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Multifactor Analysis of Variance
In many experimental situations, there are two or more factors that are of simultaneous interest.
ANOVA 2 extends the methods of ANOVA 1 to investigate such multifactor situations.
Example 2.1
In a study on automobile traffic and air pollution reported in the International
Journal of Environmental Studies, air samples taken at four different times and at
five different locations were analyzed to obtain the amount of particulate matter
present in the air (mg/rn3).
Let;
Example 1, the number of levels of factor A is I = 4, the number of levels of factor
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The Model
Includingσ², there are now I + J + 1 model parameters, so if I ≥ 3 and J ≥ 3, then there
will be fewer parameters than observations. The model specified above is called an
additive model because
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Figure 1. Mean responses for (a) an additive, and (b) a nonadditive model
Example 2(Example 1 continued)
If we plot the observed xij’s in a manner analogous to that of Figure 1, the result is shown in Figure
2. While there is some “crossing over” in the observed xij’s, the configuration is reasonably
representative of what would be expected under additivity with just one observation per treatment.
Figure 2 Plot of data from Example 2
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(
For the
The interpretation of the parameters of (3) is straightforward: μ is the true grand mean),
αi is the effect of factor A at level i (measured as a deviation from μ ), and βj is the effect
of factor B at level j. Unbiased (and maximum likelihood) estimators for these
parameters are;
(
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Example 3
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ANOVA table (Table 2.1) summarizes further calculations.
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Example 4 (Example 3 continued)
Randomized Block Experiments
It frequently happens, though, that subjects or experimental units exhibit heterogeneity with
respect to other variables that may affect the observed responses. This was the reason for
introducing a paired experiment in C.I section. The analogy to a paired experiment when I > 2 is
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called a randomized block experiment. An extraneous factor, “blocks,” is constructed by dividing the
IJ units into J groups with I units in each group. This grouping or blocking is done in such a way
that within each block, the I units are homogeneous with respect to other factors thought to affect
the responses. Then within each homogeneous block, the I treatments are randomly assigned to the
I units or subjects in the block.
Example 5
A consumer product-testing organization wished to compare the annual power consumption
for five different brands of dehumidifier. Because power consumption depends on the prevailing
humidity level, it was decided to monitor each brand at four different levels ranging from moderate
to heavy humidity (thus blocking on humidity level). Within each level, brands were randomly
assigned to the five selected locations. The resulting amount of power consumption (annual kwh)
appears in Table 2.
Table 2 Power consumption data for Example 5
Table .3 ANOVA table for Example .5
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Figure 3 SAS output for power consumption data
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Models for Random Effects
In a two-factor situation, a random effects model is appropriate. The case in which the levels of one
factor are the only ones of interest and the levels of the other factor are selected from a population
of levels leads to a mixed effects model. The two-factor random effects model when Kij = 1 is
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Exercises
Continuation
of question 4
Continuation
of question 2
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Solution for question 2
Solution for question 3
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Solution for question 4
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2.2
To obtain valid test procedures, the μij’s were assumed to have an additive structure with
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Parameters for the Fixed Effects Model with Interaction
Let
(2.7)
(2.8)
(2.9)
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(2.10)
Example 2.6
Three different varieties of tomato (Harvester, Pusa Early Dwarf, and Ife No. 1) and four different
plant densities (10, 20, 30, and 40 thousand plants per hectare) are being considered for planting in
a particular region. To see whether either variety or plant density affects yield, each combination of
variety and plant density is used in three different plots, resulting in the data on yields in Table 2.6.
Table 2.6 Yield data for Example 2.6
To test the hypotheses of interest, we again define sums of squares and present computing formulas:
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Total variation is thus partitioned into four pieces: unexplained (SSE) and three pieces that may be
explained by the truth or falsity of the three H0’s. Each of four mean squares is defined by MS = SS/df.
The expected mean squares suggest that each set of hypotheses should be tested using the appropriate
ratio of mean squares with MSE in the denominator:
Each of the three mean square ratios can be shown to have an F distribution when the associated H0 is
true, which yields the following level α test procedures:
As before, the results of the analysis are summarized in an ANOVA table.
