor betweensubjects ANOVA, there are three questions we are interested
in answering: (1) Is there a main effect of factor
A
?
(2) Is there a main effect of factor
B
?
(3) Is
there an interaction between factors
A
and
B
(i.e., does factor
A
affect the dependent variable
differently at different levels of factor
B
)?
Each of these questions is answered separately by a
different hypothesis test, shown in the following table for
a
= 3 and
b
= 2.
Thus, we calculate
three
F
values, one for the main effect of
A
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 Spring '10
 Chen
 Statistics, Mean, main effect

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