Split plot design

In Sections 2.2 and 2.3 we considered the application of response surface methodology to the investigation of the robustness of a product or process to environmental variation. The response surface designs discussed in those sections are appropriate if all of the experimental runs can be conducted independently so that the experiment is completely randomized. This section will consider the application of an alternative class of experimental designs, called split-plot designs, to the study of robustness to environmental variation. A characteristic of these designs is that, unlike the response surface designs, there is restricted randomization of the experiment. 

Split-plot designs occur in a wide range of applications of experimental design. One application area for split-plot designs is when there are some variables that can be applied only to experimental units that are larger than the units to which the other variables can be applied. 

An excellent exposition of split-plot experimental designs can be found in D.R. Cox s book, Planning of Experiments [42]. He states that split-plot designs are particularly useful when one (or more) factors are what he calls classification factors. These factors are included in the experiment to determine whether they modify the effect of the other factors or indicate how the other factors work. The classification factors are included to examine their possible interaction with the other factors. Lower precision is tolerated for comparisons of the classification factors, in order that the precision of the other factors and the interactions can be increased. In the standard terminology associated with split-plot experiments, the classification factors are called whole-plot factors and are applied to the larger experimental units. The smaller experimental units are called subplots. 

Strip-block experiments, such as the one described in this section, are clearly considerably easier to run than either the completely randomized product design or either of the split-plot designs described above, that is, arrangements (I) and (II). 

Adaptations of the split-plot methodology have been suggested by many authors (see, for example, Kempthome [18], Cochran and Cox [44]). These authors describe various blocking arrangements to control for other sources of variation in split-plot experiments. The relevance of some of these arrangements to split-plot designs that investigate the influence of environmental variation is discussed in Box and Jones [5]. 

Analysis of split-plot designs for robust experimentation... 

The appropriate analysis of data obtained from an experiment should be determined by the experimental design used to obtain those data. The fundamental characteristic of split-plot designs is that there are experimental units of different sizes and consequently multiple sources of variation. The analysis needs to take account of this structure and include multiple error terms and to test the significance of effects and interactions against the appropriate error term. This has been illustrated above with the three experimental arrangements for split-plot and strip-block designs. 

EXAMPLE OF A SPLIT-PLOT DESIGN USING A FRACTIONAL... 

Therefore, split-plot designs and strip-block designs are of tremendous value in robust design experiments since they permit the precise estimation of the interactions of interest and can be considerably easier to run than the cross-product design that have traditionally been advocated. 

In this chapter the use of statistical experimental designs in designing products and processes to be robust to environmental conditions has been considered. The focus has been on two classes of experimental design, response surface designs and split-plot designs. 

The choice of an appropriate experimental design depends on the experimental circumstances. Box and Draper [12] (p. 502, 305) list a series of experimental circumstances that should be considered by the investigator when selecting a response surface design. Many of these considerations also apply to split-plot designs, and to experimental design in general. 

As has been noted in this chapter, any restriction on the randomization of the experiment will lead the investigator to conduct one of the split-plot designs that were described in Section 2.4. In that section it was shown that the split-plot type designs can be a more efficient way to run robust design experiments than the cross-product arrays of Taguchi. Furthermore, the standard methods of analysis of split-plot experiments, that seek to... 

G.E.P. Box and S.P. Jones, Split-plot designs for robust product experimentation. 

This, replicating measurements only improves the sampling and measurement precision within the experiment. The individual measurements are not complete replicates and should not be treated as separate experiments. In general, the mean value of the measurements is treated as a single datum in the statistical analysis. See the following section and also the section in chapter 7 on the dependence of the residuals on combinations of 2 or more factors (split-plot designs). 

They were not completely replicated. This is often observed if the experimentation takes place in several stages and it is only the last stage that is repeated split-plot design) or even that it is only the... 

This is an example of the split-plot design (15), discussed briefly in chapter 7, with reference to correlated errors. 

D-optimal designs were determined between 12 and 28 experiments. There was a maximum in M I for 22 experiments, so it was this design that was carried out. For 5 of the 6 combinations of granulating conditions (batches), there were 4 combinations (sub-batches) of the other conditions. The remaining batch was split into only 2 sub-batches. This is therefore another example of the split-plot design. 

The term completely randomized design (CRD) means that we determine the total number of experimental units needed in the experimentation, and then select experimental units randomly to be executed first or last. Consider, for instance, that in lithographic nanofabrication experimentation, an engineer would like to study the output from using two levels of a chemical applied to three nanoparticle types and deposited on four sizes of mould. Therefore, a total of 24 runs must be executed. This, in turn, implies that the experimenter would have to make 24 slurry preparations and apply each to 24 moulds. If experimenters make only six slurry preparations and then divide the slurry to four portions and then deposit on the different moulds, this procedure is not a CRD. (To overcome this situation in practice, we suggest the use of a split plot design and its variants.)... 

Most papers did not directly discuss the randomization principle the experiment may have been completely randomized or completely randomized in blocks, but the choice was not clearly stated. There may have been restriction on randomization, but it was unclear whether the experimenters knew this concept. Failure to obey the randomization principle might lead to misinterpretation of the results. When randomization is not practical, a split plot design, which will be discussed in Sections 8.4.1 and 8.4.2, can often be used. 

The designs that we have previously discussed are based on the complete randomization principle. However, in many situations, it is impossible to randomize all treatment combinations. In such cases, the split plot design may be used. The name split plot comes from the agricultural experiment in which the whole plots are considered for a large plot of land and the sub-plots are used to represent a small plot of land within the large area. 

The standard split plot design is a design which has a two-factor factorial arrangement. For example, Factor A with a level, is designed as a randomized complete design the levels of Factor A treatments are called a whole plot experimental unit. Each experimental unit is divided into b split plot experimental units of Factor B. 

The strip block design is another type of design which is a bit different from the split plot design. This design has two factors, Factor A with a level and Factor 5 with b level. The levels of Factor A are randomly assigned to the whole plot experimental nnit. Then the B experimental units are formed perpendicular to the A experimental units, and the b levels of Factor B are randomly allocated to the second set of b whole plot units in each of the complete blocks. 

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