This section is divided into the following subsections:
In the case of replicated experiments, it is important to note the difference
between replicated runs and repeated runs. Both repeated
and replicated runs are multiple response readings taken at the same factor
levels. However, repeated runs are response observations taken at the
same time or in succession. Replicated runs are response observations
recorded in a random order. Therefore, replicated runs include more variation
than repeated runs. For example, a baker, who wants to investigate the
effect of two factors on the quality of cakes, will have to bake four
cakes to complete one replicate of a 2
design. Assume that the baker bakes
eight cakes in all. If, for each of the four treatments of the 2
design, the baker selects one treatment
at random and then bakes two cakes for this treatment at the same time
then this is a case of two repeated runs. If, however, the baker bakes
all the eight cakes randomly, then the eight cakes represent two sets
of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into DOE++ as shown in Figure 7.11 (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in Figure 7.11 (b).
|
Figure 7.11: Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs. |
Sometimes it is only possible to run a single replicate of the 2
design because of constraints on resources
and time. As stated in Chapter 6, in an unreplicated experiment, the error
sum of squares cannot be obtained as the model fits the data perfectly
and no degrees of freedom are available to calculate the error sum of
squares. In the absence of the error sum of squares, hypothesis tests
to identify significant factors cannot be conducted. A number of methods
of analyzing information obtained from unreplicated 2
designs are available. These include
pooling higher order interactions, using the normal probability plot of
effects or including center point replicates in the design.
One of the ways to deal with unreplicated 2
designs is to use the sum of squares
of some of the higher order interactions as the error sum of squares provided
these higher order interactions can be assumed to be insignificant. By
dropping some of the higher order interactions from the model, the degrees
of freedom corresponding to these interactions can be used to estimate
the error mean square. Once the error mean square is known, the test statistics
to conduct hypothesis tests on the factors can be calculated.
Another way to use unreplicated 2
designs to identify significant effects
is to construct the normal probability plot of the effects. As mentioned
in Chapter 7, Special
Features, the standard error for all effect coefficients in
the 2
designs is the same. Therefore, on
a normal probability plot of effect coefficients, all non-significant
effect coefficients (with 
) will fall along the straight line
representative of the normal distribution, N(
). [Note]
Effect coefficients that show large deviations from this line will be
significant since they do not come from this normal distribution. Similarly,
since effects 
effect coefficients, all non-significant
effects will also follow a straight line on the normal probability plot
of effects. For replicated designs, the Effects Probability plot of DOE++
plots the normalized effect values (or the T Values) on the standard normal
probability line, N(0,1). However, in the case of unreplicated 2
designs, 
remains unknown since 
cannot be obtained. Lenth's method
is used in this case to estimate the variance of the effects.[11] DOE++ then uses this variance
value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.
Example 7.2
Vinyl panels, used as instrument panels in a certain automobile, are
seen to develop defects after a certain amount of time. To investigate
the issue, it is decided to carry out a two level factorial experiment.
Potential factors to be investigated in the experiment are vacuum rate
(factor 
), material temperature (factor 
), element intensity (factor 
) and pre-stretch (factor 
). The two levels of the factors used
in the experiment are as shown in Table 7.1. The factor properties entered
in DOE++ using this table are shown in Figure 7.12.
With a 2
design requiring 16 runs per replicate
it is only feasible for the manufacturer to run a single replicate.
|
Table 7.1: Factors to investigate defects in vinyl panels. |
|
Figure 7.12: Factor properties for the experiment in Example 7.2. |
The experiment design and data, collected as percent defects, are shown
in Figure 7.13. Since the present experiment design
contains only a single replicate, it is not possible to obtain an estimate
of the error sum of squares, 
. It is decided to use the normal probability
plot of effects to identify the significant effects. The effect values
for each term are obtained as shown in Figure 7.14.
Lenth's method uses these values to estimate the variance. If all effects
are arranged in ascending order, using their absolute values, then 
is defined as 1.5 times the median
value:[11]
|
Figure 7.13: Experiment design for Example 7.2. |
Using 
, the "pseudo standard error"
(
) is calculated as 1.5 times the median
value of all effects that are less than 2.5
:
Using 
as an estimate of the effect variance,
the effect variance is 2.25. Knowing the effect variance, the normal probability
plot of effects for the present unreplicated experiment can be constructed
as shown in Figure 7.15. The line on this plot
is the line N(0, 2.25). The plot shows that the effects 
, 
and the interaction 
do not follow the distribution represented
by this line. Therefore, these effects are significant.
