8.1 Principles of split-plot designs

Split-plot designs are a powerful tool in experimental research, combining two factors with different sizes of experimental units. They're perfect when one factor is harder to change or apply than the other, allowing for efficient resource use and precise measurements.

These designs have a unique structure with whole plots and subplots, each with their own factors and randomization. This setup leads to two error terms and requires special analysis methods, but it offers valuable insights into main effects and interactions between factors.

Experimental Units and Factors

Experimental Units and Levels

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• Whole plot experimental units are the larger units to which the levels of the whole plot factor are randomly assigned
• Subplot experimental units are the smaller units within each whole plot to which the levels of the subplot factor are randomly assigned
• Levels of the whole plot factor are randomly assigned to whole plots while levels of the subplot factor are randomly assigned to subplots within each whole plot
• Having two sizes of experimental units (whole plots and subplots) is a key characteristic of split-plot designs

Factors and Levels

• Whole plot factor is the factor whose levels are randomly assigned to the whole plots
  • Levels of the whole plot factor are applied to the larger experimental units (whole plots)
  • Example: In an agricultural experiment, the whole plot factor could be irrigation method (drip or sprinkler)
• Subplot factor is the factor whose levels are randomly assigned to the subplots within each whole plot
  • Levels of the subplot factor are applied to the smaller experimental units (subplots) within each whole plot
  • Example: In the same agricultural experiment, the subplot factor could be fertilizer type (organic or synthetic)

Design Structure

Restricted Randomization

• Split-plot designs involve restricted randomization where the randomization of the subplot factor is restricted within each whole plot
• Randomization occurs at two levels:
  1. Whole plot factor levels are randomly assigned to whole plots
  2. Subplot factor levels are randomly assigned to subplots within each whole plot
• This restricted randomization structure is a defining feature of split-plot designs and distinguishes them from other designs like randomized complete block designs

Hierarchical Structure and Error Terms

• Split-plot designs have a hierarchical structure with subplots nested within whole plots
• This hierarchical structure leads to two different error terms:
  1. Whole plot error: Variation among whole plots that have been assigned the same level of the whole plot factor
  2. Subplot error: Variation among subplots within a whole plot that have been assigned the same level of the subplot factor (also called split-plot error)
• Having two error terms is another key characteristic of split-plot designs and affects the analysis and interpretation of results

Effects

Main Effects

• In split-plot designs, there are two types of main effects:
  1. Main effect of the whole plot factor: Compares the mean responses between levels of the whole plot factor, averaged across all levels of the subplot factor
  2. Main effect of the subplot factor: Compares the mean responses between levels of the subplot factor, averaged across all levels of the whole plot factor
• Interpreting main effects in split-plot designs requires considering the hierarchical structure and error terms
  • Example: The main effect of irrigation method would compare the mean yield between drip and sprinkler irrigation, averaged across both fertilizer types

Interaction Effects

• Interaction effect assesses whether the effect of one factor depends on the level of the other factor
• In split-plot designs, the interaction of interest is usually between the whole plot factor and the subplot factor
  • Determines if the effect of the subplot factor is consistent across all levels of the whole plot factor, or if it varies depending on the whole plot factor level
• Interpreting interaction effects requires examining the pattern of mean responses across all combinations of factor levels
  • Example: A significant interaction between irrigation method and fertilizer type would suggest that the effect of fertilizer type on yield differs depending on whether drip or sprinkler irrigation is used