A1260609308KwnUI

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CEP 933 Two-way ANOVA Expected Mean Squares
Here is the ANOVA table for the fixed-effects
model. Full formulas for the SS are in the
previous slide set (ch13). Source
df SS MS F ------------------------------------
--------------------------------------------------
---------- A a-1 S ni. ai2 SSA/(a-1) MSA/MSW B
b-1 S n.j bj 2 SSB/(b-1) MSB/MSW AB (a-1)(b-1)
SS nij abij2 SSAB/(a-1)(b-1)
MSAB/MSW Error n - ab SSS (Yijk-
)2 SSW/(n - ab) ------------------------------
--------------------------------------------------
---------------- Total n-1 SSS (Yijk -
)2
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CEP 933 Two-way ANOVA Expected Mean Squares
If all cell sizes are equal and equal to "m"
then njk m ni. bm n.j am
N abm Expected mean squares for the
fixed-effects model are E(MSA) s2e bm S
ai2/(a-1) s2e bm s2a E(MSB) se2 am S
bj2/(b-1) s2e am s2b E(MSAB) se2 mSS
abij2/(a-1)(b-1) s2e m s2ab E(MSW)
se2 Again as for the one-way the F tests get big
when the population treatment effects are nonzero
and the MSA, MSB and MSAB are larger than MSW.
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CEP 933 Two-way ANOVA Expected Mean Squares
Expected mean squares for the random-effects
model (with both factors random) include the
interaction variance in the main-effects EMSs.
E(MSW) and E(MSAB) are the same for the two
models. Expected mean squares for both the
random and fixed-effects models
are Random-effects model Fixed-effects
model E(MSA) s2e m s2ab bm s2a s2e
bm s2a E(MSB) se2 m s2ab am s2b s2e
am s2b E(MSAB) se2 m s2ab s2e m
s2ab E(MSW) se2 se2 Thus to compute the
F tests in the random model we use MSAB in the
denominator of the A and B main-effects F tests.
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CEP 933 Two-way ANOVA Expected Mean Squares
Here is the ANOVA table for the random-effects
model. Source df SS MS F ------------------
--------------------------------------------------
---------------------------- A a-1 S ni.
ai2 SSA/(a-1) MSA/MSAB B b-1 S n.j bj
2 SSB/(b-1) MSB/MSAB AB (a-1)(b-1) SS nij
abij2 SSAB/(a-1)(b-1) MSAB/MSW Error n - ab
SSS (Yijk- )2 SSW/(n -
ab) ----------------------------------------------
--------------------------------------------------
Total n-1 SSS (Yijk - )2 Note that
SPSS computes this table when you click a factor
name into the "random factors" window when
running the ANOVA.
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CEP 933 Two-way ANOVA Expected Mean Squares Rules
The differences in EMS arise because of the
numbers of levels (out of the total possible
number) observed for each factor. We will
denote the total number of levels for factors A
and B are A and B, and the observed numbers of
levels as a and b. In the fixed case we observe
ALL levels of each factor. So if A and B are
both fixed factors then a A, a/A 1
and (1 - a/A) 0, and b B, b/B 1
and (1- b/B) 0. For random factors we
observe a sample of levels (often a very small
sample). If we observe a sample of a5 out of
A5000 possible levels, then a/A 5/5000
.001 so 1- a/A .999 ? 1 so we may treat the
fraction (1 - a/A) as if it were 1.
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CEP 933 Two-way ANOVA Expected Mean Squares Rules
The general formulas below for the EMS include
the sampling fractions (1-b/B) in E(MSA) and
(1-a/A) in EMS(B), so the highlighted terms
involving these fractions and s2ab disappear when
the corresponding (i.e., opposite) factor is
fixed. If factor A is random and its number of
observed levels is small we treat (1-a/A) as if
it were1, so the highlighted terms involving the
fractions (which are assumed then to equal 1) and
s2ab stay in the EMS for B. Also when factor B
is random, the EMS(A) contains the interaction
term! E(MSA) s2e m (1- b/B) s2ab bm s2a
E(MSB) se2 m (1- a/A) s2ab am
s2b E(MSAB) se2 m s2ab E(MSW) se2
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CEP 933 Two-way ANOVA Contrasts and Post Hoc
Tests
As for the one-way ANOVA we can compute contrasts
and post-hoc tests for each of the main-effects
factors. These tests will be more powerful in
the two-way case than if we ran a one-way for
each factor IF the added second factor has
accounted for some of the variance that was error
in the one-way model. The tests are
essentially the same as for the one-way case,
except that the new (reduced) MSW is used to
compute the standard errors of the contrasts.
This means we will be more likely to be able to
distinguish among group means in the two-way
model, even if the means are exactly the same as
for the one-way model, because the precision of
the means is greater in the two-way case. We
will see this in our next slide.
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CEP 933 Two-way ANOVA Contrasts, Post Hoc Tests
Power
Here is SPSS output for the Tukey test on all
pairs of f1txcomp (achievement) school means. The
factor in the one-way model is region, and
school type (public vs. private) is the second
factor. Note that the two-way MSE is much smaller
(32.7 vs 38.5), and the West and N'central means
are in different groupings in the 2-way
model. One-way Model Two-way Model
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CEP 933 Two-way ANOVA Contrasts, Post Hoc Tests
Power
Similarly we can compute power for either main
effect or the interaction using Cohen's tables
with f, or the tables in our book with f f' ?
n f ? n. Note that fA S (mi. - m)2
/a1/2 /se fB S (m.j - m)2 /b1/2 /se fAB
S (mij - mi. - m.j m)2 /ab1/2 /se Or, of
course, we can be lazy and use the rules of thumb
discussed before if we do not have pilot data
that would allow us to properly estimate these
quantities.
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CEP 933 Two-way ANOVA Effect Magnitude Measures
Finally we should return to the indices we can
use to assess the contributions of the factors in
the model. Often we estimate the variance
explained by each one. As before we can use
the more complex and accurate variance-explained
measure w2 (see Howell p. 447) or the
quick-and-dirty Eta-squared. E2 is computed as
each SS term over the SSTotal. E2A SSA/SST
and E2B SSB/SST will be the same as if we
ran separate one-way models (that is, as long as
the sample sizes are equal or nearly equal and we
have complete data). Also we can look at E2AB
SSAB/SST and E2Model (SSA SSB
SSAB)/SST For quick decisions during
data-analysis E2 is fine, but for reporting
results in a journal it is more accurate to use
w2.