11
Multifactor Analysis
of Variance
Copyright © Cengage Learning. All rights reserved.
11.3
Three-Factor ANOVA
Copyright © Cengage Learning. All rights reserved.
Three-Factor ANOVA
To indicate the nature of models and analyses when
ANOVA experiments involve more than two factors, we will
focus here on the case of three fixed factors—A, B, and C.
The numbers of levels of these factors will be denoted by
I, J, and K, respectively, and Lijk = the number of
observations made with factor A at level i, factor B at level j,
and factor C at level k.
The analysis is quite complicated when the Lijk’s are not all
equal, so we further specialize to Lijk = L.
3
Three-Factor ANOVA
Then Xijkl and xijkl denote the observed value, before and
after the experiment is performed, of the lth replication
(l = 1, 2,…, L) when the three factors are fixed at levels i, j,
and k.
To understand the parameters that will appear in the threefactor ANOVA model, first recall that in two-factor ANOVA
with replications, E(Xijk) = ij =  + i + j + γij , where the
restrictions ii = jj = 0, iγij = 0 for every j, and iγij = 0
for every i were necessary to obtain a unique set of
parameters.
4
Three-Factor ANOVA
If we use dot subscripts on the ij’s to denote averaging
(rather than summation), then
is the effect of factor A at level i averaged over levels of
factor B, whereas
is the effect of factor A at level i specific to factor B at level j.
5
Three-Factor ANOVA
When the effect of A at level i depends on the level of B,
there is interaction between the factors, and the γij ’s are not
all zero.
In particular,
(11.11)
6
The Fixed Effects Model and Test
Procedures
7
The Fixed Effects Model and Test Procedures
The fixed effects model for three-factor ANOVA with
Lijk = L is
(11.12)
where the ijkl ’s are normally distributed with mean 0 and
variance  2, and
(11.13)
The restrictions necessary to obtain uniquely defined
parameters are that the sum over any subscript of any
parameter on the right-hand side of (11.13) equal 0.
8
The Fixed Effects Model and Test Procedures
The parameters ,
and
are called two-factor
interactions, and
is called a three-factor interaction; the
i’s, j ’s, and k ’s are the main effects parameters.
For any fixed level k of the third factor, analogous to
(11.11),
is the interaction of the ith level of A with the jth level of B
specific to the kth level of C, whereas
9
The Fixed Effects Model and Test Procedures
Is the interaction between A at level i and B at level j
averaged over levels of C. If the interaction of A at level i
and B at level j does not depend on k, then all ’s equal 0.
Thus nonzero ’s represent nonadditivity of the two-factor
’s over the various levels of the third factor C.
If the experiment included more than three factors, there
would be corresponding higher-order interaction terms with
analogous interpretations.
10
The Fixed Effects Model and Test Procedures
Note that in the previous argument, if we had considered
fixing the level of either A or B (rather than C, as was done)
and examining the ’s, their interpretation would be the
same; if any of the interactions of two factors depend on
the level of the third factor, then there are nonzero ’s.
When L > 1, there is a sum of squares for each main effect,
each two-factor interaction, and the three-factor interaction.
11
The Fixed Effects Model and Test Procedures
To write these in a way that indicates how sums of squares
are defined when there are more than three factors, note
that any of the model parameters in (11.13) can be
estimated unbiasedly by averaging Xijkl over appropriate
subscripts and taking differences.
Thus
with other main effects and interaction estimators obtained
by symmetry.
12
The Fixed Effects Model and Test Procedures
Definition
Relevant sums of squares are
df = IJKL – 1
df = I – 1
df = (I – 1)(J – 1)
13
The Fixed Effects Model and Test Procedures
df = (I – 1)(J – 1)(k – 1)
df = IJK(L – 1)
with the remaining main effect and two-factor interaction
sums of squares obtained by symmetry.
SST is the sum of the other eight SSs.
14
The Fixed Effects Model and Test Procedures
Each sum of squares (excepting SST) when divided by its
df gives a mean square. Expected mean squares are
E(MSE) =  2
with similar expressions for the other expected mean
squares. Main effect and interaction hypotheses are tested
by forming F ratios with MSE in each denominator.
15
The Fixed Effects Model and Test Procedures
Null Hypothesis
Test Statistic Value
Rejection Region
Usually the main effect hypotheses are tested only if all
interactions are judged not significant.
This analysis assumes that Lijk = L > 1. if L = 1, then as in the
two-factor case, the highest-order interactions must be
assumed absent to obtain an MSE that estimates  2.
16
The Fixed Effects Model and Test Procedures
Setting L = 1 and disregarding the fourth subscript
summation over l, the foregoing formulas for sums of
squares are still valid, and error sum of squares is
SSE =
with
= Xijk in the expression for .
17
Example 10
The following observations (body temperature –100°F)
were reported in an experiment to study heat tolerance of
cattle (“The Significance of the Coat in Heat Tolerance of
Cattle,” Australian J. Agric. Res., 1959: 744–748).
18
Example 10
cont’d
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  2  4 = 32 combinations of levels of the three factors.
