and 
, on the response is being investigated.
Let there be 
levels of factor 
and 
levels of factor 
. The ANOVA model for this experiment
can be stated as:This section is divided into the following subsections:
In DOE++, factorial
experiments are referred to as factorial designs. The experiments explained
in this section are referred to as general factorial designs. This is
done to distinguish these experiments from the other factorial designs
supported by DOE++ (see Figure 6.10). The other
designs (such as the 2 level full factorial designs that are explained
in Chapter 7) are special
cases of these experiments in which factors are limited to a specified
number of levels. The ANOVA model for the analysis of factorial experiments
is formulated as shown next. Assume a factorial experiment in which the
effect of two factors, and , on the response is being investigated.
Let there be levels of factor and levels of factor . The ANOVA model for this experiment
can be stated as:
(16)
where:
represents the overall mean effect.
is the effect of the 
th level of factor 
(
).
is the effect of the 
th level of factor 
(
).
represents the interaction effect
between 
and 
.
represents the random error terms
(which are assumed to be normally distributed with a mean of zero and
variance of 
).
the subscript 
denotes the 
replicates (
).
|
Figure 6.10: Factorial experiments available in DOE++. |
Since the effects 
, 
and 
represent deviations from the overall
mean, the following constraints exist: [Note]
These tests are used to check whether each of the factors investigated
in the experiment is significant or not. For the previous example, with
two factors, 
and 
, and their interaction, 
, the statements for the hypothesis
tests can be formulated as follows:
The test statistics for the three
tests are as follows:
The tests are identical to the partial
test explained in Chapter
5. The sum of squares for these tests (to obtain the mean squares)
are calculated by splitting the model sum of squares into the extra sum
of squares due to each factor. The extra sum of squares calculated for
each of the factors may either be partial or sequential. [Note]
For the present example, if the extra sum of squares used is sequential,
then the model sum of squares can be written as:
where 
represents the model sum of squares,
represents the sequential sum of squares
due to factor 
, 
represents the sequential sum of squares
due to factor 
and 
represents the sequential sum of squares
due to the interaction 
.
The mean squares are obtained by dividing the sum of squares by the
associated degrees of freedom. Once the mean squares are known the test
statistics can be calculated. For example, the test statistic to test
the significance of factor 
(or the hypothesis 
) can then be obtained as:
Similarly the test statistic to test
significance of factor 
and the interaction 
can be respectively obtained as:
It is recommended to conduct the
test for interactions before conducting the test for the main effects.
This is because, if an interaction is present, then the main effect of
the factor depends on the level of the other factors and looking at the
main effect is of little value. However, if the interaction is absent
then the main effects become important.
Consider an experiment to investigate the effect of speed and type of fuel additive used on the mileage of a sports utility vehicle. Three speeds and two types of fuel additives are investigated. Each of the treatment combinations are replicated three times. The mileage values observed are displayed in Table 6.5.
|
Table 6.5: Mileage data for different speeds and fuel additive types. |
The experimental design for the data in Table 6.5 is shown in Figure
6.11. In the figure, the factor Speed is represented
as factor 
and the factor Fuel Additive is represented
as factor 
. The experimenter would like to investigate
if speed, fuel additive or the interaction between speed and fuel additive
affects the mileage of the sports utility vehicle. In other words, the
following hypotheses need to be tested:
The test statistics for the three tests are:
|
Figure 6.11: Experimental design for the data in Table 6.5. |
The ANOVA model for this experiment can be written as:
(17)where 
represents the 
th treatment of factor 
(speed) with 
=1, 2, 3; 
represents the 
th treatment of factor 
(fuel additive) with 
=1, 2; and 
represents the interaction effect.
In order to calculate the test statistics, it is convenient to express
the ANOVA model of Eqn. (17) in the form 
. This can be done as explained next.
Since the effects 
, 
and 
represent deviations from the overall
mean, the following constraints exist.
Constraints on 
are:
(18)
Therefore, only two of the 
effects are independent. Assuming
that 
and 
are independent, 
. (The null hypothesis to test the
significance of factor 
can be rewritten using only the independent
effects as 
.) DOE++ displays only the independent
effects because only these effects are important to the analysis. The
independent effects, 
and 
, are displayed as A[1] and A[2] respectively
because these are the effects associated with factor 
(speed).
Constraints on 
are:
(19)
Therefore, only one of the 
effects are independent. Assuming
that 
is independent, 
. (The null hypothesis to test the
significance of factor 
can be rewritten using only the independent
effect as 
.) The independent effect 
is displayed as B:B in DOE++.
Constraints on 
are:
(20)
(21)
(22)
(23)
(24)
Eqns. (20) to (24) represent
four constraints as only four of these five equations are independent.
Therefore, only two out of the six 
effects are independent. Assuming
that 
and 
are independent, the other four effects
can be expressed in terms of these effects. (The null hypothesis to test
the significance of interaction 
can be rewritten using only the independent
effects as 
.) The effects 
and 
are displayed as A[1]B and A[2]B respectively
in DOE++.