The significant effects can also be identified by comparing individual
effect values to the margin of error or the threshold value
using the pareto chart (see Figure 7.16). If the
required significance is 0.1, then:
The 
statistic, 
, is calculated at a significance of
(for the two-sided hypothesis) and
degrees of freedom 
number of effects
. Thus:
The value of 4.534 is shown as the critical value line in Figure 7.16. All effects with absolute values greater than the
margin of error can be considered to be significant. These effects are
, 
and the interaction 
. Therefore, the vacuum rate, the pre-stretch
and their interaction have a significant effect on the defects of the
vinyl panels.
|
Figure 7.14: Effect values for the experiment in Example 7.2. |
|
Figure 7.15: Normal probability plot of effects for the experiment in Example 7.2. |
|
Figure 7.16: Pareto chart for the experiment in Example 7.2. |
Another method of dealing with unreplicated 2
designs that only have quantitative
factors is to use replicated runs at the center point. [Note]
The center point is the response corresponding to the treatment exactly
midway between the two levels of all factors. Running multiple replicates
at this point provides an estimate of pure error. Although running multiple
replicates at any treatment level can provide an estimate of pure error,
the other advantage of running center point replicates in the 2
design is in checking for the presence
of curvature. The test for curvature investigates whether the model between
the response and the factors is linear and is discussed below in Using
Center Points to Test Curvature.
Example 7.3
Consider a 2
experiment design to investigate the
effect of two factors, 
and 
, on a certain response. The energy
consumed when the treatments of the 2
design are run is considerably larger
than the energy consumed for the center point run (because at the center
point the factors are at their middle levels). Therefore, the analyst
decides to run only a single replicate of the design and augment the design
by five replicated runs at the center point as shown in Figure 7.17.
[Note]
The design properties for this experiment are shown in Figure 7.18.
The complete experiment design is shown in Figure 7.19.
The center points can be used in the identification of significant effects
as shown next.
|
Figure 7.17: 2 |
|
Figure 7.18: Design properties for the experiment in Example 7.3. |
|
Figure 7.19: Experiment design for Example 7.3. |
Since the present 2
design is unreplicated, there are
no degrees of freedom available to calculate the error sum of squares.
By augmenting this design with five center points, the response values
at the center points, 
, can be used to obtain an estimate
of pure error, 
. Let 
represent the average response for
the five replicates at the center. Then:
Then the corresponding mean square is:
Alternatively, 
can be directly obtained by calculating
the variance of the response values at the center points:
Once 
is known, it can be used as the error
mean square, 
, to carry out the test of significance
for each effect. For example, to test the significance of the main effect
of factor 
the sum of squares corresponding to
this effect is obtained in the usual manner by considering only the four
runs of the original 2
design.
Then, the test statistic to test the significance of the main effect
of factor 
is:
The 
value corresponding to the statistic,
, based on the 
distribution with one degree of freedom
in the numerator and eight degrees of freedom in the denominator is:
Assuming that the desired significance is 0.1, since 
value < 0.1, it can be concluded
that the main effect of factor 
significantly affects the response.
This result is displayed in the ANOVA table as shown in Figure 7.20.
Test for the significance of other factors can be carried out in a similar
manner.
|
Figure 7.20: Results for the experiment in Example 7.3. |
Center point replicates can also be used to check for curvature in replicated
or unreplicated 2
designs. The test for curvature investigates
whether the model between the response and the factors is linear. The
way DOE++ handles center point replicates is similar to its handling of
blocks. The center point replicates are treated as an additional factor
in the model. The factor is labeled as Curvature in the results of DOE++.
If Curvature turns out to be a significant factor in the results, then
this is an indication of the presence of curvature in the model.
Example 7.4
To illustrate the use of center point replicates in testing for curvature,
consider again the data of the single replicate 2
experiment from Figure 7.17.
Let 
be the indicator variable to indicate
if the run is a center point:
If 
and 
are the indicator variables representing
factors 
and 
, respectively, then the model for
this experiment is:
To investigate the presence of curvature, the following hypotheses need
to be tested:
The test statistic to be used for this test is:
where 
is the mean square for Curvature and
is the error mean square.
The 
matrix and 
vector for this experiment are:
The sum of squares can now be calculated. For example, the error sum
of squares is:
where 
is the identity matrix and 
is the hat matrix. It can be seen
that this is equal to 
(the sum of squares due to pure error)
because of the replicates at the center point, as obtained in the Example
7.3. The number of degrees of freedom associated with 
, 
is four. [Note
]
The extra sum of squares corresponding to the center point replicates
(or Curvature) is:
where 
is the hat matrix and 
is the matrix of ones. The matrix
can be calculated using 
where 
is the design matrix, 
, excluding the second column that
represents the center point. Thus, the extra sum of squares corresponding
to Curvature is:
This extra sum of squares can be used to test for the significance of
curvature. The corresponding mean square is:
Knowing the mean squares, the statistic to check the significance of
curvature can be calculated.
The 
value corresponding to the statistic,
, based on the 
distribution with one degree of freedom
in the numerator and four degrees of freedom in the denominator is:
Assuming that the desired significance is 0.1, since 
value > 0.1, it can be concluded
that curvature does not exist for this design. This result is shown in
the ANOVA table in Figure 7.20. The surface of
the fitted model based on these results, along with the observed response
values, is shown in Figure 7.21.
|
Figure 7.21: Model surface and observed response values for the design in Example 7.4. |
design augmented by five center point runs.