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Consider a batch process that uses 7 monitor wafers in each run. The plan further calls for measuring a [[response variable]] on each wafer at each of 9 sites. The organization of the [[sampling plan]] has a hierarchical or nested structure: the batch run is the topmost level, the second level is an individual wafer, and the third level is the site on the wafer.
The total amount of data generated per batch run will be 7
Analyzing the data as suggested above is not absolutely incorrect, but doing so loses information that one might otherwise obtain. For example, site 1 on wafer 1 is physically different from site 1 on wafer 2 or on any other wafer. The same is true for any of the sites on any of the wafers. Similarly, wafer 1 in run 1 is physically different from wafer 1 in run 2, and so on. To describe this situation one says that sites are nested within wafers while wafers are nested within runs.
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In many industrial experiments, three situations often occur:
# some of the factors of interest may be 'hard to vary' while the remaining factors are easy to vary. As a result, the order in which the treatment combinations for the experiment are run is determined by the ordering of these 'hard-to-vary' factors
# experimental units are processed together as a batch for one or more of the factors in a particular treatment combination
# experimental units are processed individually, one right after the other, for the same treatment combination without resetting the factor settings for that treatment combination.
===Split-plot experimental examples===
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Consider running the experiment under the first condition listed above, with the factor solution concentration of the plating agent (S) being hard to vary. Since this factor is hard to vary, the experimenter would like to randomize the treatment combinations so that the solution concentration factor has a minimal number of changes. In other words, the randomization of the treatment runs is restricted somewhat by the level of the solution concentration factor.
As a result, the treatment combinations might be randomized such that those treatment runs corresponding to one level of the concentration (
Once one complete replicate of the experiment has been completed, a second replicate is performed with a set of four copper strips processed for a given level of solution concentration before changing the concentration and processing the remaining four strips. Note that the levels for the remaining two factors can still be randomized. In addition, the level of concentration that is run first in the replication runs can also be randomized.
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====Example: batch process====
Consider running the experiment under the second condition listed above (i.e., a batch process) for which four copper strips are placed in the solution at one time. A specified level of current can be applied to an individual strip within the solution. The same 16 treatment combinations (a replicated 2<sup>3</sup> factorial) are run as were run under the first scenario. However, the way in which the experiment is performed would be different. There are four treatment combinations of solution temperature and solution concentration: (
Running the experiment in this way also results in a split-plot design in which the whole-plot factors are now solution concentration and solution temperature, and the subplot factor is current.
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The smaller size experimental unit is again referred to as the subplot experimental unit. There are 16 subplot experimental units for this experiment. Current is the subplot factor in this experiment.
The larger-size experimental unit is the whole-plot experimental unit. There are four whole plot experimental units in this experiment and solution concentration and solution temperature are the whole plot factors in this experiment.
There are two sizes of experimental units and there are two error terms in the model: one that corresponds to the whole-plot error or whole-plot experimental unit, and one that corresponds to the subplot error or subplot experimental unit.
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Consider the following example from the semiconductor industry. An experiment requires an implant step and an anneal step. At both the anneal and the implant steps there are three factors to test. The implant process accommodates 12 wafers in a batch, and implanting a single wafer under a specified set of conditions is not practical nor does doing so represent economical use of the implanter. The anneal furnace can handle up to 100 wafers.
The settings for a two-level factorial design for the three factors in the implant step are denoted (A, B, C), and a two-level factorial design for the three factors in the anneal step are denoted (D, E, F). Also present are [[interaction effects]] between the implant factors and the anneal factors. Therefore, this experiment contains three sizes of experimental units, each of which has a unique error term for estimating the significance of effects.
To put actual physical meaning to each of the experimental units in the above example, consider each combination of implant and anneal steps as an individual wafer. A batch of eight wafers goes through the implant step first. Treatment combination 3 in factors A, B, and C is the first implant treatment run. This implant treatment is applied to all eight wafers at once. Once the first implant treatment is finished, another set of eight wafers is implanted with treatment combination 5 of factors A, B, and C. This continues until the last batch of eight wafers is implanted with treatment combination 6 of factors A, B, and C. Once all of the eight treatment combinations of the implant factors have been run, the anneal step starts. The first anneal treatment combination to be run is treatment combination 5 of factors D, E, and F. This anneal treatment combination is applied to a set of eight wafers, with each of these eight wafers coming from one of the eight implant treatment combinations. After this first batch of wafers has been annealed, the second anneal treatment is applied to a second batch of eight wafers, with these eight wafers coming from one each of the eight implant treatment combinations. This is continued until the last batch of eight wafers has been implanted with a particular combination of factors D, E, and F.
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==Further reading==
For a more detailed discussion of these designs and the appropriate analysis procedures, see:
* {{cite book
|author = Milliken, G. A.
|coauthors = Johnson, D. E.
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|publisher = Van Nostrand Reinhold
|___location = New York}}
* {{cite journal
|author = Miller, A.
|year = 1997
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==External links==
* [http://www.itl.nist.gov/div898/handbook/pri/section5/pri55.htm How can I account for nested variation (restricted randomization)?]
{{Statistics}}
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