Chapter 6 - Split-Plot and Strip-Plot Designs - Analysis of Variance and Analysis of Covariance

Oct 13, 2026

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Explains how split-plot and strip-plot designs extend factorial experimentation when one factor requires large, costly equipment while another can be applied to smaller plots. Describes the two-stratum randomization in split-plot designs, including whole-plot assignment and split-plot assignment, and the resulting impact on variance and interaction contrasts. Provides motivating industrial and cake-baking examples, then introduces the statistical model (two randomization terms) and the corresponding analysis of variance framework for a CRD arrangement of whole-plots.

LanguageEnglish

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Pages28

Reading Time10 min

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Chapter 6: Split-Plot and Strip-Plot Designs

Analysis of Variance and Analysis of Covariance

Motivation for Split-Plot Designs

Two Randomizations in Split-Plot Designs

Examples (Cake Baking and Packaging Material)

Statistical Analysis of Split-Plot Experiments

Statistical Model

Analysis of Variance

Chapter 6 Split-Plot and Strip-Plot Designs Analysis of Variance and Analysis of Covariance An example to motivate the use of split plot design is as follows. Suppose we want to study two factors, say methods of cultivation and varieties of wheat. Suppose, the first factor has t levels and second factor has s varieties. The first factor requires the use of large complex equipment and consequently, relatively large plots of land are needed. This will require higher cost and puts a restriction on the number of plots to be used. Because of the nature of the equipment used for planting the wheat, the second factor can be accommodated in much smaller plots. To achieve this, the large plots are split into smaller plots at the planting stage. This means that since the plots are close together, so less variability is expected among the plots and in turn, more plots and less variability among plots is expected which implies that the contrasts will have more information in terms of smaller standard errors. This suggests that the experiment can be conducted with two strata. The whole-plot stratum consists of large plots in which the plots can be assigned as per any standard design, e.g. CRD, RBD, or Latin square design. Next stratum is the split-plot stratum which consists of the split-plots. There are the smaller plots that are obtained by splitting each of the large plots into s parts. The treatments assigned to the large whole-plots are replicated r times, and treatments assigned to the split-plots are replicated rt times. Now much more information on the split-plot factor is available because of the extra replication, and in turn, a smaller splitplot-to-split-plot variance is expected. The interaction contrasts between whole-and split-plot treatment also fall into the split-plot stratum and benefit due to smaller variance. There are two distinct randomizations in the split plot designs: (i) The first randomization takes place in stratum 1, when the levels of the whole-plot treatment are randomly assigned to the whole–plots. (ii) The second randomization takes place in stratum 2 where the levels in the split-plot treatment are randomly assigned in the split-plot. Many split-plot plans can easily be modified to become strip-plot experiments. These have their own advantages and disadvantages. Examples Following examples have been opted from Giesbrecht and Gumpertz (2004) . “Consider a hypothetical cake baking study in an industry. Assume that there are r recipes and c baking conditions are to be studied. A simple split-plot experiment with the recipes as a whole-plot factor and the baking condition as a split-plot factor can be set up if cake batters are made up using recipes in random order. Each batch of batter is then split into c portions. The portion are then baked under the c conditions. Anew random baking order is selected for each batter. Replication is provided by repeating recipes. Another option is to make up enough batter to make one cake from each of the r recipes. All cakes based on r recipes are then baked at one time in an oven at one of the c conditions. Now we have an experiment with baking conditions as a a whole-plot factor and recipe as a split-plot factor. In case of a strip-plot design, the experimenter would make up batches of each of the batters large enough. Then partition each batch into c cakes and then bake the cakes in sets, with one cake of each recipe in each set. In terms of row-column structure of design, the rows represent recipes and the columns represent baking conditions. The advantage here is that in the absence of replication, only r batches of batter need be mixed and the oven need only be set up c times. In another example of a split-plot experiment in industrial quality research is as follows: The object of the project is to develop a packaging material that would give a better seal under the wide range of possible sealing process conditions used by potential custo

FAQs

Why split-plot designs are used instead of fully randomized designs?

They arise when one factor needs large, complex equipment that forces larger plots, while the other factor can be applied in smaller plots by splitting the large plots at planting. This structure reduces expected variability within neighboring plots and improves information through more informative contrasts.

What are the two randomization steps in a split-plot design?

First, whole-plot treatment levels are randomly assigned within stratum 1 (whole-plots). Second, split-plot treatment levels are randomly assigned within stratum 2 (split-plots produced by splitting each whole-plot).

How does the split-plot statistical model reflect the design structure?

The model includes two terms corresponding to the two randomization steps, one in each stratum. It uses whole-plot effects and whole-plot error in the first stratum, and split-plot effects, the interaction between whole- and split-plot treatments, and split-plot error in the second stratum, with random errors having different variances.

Oct 13, 2026

Upload by OmBimo