Split-split plot designs are a powerful tool for studying three factors simultaneously in experimental research. They allow researchers to investigate main effects and interactions between factors at different levels of experimental units.

This design is particularly useful when some factors are harder to change than others. It provides a structured approach to analyzing complex experiments, helping researchers uncover intricate relationships between variables.

Experimental Design

Three-Factor Experiments

• Three-factor experiments involve studying the effects of three independent variables (factors) on a response variable
• Each factor has two or more levels, and the experiment is designed to investigate the main effects of each factor and their interactions
• The three factors are typically referred to as the whole plot factor, split-plot factor, and split-split plot factor
• The experimental units are divided into whole plots, split plots, and split-split plots to accommodate the three factors

Factors and Plots

• The whole plot factor is the first factor applied to the experimental units
  • Whole plots are the largest experimental units to which the levels of the whole plot factor are randomly assigned
• The split-plot factor is the second factor applied to the experimental units
  • Each whole plot is divided into smaller units called split plots, and the levels of the split-plot factor are randomly assigned to these split plots
• The split-split plot factor is the third factor applied to the experimental units
  • Each split plot is further divided into even smaller units called split-split plots, and the levels of the split-split plot factor are randomly assigned to these split-split plots

Data Structure

Nested Structure

• Split-split plot designs have a nested structure, where the split-split plots are nested within the split plots, which are nested within the whole plots
• This nested structure reflects the hierarchical nature of the experimental design
• The nesting of the experimental units leads to different sources of variation and error terms in the analysis

Interactions and Analysis

• In split-split plot designs, the primary interest is often in the triple interaction among the three factors
  • The triple interaction represents the combined effect of all three factors on the response variable
• The analysis of variance (ANOVA) for split-split plot designs includes main effects for each factor, two-way interactions between pairs of factors, and the triple interaction
• The ANOVA table for a split-split plot design is more complex than for simpler designs due to the nested structure and multiple error terms

Results

Error Terms

• Split-split plot designs have three different error terms, corresponding to the three levels of the design
  • The whole plot error is used to test the significance of the whole plot factor and its interactions with other factors
  • The split plot error is used to test the significance of the split-plot factor and its interactions with other factors
  • The split-split plot error is used to test the significance of the split-split plot factor and its interactions with other factors
• The correct error term must be used for each hypothesis test to ensure valid inferences

Interpretation of Results

• Interpreting the results of a split-split plot analysis involves examining the significance of the main effects, two-way interactions, and the triple interaction
• Significant main effects indicate that the levels of a factor have different effects on the response variable, averaged over the levels of the other factors
  • For example, if the main effect of the whole plot factor is significant, it means that the response variable differs across the levels of the whole plot factor, regardless of the levels of the split-plot and split-split plot factors
• Significant two-way interactions indicate that the effect of one factor depends on the level of another factor
  • For instance, a significant interaction between the whole plot and split-plot factors suggests that the effect of the split-plot factor varies across the levels of the whole plot factor
• A significant triple interaction indicates that the combined effect of all three factors on the response variable is not additive
  • In other words, the effect of one factor depends on the levels of the other two factors simultaneously
  • Interpreting a significant triple interaction can be complex and may require further investigation, such as examining interaction plots or conducting post-hoc tests