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Example 2.7 (continuation of example 2.6)
Table 2.7 ANOVA table for Example 2.7
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Example 2.8 (continuation of example 2.7)
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Models with Mixed and Random Effects
In some problems, the levels of either factor may have been chosen from a large population of
possible levels, so that the effects contributed by the factor are random rather than fixed. If both
factors contribute random effects, the model is referred to as a random effects model, while if one
factor is fixed and the other is random, a mixed effects model results. We will consider here the
analysis for a mixed effects model in which factor A (rows) is the fixed factor and factor B
(columns) is the random factor. The mixed effects model in this situation is
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Example 2.9
A study was carried out to compare the writing lifetimes of four premium brands of
pens. It was thought that the writing surface might affect lifetime, so three different surfaces were
randomly selected. A writing machine was used to ensure that conditions were otherwise
homogeneous (e.g., constant pressure and a fixed angle). Table 2.8 shows the two lifetimes (mm)
obtained for each brand—surface combination.
Table 2.8 Lifetime data for Example 2.9
Table 2.9 ANOVA table for Example 2.9
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Exercises
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Solution for question 15
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Solution for question 16
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e)-
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Three-Factor ANOVA
If we use dot subscripts on the μij’s to denote averaging (rather than summation), then
(2.11)
(2.12)
(2 11)
(2.13)
(2 11)
(2.13)
(2.11)
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factors, there would be corresponding higher-order interaction terms with analogous
interpretations.
The Analysis of a Three-Factor Experiment
When L > 1, there is a sum of squares for each main effect, note that any of the model parameters
in (2.13) can be estimated unbiasedly by averaging Xijkl over appropriate subscripts and taking
differences. Thus,
Even the computational formulas for these SSs are quite tedious to use, so we eschew them in favour
of output from a statistical computer package. The current version of MINITAB, for example, will fit a
three-factor model with fixed, mixed, or random effects. Each sum of squares (excepting SST) when
divided by its df gives a mean square, with
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Main effect and interaction hypotheses are tested by forming F ratios with MSE in each denominator:
Example 2.10
The following observations (body temperature — 100°F) were reported in an
experiment to study heat tolerance of cattle. Measurements were made at four
different periods (factor A, with I = 4) on two different strains of cattle (factor B,
with J = 2) having four different types of coat (factor C, with K = 4); L = 3
observations were made for each of the 4 x 2 x 4 = 32 combinations of levels of
the three factors.
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Figure 2.4
Figure 2.4 Plots of Xijk. for Example 2.10
Table 2.10 ANOVA table for Example 2.10
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Latin Square Designs
When several factors are to be studied simultaneously, an experiment in which
there is at least one observation for every possible combination of levels is referred
to as a complete layout. If the factors are A, B, and C with I, J, and K levels,
respectively, a complete layout requires at least IJK observations.
A three-factor experiment in which fewer than JfK observations are made is
called an incomplete layout. There are some incomplete layouts in which the
pattern of combinations of factors is such that the analysis is straightforward. One
such three-factor design is called a Latin square. It is appropriate when I = J = K
(e.g., four display configurations, four stores, and four time periods) and all two- and
three-factor interaction effects are assumed absent. If the levels of factor A are
identified with the rows of a two-way table and the levels of B with the columns of
the table, then the defining characteristic of a Latin square design is that every
level of factor C appears exactly once in each row and exactly once in each column.
Pictured in Figure 2.5 are examples of 3 x 3, 4 x 4, and 5 x 5 Latin squares. There
are 12 different 3 x 3 Latin squares, and the number of different N x N Latin squares
increases rapidly with N (e.g., every permutation of rows of a given Latin square
yields a Latin square, and similarly for column permutations).
Figure 2.5 Examples of Latin squares
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The Model and Analysis for Latin Squares
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Example 2.11
In an experiment to investigate the effect of relative humidity on abrasion
resistance of leather cut from a rectangular pattern, a 6 x 6 Latin square was
used to control for possible variability due to row and column position in the
pattern. The six levels of relative humidity studied were 1 = 25%, 2 = 37%, 3
= 50%, 4 = 62%, 5 = 75%, and 6 = 87%, with the following results:
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Table 2.10 ANOVA table for Example 2.11
Exercises
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Solution of question 27
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Solution of question 28
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