The table of cell totals (xijk.’s) for all combinations of the
three factors is
19
Example 10
cont’d
Figure 11.8 displays plots of the corresponding cell means .
We will return to these plots after considering
tests of various hypotheses.
Plots of xijk. for Example 10
Figure 11.8
20
Example 10
cont’d
The basis for these tests is the ANOVA table given in Table
11.8.
ANOVA Table for Example 10
Table 11.8
21
Example 10
cont’d
Since F.01,9,64  2.70 and fABC = MSABC/MSE = .704 does
not exceed 2.70, we conclude that three-factor interactions
are not significant, However, although the AB interactions
are also not significant, both AC and BC interactions as
well as all main effects seem to be necessary in the model.
When there are no ABC or AB interactions, a plot of the
separately for each level of C should reveal
no substantial interactions (if only the ABC interactions are
zero, plots are more difficult to interpret.)
22
Latin Square Designs
23
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.
Frequently an experiment of this size is either impracticable
because of cost, time, or space constraints or literally
impossible.
24
Latin Square Designs
For example, if the response variable is sales of a certain
product and the factors are different display configurations,
different stores, and different time periods, then only one
display configuration can realistically be used in a given
store during a given time period.
A three-factor experiment in which fewer than IJK
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.
25
Latin Square Designs
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 twoway 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.
26
Latin Square Designs
Figure 11.9 shows examples of 3  3, 4  4, and 5  5 Latin
squares.
Examples of Latin squares
Figure 11.9
There are 12 different 3  3 Latin squares, and the number
of different Latin squares increases rapidly with the number
of levels. (e.g., every permutation of rows of a given Latin
square yields a Latin square, and similarly for column
permutations).
27
Latin Square Designs
It is recommended that the square used in a an actual
experiment be chosen at random from the set of all possible
squares of the desired dimension; for further details, consult
one of the chapter references.
The letter N will denote the common value of I, J, and K.
Then a complete layout with one observation per
combination would require N3 observations, whereas a Latin
square requires only N2 observations.
28
Latin Square Designs
Once a particular square has been chosen, the value of k
(the level of factor C) is completely determined by the
values of i and j.
To emphasize this, we use xij(k) to denote the observed
value when the three factors are at levels i, j, and k,
respectively, with k taking on only one value for each i, j
pair.
The model equation for a Latin square design is
Xij(k) =  + i + j + k + ij(k)
i, j, k = 1,…, N
where i = j = k = 0 and ij(k) ’s the are independent
and normally distributed with mean 0 and variance  2.
29
Latin Square Designs
We employ the following notation for totals and averages:
Note that although Xi.. previously suggested a double
summation, now it corresponds to a single sum over all j
(and the associated values of k).
30
Latin Square Designs
Definition
Sums of squares for a Latin square experiment are
df = N2 – 1
df = N = 1
df = N – 1
df = N – 1
31
Latin Square Designs
df = N – 1
df = (N – 1)(N – 2)
SST = SSA + SSB + SSC + SSE
Each mean square is, of course, the ratio SS/df. For testing
H0C : 1 = 2 =    =N = 0, the test statistic value is
fC = MSC/MSE, with H0C rejected if fc  F ,N – 1,(N – 1)(N – 2).
32
Latin Square Designs
The other two main effect null hypotheses are also rejected
if the corresponding F ratio is at least F ,N – 1,(N – 1)(N – 2).
If any of the null hypotheses is rejected, significant
differences can be identified by using Tukey’s procedure.
After computing
pairs of sample
means (the xi..’s, x.j.’s, or x..k’s ) differing by more than w
correspond to significant differences between associated
factor effects (the i’s, j’s, or k’s ).
The hypothesis H0C is frequently the one of central interest.
33
Latin Square Designs
A Latin square design is used to control for extraneous
variation in the A and B factors, as was done by a
randomized block design for the case of a single
extraneous factor.
Thus in the product sales example mentioned previously,
variation due to both stores and time periods is controlled
by a Latin square design, enabling an investigator to test
for the presence of effects due to different product-display
configurations.
34
Example 11
In an experiment to investigate the effect of relative
humidity on abrasion resistance of leather cut from a
rectangular pattern
(“The Abrasion of Leather,” J. Inter. Soc. Leather Trades’
Chemists, 1946: 287), a 6  6 Latin square was used to
control for possible variability due to row and column
position in the pattern.
35
Example 11
cont’d
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:
Also, x..1 = 46.10, x..2 = 40.59, x..3 = 39.56, x..4 = 35.86,
x..5 = 32.23, x..6 = 32.64, x… = 226.98.
36
Example 11
cont’d
Further computations are summarized in Table 11.9.
ANOVA Table for Example 11
Table 11.9
Since F.05,5,20 = 2.71 and 26.89  2.71, H0C is rejected in
favor of the hypothesis that relative humidity does on
average affect abrasion resistance.
37
Example 11
cont’d
To apply Tukey’s procedure,
Ordering the x  k’s and underscoring yields
In particular, the lowest relative humidity appears to result
in a true average abrasion resistance significantly higher
than for any other relative humidity studied.
38