22.0 Two-Way ANOVA

34 Pages·2015·0.13 MB·English

22.0 Two-Way ANOVA
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Answer Questions
1
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Fixed, Random, and Mixed Effects
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Fractional Factorial Designs
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Filtration Example
22.1 Fixed, Random, and Mixed Effects
In the previous two-way ANOVA examples, we assumed (without comment)
that the two treatments had fixed effects. A factor has fixed effects if the
treatment levels are not chosen at random. In contrast, a factor has random
effects if the treatment levels are chosen at random.
2
In the example on the compressive strength of concrete cylinders, recall that
we had two factors: drying type, and batch. The three drying types were
specifically chosen, and thus fixed. If we had had more than one observation
for each combination of drying type and batch, then we could have done a
two-way ANOVA with interaction analysis. In that case, the batch factor
would be random, since we drew five batches from the company’s history of all
batches.
In a two-way ANOVA with interaction, if both factors have non-random
levels, then it is called a fixed effects design. If both factors have levels that
are chosen at random, then it is called a random effects design. And if one
factor has random levels and the other has non-random levels, then it is a
mixed effects design.
In the concrete drying example, if analyzed as a two-way ANOVA with
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interaction, we would have a mixed effects model. This is because the
methods of drying are three non-randomly chosen industrial processes, but the
five batches are random.
These distinctions are important because the test statistics used to test
hypotheses in the random and mixed effects models are different from the test
statistics used in the fixed effects model.
Recall that for the fixed effects model, the test statistics are:
−
SS /(I 1)
A
ts = = MS /MSerr
A − A
SSerr/(n IJ)
...
−
SS /(J 1)
B
ts = = MS /MSerr
B − B
SSerr/(n IJ)
...
− −
SS /(I 1)(J 1)
AB
ts = = MS /MSerr
AB − AB
SSerr/(n IJ)
...
4 In contrast, for the random effects model, the test statistics are
−
SS /(I 1)
A
ts = = MS /MS
A − − A AB
SS /(I 1)(J 1)
AB
−
SS /(J 1)
B
ts = = MS /MS
B − − B AB
SS /(I 1)(J 1)
AB
− −
SS /(I 1)(J 1)
AB
ts = = MS /MSerr
AB − AB
SSerr/(n IJ)
...
For the mixed effects model, assume that factor A is fixed and factor B is
random. Then the test statistics are
−
SS /(I 1)
A
ts = = MS /MS
A − − A AB
SS /(I 1)(J 1)
AB
−
SS /(J 1)
B
ts = = MS /MSerr
B − B
SSerr/(n IJ
...
− −
SS /(I 1)(J 1)
AB
5 ts = = MS /MSerr
AB − AB
SSerr/(n IJ)
...
The interaction test is always the same. For random effect tests, use the mean
squared error as the denominator; for fixed effect tests, use the mean square
for interaction as the denominator.
The situation is complex when there are more than two factors, or when
factors have both random and fixed levels. Satterthwaite’s approximation is
needed to create synthetic mean squared terms to make the tests.
The reason for these different test statistics is the expected value of the Mean
Squared term. For simplicity, assume that n = n. Then, using distribution
ij.
theory we don’t have in this class:
2
IE[MSerr] = σ
and if A and B are fixed-effect factors
I
Jn
2 2
IE[MS ] = σ + α
6 A − i
I 1
i=1
X
J
In
2 2
IE[MS ] = σ + β
B − j
J 1
j=1
X
I J
n
2 2
IE[MS ] = σ + (αβ)
AB − − ij
(J 1)(I 1)
i=1 j=1
X X
2
where σ is an estimate of the variance of ǫ.
But if both A and B are random-effect factors,
2 2 2
IE[MS ] = σ + nJσ + nσ
A err A AB
2 2 2
IE[MS ] = σ + nIσ + nσ
B err B AB
2 2
IE[MS ] = σ + nσ
AB err AB
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2 2 2
where σ , σ , σ are the variances of the effect sizes for factors A, B, and
A B AB
the interactions, respectively.
Thus the F-statistics are always being formed as the ratio of two Mean Squares
which, when the null hypothesis is true, are estimating the same thing.
22.2 Mixed-Effects Example
Consider an experiment to test four different lesson plans for the same topic,
as taught by five instructors. Each instructor taught each set of lesson four
times, to different classes. The response is the average score of the class on a
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quiz at the end of the session.
The lesson plans are fixed effects. There are four, and they were not developed
at random.
The instructors are random effects. Conceptually, they are a random draw
from the pool of all possible instructors.
The numbers reported are from real data, but we know how to calculate them
from previous formulae.
Mixed Effects Two-Way ANOVA
Source df SS MS EMS F
2 2 nJ 2
Lesson 3 42 14.0 σ + nσ + α 14.0/3.9= 3.59
LI I−1 i
2 2
9 Inst. 4 54 13.5 σ + nIσ 13.5/2.1=6.43
P
I
2 2
L*I 12 47 3.9 σ + nσ 3.9/2.1=1.86
LI
2
error 60 126 2.1 σ
total 79 269
At the .05 level, the F cv is 1.92, the F cv is 2.53, and the F cv is
12,60 4,60 3,12
3.49. What do we conclude?
22.3 Factorial Designs
A factorial design is one in which every possible combination of treatment
levels for different factors appears.
1 The two-way ANOVA with interaction we considered was a factorial design.
0
We had n observations on each of the IJ combinations of treatment levels.
If there are, say, a levels of factor A, b levels of factor B, c levels of factors
C, then a factorial design requires at least abc observations, and more if one
wants to estimate the three way interaction among the factors. This can get
expensive when experiments have many different factors.