Based on the ANOVA model and the constraints of Eqns. (18)
to (24), the regression version of the ANOVA model
can be obtained using indicator variables, similar to the case of the
single factor experiment in Chapter 6, Fitting
ANOVA Models. Since factor 
has three levels, two indicator variables,
and 
, are required which need to be coded
as shown next:
Factor 
has two levels and can be represented
using one indicator variable, 
, as follows:
The 
interaction will be represented by
all possible terms resulting from the product of the indicator variables
representing factors 
and 
. There are two such terms here - 
and 
. The regression version of the ANOVA
model can finally be obtained as:
(25)In matrix notation this model
can be expressed as:
where:
The vector 
can be substituted with the response
values from Table 6.5 to get:
Knowing 
, 
and 
, the sum of squares for the ANOVA
model and the extra sum of squares for each of the factors can be calculated.
These are used to calculate the mean squares that are used to obtain the
test statistics.
The model sum of squares, 
, for the model of Eqn. (25)
can be obtained as:
where 
is the hat matrix and 
is the matrix of ones. Since five
effect terms (
, 
, 
, 
and 
) are used in the model, the number
of degrees of freedom associated with 
is five (
).
The total sum of squares, 
, can be calculated as:
Since there are 18 observed response
values, the number of degrees of freedom associated with the total sum
of squares is 17 (
). The error sum of squares can now
be obtained:
Since there are three replicates of
the full factorial experiment, all of the error sum of squares is pure
error. (This can also be seen from Figure 6.11 where
each treatment combination of the full factorial design is repeated three
times.) The number of degrees of freedom associated with the error sum
of squares is:
The sequential sum of squares for factor 
can be calculated as: [Note]
where 
and 
is the matrix containing only the
first three columns of the 
matrix. Thus:
Since there are two independent effects (
, 
) for factor 
, the degrees of freedom associated
with 
are two (
).
Similarly, the sum of squares for factor 
can be calculated as:
Since there is one independent effect, 
, for factor 
, the number of degrees of freedom
associated with 
is one (
).
The sum of squares for the interaction 
is:
Since there are two independent interaction
effects, 
and 
, the number of degrees of freedom
associated with 
is two (
).
Knowing the sum of squares, the test statistic for each of the factors
can be calculated. Analyzing the interaction first, the test statistic
for interaction 
is:
The 
value corresponding to this statistic,
based on the 
distribution with 2 degrees of freedom
in the numerator and 12 degrees of freedom in the denominator, is: [Note]
Assuming that the desired significance
level is 0.1, since 
value > 0.1, we fail to reject
and conclude that the interaction
between speed and fuel additive does not significantly affect the mileage
of the sports utility vehicle. DOE++ displays this result in the ANOVA
table, as shown in Figure 6.12. In the absence
of the interaction, the analysis of main effects becomes important.
The test statistic for factor 
is:
The 
value corresponding to this statistic
based on the 
distribution with 2 degrees of freedom
in the numerator and 12 degrees of freedom in the denominator is:
Since 
value < 0.1, 
is rejected and it is concluded that
factor 
(or speed) has a significant effect
on the mileage.
The test statistic for factor 
is:
The 
value corresponding to this statistic
based on the 
distribution with 2 degrees of freedom
in the numerator and 12 degrees of freedom in the denominator is:
Since 
value < 0.1, 
is rejected and it is concluded that
factor 
(or fuel additive type) has a significant
effect on the mileage.
Therefore, it can be concluded that speed and fuel additive type affect the mileage of the vehicle significantly. The results are displayed in the ANOVA table of Figure 6.12.
|
Figure 6.12: Analysis results for the experiment in Table 6.5. |
Calculation of Effect Coefficients
Results for the effect coefficients of the model of Eqn. (25)
are displayed in the Regression Information table in Figure 6.12.
Calculations of the results in this table are discussed next. The effect
coefficients can be calculated as follows:
Therefore, 
, 
, 
etc. As mentioned previously, these
coefficients are displayed as Intercept, A[1] and A[2] respectively depending
on the name of the factor used in the experimental design. The standard
error for each of these estimates is obtained using the diagonal elements
of the variance-covariance matrix 
.
For example, the standard error for 
is:
Then the 
statistic for 
can be obtained as:
The 
value corresponding to this statistic
is:
Confidence intervals on 
can also be calculated. The 90% limits
on 
are:
Thus, the 90% limits on 
are 
and 
respectively. Results for other coefficients
are obtained in a similar manner.
The estimated mean response corresponding to the 
th level of any factor is obtained using
the adjusted estimated mean which is also called the least
squares mean. For example, the mean response corresponding to the
first level of factor 
is 
. An estimate of this is 
or (
). Similarly, the estimated response
at the third level of factor 
is 
or 
or (
).
As in the case of single factor experiments, plots of residuals can also be used to check for model adequacy in factorial experiments. Box-Cox transformations are also available in DOE++ for factorial experiments.
Factorial Experiments with a Single Replicate
Tests on Subsets of Regression Coefficients (Partial F Test)
