Goos (2002) presents an example involving a pastry dough mixing experiment. The purpose of the experiment is to understand how certain properties of the dough depend on three factors: feed flow rate, initial moisture content, and rotational screw speed. Since it was possible to conduct only four runs a day, the experiment required several days to run. It is likely that random day-to-day differences in environmental variables have some effect on all of the runs that are performed on a given day. To account for the day-to-day variation, the runs were grouped into blocks of size four so that this variation would not compromise the information about the three factors.
The blocking factor, Day, consists of each day's runs. The days on which the trials were conducted are representative of a large population of days with different environmental conditions. It follows that Day is a random blocking factor.
To create a random block design, use the Custom Design platform to enter responses and factors and define your model as usual. In the Design Generation outline, select the Group runs into random blocks of size option and enter the number of runs you want in each block. See Design Structure Options.
Note: To define a fixed blocking factor, enter a blocking factor in the Factors outline. To define a random blocking factor, do not enter a blocking factor in the Factors outline. Instead, select the Group runs into random blocks of size option under Design Generation.
Box et al. (2005) presents an experiment to study the corrosion resistance of steel bars. The bars are placed in a furnace for curing. Afterward, a coating is applied to increase resistance to corrosion. The two factors of interest are:
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Furnace Temp in degrees centigrade, with levels 360, 370, and 380
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Coating, with levels C1, C2, C3, and C4 depicting four different types of coating
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Furnace Temp is a hard-to-change factor, due to the time it takes to reset the temperature in the furnace. For this reason, four bars are processed for each setting of furnace temperature. At a later stage, the four coatings are randomly assigned to the four bars.
The experimental units are the bars. Furnace Temp is a hard-to-change factor whose levels define whole plots. Within each whole plot, the Coating factor is randomly assigned to the experimental units to which the whole plot factor was applied.
The Factors outline for the corrosion experiment has Changes set to Hard for Furnace Temp and Easy for Coating. The 15-run design consists of five whole plots, within which the settings of Temperature are held constant.
When a custom design involves only easy-to-change and hard-to-change factors, the runs of the hard-to-change factors are grouped using a new factor called Whole Plots. The values of Whole Plots designate blocks of runs with identical settings for the hard-to-change factors. The Model script that is saved to the design table treats Whole Plots as a random effect. For details, see Changes and Design Structure Options.
For an example of creating a split-plot design and analyzing the experimental data, see Split-Plot Experiment in Examples of Custom Designs.
A split-split-plot design is used when there are two levels of factors that are hard-to-change. In industry, such designs often occur when batches of material or experimental units from one processing stage pass to a second processing stage. Factors are applied to batches of material at the first stage. Then those batches are divided for second-stage processing, where additional factors are studied. The first stage factors are considered very-hard-to-change, and the second-stage factors are considered hard-to-change. Additional factors can be applied to experimental units after the second processing stage. These factors are considered easy-to-change.
Schoen (1999) presents an example of a split-split-plot design that relates to cheese quality. The factors are given in the Cheese Factors.jmp data table found in the Design Experiment folder. The experiment consists of three stages of processing:
The default number of whole plots is 5 and the default number of subplots is 10.The number of runs is set to 22. If only 10 subplots are used the design does not have enough whole plots to estimate the subplot variance. Change the number of subplots to 11 and click Make Design to see a 22-run design.
Figure 3.31 Split-Split-Plot Design for Cheese Scenario
The five whole plots correspond to the storage factors; storage 1 and storage 2. The settings of the storage factors are constant within a whole plot. If consecutive whole plots have the same setting for a whole plot factor, the factor should be reset between the plots. For example, you should reset the level for storage 1 between runs 10 and 11 and between runs 14 and 15, and you should reset the level for storage 2 between runs 18 and 19. Resetting the factor between whole plots, even when the specified settings are the same, is required in order to capture whole plot variation.
The 11 subplots correspond to the curds factors. Within a subplot, the settings of the curds factors are constant. Each level of Subplots appears only within one level of Whole Plots, indicating that the levels of Subplots are nested within the levels of Whole Plots.
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A factor called Whole Plots represents the blocks of constant levels of the factors with Changes set to Very Hard.
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A factor called Subplots represents the blocks of constant levels of the factors with Changes set to Hard.
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The factor Subplots reflects the nesting of the levels of the factors with Changes set to Hard within the levels of the factors with Changes set to Very Hard.
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The factors Whole Plots and Subplots are treated as random effects in the Model script that is saved to the design table.
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In contrast to a split-split-plot design, the second-stage factors are not nested within the first-stage factors. After the first stage, the batches are subdivided and formed into new batches. Therefore, both the first- and second-stage factors are applied to whole batches.
Although factors at both stages might be equally hard-to-change, to distinguish these factors, JMP denotes the first stage factors as very-hard-to-change, and the second-stage factors as hard-to-change. Additional factors applied to experimental units after the second processing stage are considered easy-to-change.
Vivacqua and Bisgaard (2004) describe an experiment to improve the open circuit voltage in battery cells. Two stages of processing are of interest:
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A full factorial design with all factors at two levels would require 26 = 64 runs, and would require a prohibitive 64*5 = 320 days. Also, it is not practical to vary assembly conditions for individual batteries. However, assembly conditions can be changed for large batches, such as batches of 2000 batteries.
To distinguish between the first- and second-stage factors, you designate the Changes for the first-stage factors as Very Hard, and the Changes for the second-stage factors as Hard. See Figure 3.32. Also, under Design Generation, note the following option: Hard to change factors can vary independently of Very Hard to change factors. If this is not checked, the design is treated as a split-split-plot design, with nesting of factors at the two levels. Check this option to create a two-way split-plot design.
The default number of whole plots is 7; the default number of subplots is 14. Click Make Design to see the 28-run design.
Figure 3.33 Two-Way Split-Plot Design for Battery Cells
The seven whole plots correspond to the first-stage factors, X1, X2, X3, and X4. The settings of these factors are constant within a whole plot. The 14 subplots correspond to the second-stage factors, X5 and X6. For example, the sub-batches for runs 1 and 15 (from different whole plots) are subject to the same subplot treatment, where X5 is set at 1 and X6 at -1.
A two-way split-plot design requires factors with Changes set to Very Hard and to Hard. As described in Setup for a Split-Split-Plot Design, factors called Whole Plots and Subplots are created. However, in a two-way split-plot design, Subplots does not nest the levels of factors with Changes set to Hard within the levels of factors with Changes set to Very Hard. Both Whole Plots and Subplots are treated as random effects in the Model script that is saved to the design table.
You need to ensure that the factor Subplots is not nested within the factor Whole Plots. Select the option Hard to change factors can vary independently of Very Hard to change factor in the Design Generation outline (Figure 3.32). For more details, see Changes and Design Structure Options.