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ANOVA – TWO-FACTOR FACTORIAL EXPERIMENTS
In many experimental situations there are two or more factors
(treatments) of interest.
For simplicity we will describe the case for two factors: factor A
with a levels and factor B with b levels.
We have previously considered ANOVA for p different levels of a
single factor incorporated in a completely randomized design and in a
randomized block design.
A randomized block design is often called a two-way
classification of data because it has the following characteristics:
 It involves two independent variables – one factor and one
direction of blocking.
 Each level of one independent variable occurs with every
level of the other independent variable.
A two-way classification of data always permits the display
of the data in a two-way table, one containing r rows and c
columns.
The treatment selection for a two-factor experiment may also yield a
two-way classification of data.
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For example, suppose we want to relate the mean number of
defects on a finished item to two factors, type of nozzle for a
varnish spray gun and the length of spraying time.
Suppose further that we want to investigate the mean number of
defects for three types of nozzles (three levels) and two lengths
of spraying time (two levels).
If we include the treatments for the experiment to include
all combinations of the three levels of nozzle type with the
two levels of spraying time, we will obtain a two-way
classification of data.
This selection of treatments is called a complete 3 x 2
factorial experiment.
Note that the design, called a factorial design, will
contain 3 x 2 = 6 treatments.
If we were to include a third factor, say, paint type at three
levels, then a complete factorial experiment would include
all 3 x 2 x 3 = 18 combinations, and the resulting collection
of data would be termed a three-way classification of data.
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Methodology
Suppose a two-way classification represents a two-factor factorial
experiment with factor A at a levels and factor B at b levels.
Further, assume that the ab treatments of the factorial experiment are
replicated r times so that there are r observations for each of the ab
treatment combinations.
Then the total number of observations is n = abr and the total sum of
squares, SS(Total) can be partitioned into four parts: SS(A), SS(B),
SS(AB), and SSE.
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The first two sums of squares are called main effect sum of
squares to distinguish them from the interaction sum of squares.
When the number of observations per cell for a two-way factorial
experiment is the same for every cell (r per cell), the sums of squares
and the degrees of freedom for the analysis of variance are additive:
SS(Total) = SS(A) + SS(B) +SS(AB) +SSE
and
abr – 1 = (a-1) + (b-1) + (a-1)(b-1) + ab(r-1)
The corresponding ANOVA table would then appear as follows:
Note that for a factorial experiment, the number of r observations per
factor-level combination, must always be two or more; otherwise,
there will not be any degrees of freedom for SSE.
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If either A or B represents a direction of blocking, then the AB
interaction terms are deleted from the ANOVA table and the degrees
of freedom and sums of squares from lines 3 and 4 are combined to
form a source of error variation.
This is because the block-treatment interaction always
represents experimental error.
The resulting ANOVA table would then appear as shown below:
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For reference, the notation used in the formulas for the respective
sums of squares and the formulas are given below:
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To test a hypothesis for any one of the three sources of variation, we
proceed in exactly the same manner as was done previously, i.e., we
divide the appropriate mean square by the MSE and use F ratio at a
test statistic.
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ANOVA Example 3
A company that stamps gaskets out of sheets of rubber, plastic, an
other materials, wants to compare the mean number of gaskets
produced per hour for two different types of stamping machines.
Practically, the manufacturer wants to determine whether one
machine is more productive than the other, and even more
important, whether one machine is more productive in
producing rubber gaskets while the other is more productive in
producing plastic gaskets.
To answer these questions, the manufacturer decides to conduct a 2 x
3 factorial experiment using three types of gasket materials, B1, B2,
and B3, with each of the two types of stamping machines, A1 and A2.
Each machine is operated for three 1-hour periods for each of
the gasket materials, with the 18 1-hour time periods assigned to
the six machine-material combinations in random order (to
eliminate the possibility that uncontrolled environmental factors
might bias the results).
Assume that we have calculated the six treatment means.
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Two hypothetical plots of the six means are shown below - what do
each of these plots imply about the productivity of the two stamping
machines?
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The first figure suggests that machine A1 produces a larger number of
gaskets per hour regardless of the gasket material, and is therefore
superior to machine A2.
On the average, machine A1 stamps out more B1 gaskets per hour
than B2 or B3, but the difference in the mean numbers of the
gaskets produced by the two machines remains approximately
the same, regardless of the material.
Thus, the difference in the mean number of gaskets produced by
the two machines is independent of the material used in the
stamping process.
In contrast, the second figure shows the productivity of machine A1
to be greater than for machine A2 when the gasket material is B1 or
B3.
But the means are reversed for B2 such that A2 produces more on
average than A1.
Therefore, this figure illustrates a situation where the mean value of
the response variable depends on the combination of the factor levels.
When this situation occurs, we say that the factors interact.
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Thus, one of the most important objectives of a factorial experiment
is to detect factor interaction if it exists.
Tests for main effects are relevant only when no interaction
exists between factors.
Generally, the test for interaction is performed first.
If there is evidence of factor interaction, then tests to assess
main factor effects are not performed.
In Summary,
 In a factorial experiment, when the difference in the mean levels of
factor A depends on the different levels of factor B, we say that the
factors A and B interact.
 If the difference in A is independent of the levels of B, then there is
no interaction between A and B.
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ANOVA Example 4
A manufacturer, whose daily supply of raw materials is variable and
limited, can use the material to produce two different products in
various proportions.
The profit per unit of raw material obtained by producing each
of the two products depends on the length of product’s
manufacturing run and hence on the amount of raw material
assigned to it.
Other factors, such as worker productivity and machine
breakdown, affect the profit per unit as well, but their net effect
on profit is random and uncontrollable.
The manufacturer has conducted an experiment to investigate the
effect of the level of supply of raw materials (S) and the ratio of its
assignment (R) to the two product manufacturing lines on the profit y
per unit of raw material.
The ultimate goal would be to be able to choose the best ratio R
to match each day’s supply of raw materials S.
The levels of supply of the raw material chosen for the
experiment were 15, 18, and 21 tons; the levels of the ratio of
allocation to the two product lines were 0.5, 1, and 2.
The response was the profit ($) per unit of raw material supply
obtained from a single day’s production.
Three replications of a complete 3 x 3 factorial experiment were
conducted in a random sequence and the results are shown below.
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The presence of interaction tells you that the mean profit depends
upon the particular combination of levels of S and R.
Consequently, there is little point in checking to see whether the
means differ for the 3 levels of S or 3 levels of R, i.e., we will not
perform the tests for main effects.
For example, the supply level that provides the highest mean
profit (over all levels of R) might not be the same supply-ratio
level combination that produces the largest mean profit per unit
of raw material.