Split-plot vs. hierarchical design: how to choose the correct model for your experiment

 

In biological, agricultural and clinical research, we often encounter experiments where the same experimental unit is measured multiple times, or different parts of the same unit receive distinct treatments. These structures violate the assumption of independence of observations, requiring specific statistical approaches.

When the problem arises

Consider a crossover clinical trial to evaluate three treatments for cardiac arrhythmia. Each participant receives all treatments in random sequence, with washout periods between them to avoid carry-over effects. Each participant serves as a block.

 

More complex situations involve two levels of treatment: groups of units receive main treatments, while each individual unit receives multiple secondary treatments over time.

Practical example: tomato plant study

Imagine 30 tomato plants (plots) randomized to 5 fertilizer formulas (main treatments). Each plant receives two irrigation regimes (secondary treatments) in distinct periods:       

The split-plot model: when plots are heterogeneous

The split-plot design is appropriate when there is natural variability between experimental units (plots). This model explicitly considers two error levels:

·       Error (a): Variability between plots within the same main treatment

·       Error (b): Variability within plots (between subplots)

Statistical model:

ANOVA Table - Split-Plot Design                                                      

Note on F tests: In split-plot, the test for main treatments uses Residual (a) as the denominator, while the tests for secondary treatments and interaction use Residual (b). This distinction is essential for valid conclusions..

The hierarchical model: when homogeneity is assumed

In situations where plots can be considered perfectly homogeneous, the hierarchical (nested) model is more appropriate.

Practical example: coffee quality study

Evaluation of coffee quality from four different origins. From each origin, we sample four bags, and from each bag we perform three laboratory analyses:

Critical assumption: coffee within bags from the same origin is homogeneous.

Statistical model:

where tij represents the effect of the j-th secondary treatment nested within the i-th main treatment.

                                  ANOVA TABLE - nested design

Comparative Table: Split-Plot vs. Hierarchical

Practical Conclusions

1.   Choose split-plot when your plots are naturally variable biological or experimental units (animals, people, individual plants, production batches).

2.   Prefer the hierarchical model only when there is strong evidence or valid assumptions about plot homogeneity (e.g., aliquots of the same solution, subsamples of homogeneous material).

3.   Warning! Incorrect application of the hierarchical model to data with between-plot variability results in variance underestimation and falsely significant tests.

Final Considerations

The choice between these models is not merely technical but conceptual. It reflects our understanding of the nature of the experimental material and the variation structure present in the data. When in doubt, the split-plot model is generally more conservative and appropriate, as it does not assume homogeneity where it may not exist.

Historical note: This discussion dates back to classical works in experimental statistics but remains surprisingly relevant in the era of mixed models and multilevel analyses.