|
Replicated Runs and Repeated
Runs
In the context of Design of
Experiments (DOE), replicated runs and repeated runs are both
multiple response readings taken at the same factor levels. This
article explains the difference between the two and illustrates
how ReliaSoft's DOE++
software can be used to deal with the two cases.
In experiments with many
factors, each combination of levels of the different factors is
called a treatment. For example, imagine that 3 factors,
A, B and C, are investigated in an experiment. Factor A has 3
levels, factor B has 2 levels and factor C has 3 levels. The
total combinations, or treatments, for the experiment would be
3*2*3 = 18. To run the experiment completely, 18 observations
would have to be taken, 1 observation corresponding to each of
the 18 treatments.
While conducting an experiment,
observations are recorded in a random order so that the effect
of nuisance factors may get cancelled out. (Nuisance factors are
those factors that are not of interest to the experimenter but
may affect the response.) Thus, to run the given 3-factor
experiment once, 18 observations would be recorded in a random
order. At times an experiment may be run more than once to
obtain an estimate of pure error. (Pure error is the error
between observations taken at the same treatment.) If the
experiment is run completely twice, then 36 observations would
be recorded, with each treatment having 2 observations. If all
36 observations are recorded in a random order, then each
treatment of the experiment is said to have 2 replicates. If the
observations are recorded in such a way that the 2 readings
corresponding to each of the 18 treatments are taken at the same
time, then each treatment of the experiment is said to have 2
repeated runs.
In the case of repeated runs,
only 1 reading is recorded at each treatment of the experiment;
this is obtained by taking an average of all the repeated
observations at that treatment. The reason for doing this is
that since repeated runs are recorded at the same time (without
considering randomization of all experimental observations) some
of the variation that otherwise would have been captured (had
all of the observations been randomized) is considered lost.
Therefore, as far as the analysis of experiment data is
concerned, observations corresponding to repeated runs are
treated as if just 1 observation had been recorded for each
treatment of the experiment.
The difference between
replicated runs and repeated runs is further explained through
the following example.
Example
Assume that a baker wants to investigate the effects of 2
factors, A and B, on the taste of cakes made in his bakery. Each
of the 2 factors has 2 levels, with the first level of the
factors represented by "Low" and the second level by "High."
Therefore, the total number of treatments for this experiment is
4 (2*2 = 4), as shown in Table 1.
Table 1: Treatments for an Experiment with Two
Factors at Two Levels
|
Factors |
|
A |
B |
| Low |
Low |
|
High |
Low |
| Low |
High |
|
High |
High |
To run the experiment once
completely, the baker will have to bake 4 cakes, 1 corresponding
to each of the 4 treatments. Figure 1 shows the cakes
corresponding to each of the 4 treatments.
Figure 1: Cakes Corresponding
to Each Treatment of Table 1
The baker decides to bake 8
cakes, 2 for each treatment. He tastes the cakes and gives them
a rating on a scale from 0 to 10 with 0 indicating the worst
tasting cake and 10 indicating the best tasting cake.
If the baker bakes each of the
8 cakes in a random order, as shown in Figure 2, then this
represents the case of 2 replicates. The readings obtained by
the baker are entered in DOE++ as shown in Figure 3.
Figure 2: Replicated Runs All 8
Cakes Are Baked in a Random Order
Figure 3: Data Entry in DOE++
for the Case of 2 Replicates
However, if the baker bakes the
2 cakes corresponding to each of the treatments together, as
shown in Figure 4, then this represents a case of 2 repeats.
This case is effectively the same as if the baker had baked 1
large cake for each treatment instead of 2 cakes and tasted the
large cake twice to obtain the two readings.
Figure 4: Repeated Runs Cakes
Corresponding to the Same Treatment Are Baked at the Same Time
In the case of repeats, the
average of the observations at a treatment is used in the
experiment, as shown in Figure 5. In DOE++, the average
values are automatically calculated by using the option in the
Design Wizard to specify that you are using repeated
measurements, as shown in Figure 6.
Figure 5: Data Entry in DOE++
for the Case of Repeats
Figure 6: Specification of Repeated Runs in the Design Wizard
Conclusion
This article explained the difference between replicated runs
and repeated runs. The choice of replicated or repeated runs can
have a significant impact on the analysis of experiment data.
For example, the results for the previous cases of replicated
runs and repeated runs are shown in Figures 7 and 8
respectively.
Figure 7: Results for the Case
of Replicated Runs
Figure 8: Results for the Case of Repeated Runs
Although the set of observations
is the same in both the cases, the results of the analysis lead
to entirely different conclusions. In the case of replicates,
Figure 7 shows that the interaction between the two factors has
a significant impact on the taste of the baked cakes. However,
for the case of repeats shown in Figure 8, both of the factors
and the interaction are found to be insignificant. When you are
using multiple observations for all treatments of an experiment,
all the runs must be randomized in order for the runs to count
as replicated runs. |
