Split-plot

The sequence of meal consumption was determined by random assignment of diets to subjects. Statistical analysis was performed by a General Linear Models Procedure (20) using split-plot in time analysis with the following non-orthogonal contrasts ... 

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. 

In the following subsections several alternative experimental arrangements of split-plot experiments will be considered. The tablet formulation data given in the example of Table 2.1 in Section 2.1.1 will be... 

For the tablet formulation example of Table 2.1, this split-plot arrangement would require mxnxl = 9x8 = 72 tablet formulation batches to be made but only mxl = 9 operations of the climate chamber. The experiment would be conducted by placing in the climate chamber a complete set of 8 different tablet formulations at the same time. A completely randomized experiment (the cross-product experiment of... 

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]. 

The use of factorial and fractional factorial designs in split-plot arrangements has been investigated by Addelman [46], see also Daniel [47]. As an example of such an arrangement, consider a tablet formulation experiment with two environmental variables, temperature (T) and humidity (H), and five design variables. A, B, C, D, and E with all of the... 

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. 

S. Addelman, Some two-level factorial plans with split-plot confounding. Technometrics, 6 (1964) 253-258. 

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