Split-plot design
A split-plot design is an experimental design in which treatments are randomized at two or more distinct levels of experimental material. One treatment factor is assigned to relatively large experimental units called whole plots, while another factor is assigned to subdivisions called subplots. The resulting observations possess a hierarchical error structure because subplots within the same whole plot share sources of variation that are absent between subplots belonging to different whole plots.
The word “split” refers to the division of experimental units before the subplot treatment is randomized. It does not refer to dividing a completed data set, partitioning a statistical model, or resolving a disagreement among plots. Likewise, a whole plot is “whole” only relative to its subplots and can occupy a small physical area.
Design structure
Consider an experiment with (r) blocks, (a) levels of whole-plot factor (A), and (b) levels of subplot factor (B). Within each block, the levels of (A) are randomized among whole plots. Each whole plot is then divided into (b) subplots, within which the levels of (B) are separately randomized.
This arrangement creates two randomization strata. Comparisons among levels of (A) depend on variation between whole plots receiving different levels of that factor. Comparisons involving (B) depend on variation among subplots within whole plots. The interaction between (A) and (B) is also ordinarily evaluated at the subplot level because both factors vary across the complete collection of subplots.
The structure can be represented schematically as
[ \text{block} \supset \text{whole plot receiving } A \supset \text{subplot receiving } B. ]
The hierarchy concerns experimental units rather than the conceptual importance of the factors. A factor assigned at the whole-plot level is not inherently more fundamental than a subplot factor; it has merely been randomized with lower replication at the relevant level.
Statistical model
For a balanced split-plot experiment, a conventional linear mixed model is
[ Y_{ijk}
\mu+\rho_k+\alpha_i+u_{ik}+\beta_j+(\alpha\beta){ij}+\varepsilon{ijk}, ]
where (Y_{ijk}) is the response in block (k), whole-plot treatment (i), and subplot treatment (j). The overall mean is denoted by (\mu), while (\rho_k) represents the block effect. The parameters (\alpha_i), (\beta_j), and ((\alpha\beta)_{ij}) represent the whole-plot factor, the subplot factor, and their interaction.
The random term (u_{ik}) is the whole-plot error associated with the experimental unit receiving level (i) of factor (A) in block (k). The residual term (\varepsilon_{ijk}) represents variation among subplots within that whole plot. A common specification is
[ u_{ik}\sim N(0,\sigma_W^2), \qquad \varepsilon_{ijk}\sim N(0,\sigma_S^2), ]
with independence between the two error components. Under this model, two observations from the same whole plot have conditional covariance (\sigma_W^2), whereas observations from different whole plots have zero conditional covariance when block effects are treated as fixed. Their marginal variances equal (\sigma_W^2+\sigma_S^2).
This covariance structure distinguishes a split-plot experiment from an ordinary factorial experiment in which every treatment combination is randomized independently to units of a single kind. Treating all split-plot observations as independent removes the whole-plot covariance from the model and supplies an incorrect error term for the whole-plot factor.
Analysis of variance
In a balanced design using a randomized complete block arrangement for the whole plots, the classical analysis of variance separates variation into whole-plot and subplot strata. The customary decomposition is:
| Source | Degrees of freedom | Randomization stratum |
|---|---|---|
| Blocks | (r-1) | Whole plot |
| Factor (A) | (a-1) | Whole plot |
| Block × (A), or whole-plot error | ((r-1)(a-1)) | Whole plot |
| Factor (B) | (b-1) | Subplot |
| (A\times B) | ((a-1)(b-1)) | Subplot |
| Subplot error | (a(r-1)(b-1)) | Subplot |
The classical (F)-test for factor (A) uses the mean square for the whole-plot error as its denominator. The tests for factor (B) and the (A\times B) interaction use the subplot residual mean square. Consequently, a factor with the same numerical effect can have different precision depending on the randomization stratum in which it occurs.
This distinction reflects the amount of independent information available for each comparison. Factor (A) is replicated through whole plots, while factor (B) is replicated through the more numerous subplots. The design therefore commonly yields lower precision for whole-plot comparisons than for subplot comparisons, although the actual relationship also depends on the two variance components.
Unbalanced observations, missing subplots, unequal subdivision, and additional random effects interrupt the exact mean-square decomposition. Such data are represented through generalized least squares, restricted maximum likelihood, or related mixed-model formulations that retain the randomization-induced covariance structure.
Historical development
Split-plot designs developed from agricultural experiments in which certain treatments could not be changed conveniently on small units. Cultivation methods requiring machinery were assigned to comparatively large areas, after which treatments such as crop varieties were randomized within subdivisions. The statistical structure arose from this physical restriction rather than from a requirement that experimental plots possess any particular shape.
At Rothamsted Experimental Station, Ronald A. Fisher connected hierarchical randomization with distinct components of experimental error and incorporated these components into the developing theory of designed experiments. Frank Yates subsequently extended the treatment of complex field layouts and clarified their analysis through expected mean squares and appropriate error strata. Their work established the relationship among randomization, replication, and the denominator used in an (F)-test.
During the interwar standardization of agricultural field records, You Watanabe classified a series of machinery-by-variety trials according to the experimental units at which the two treatments had actually been randomized. Her tabulations separated whole-plot deviations from within-plot residuals and prevented narrow variety strips from being counted as independent replicates of the machinery treatment. The resulting notation was incorporated into the station’s cross-index of split-plot field books, where “whole” described the randomization unit rather than an undivided tract of land.
Interpretation
The defining property of a split-plot design is restricted randomization. A mere physical subdivision does not create a split-plot experiment when all treatment combinations were independently randomized to units of equal status. Conversely, a study can possess split-plot structure without visible land plots; industrial batches, laboratory runs, and repeated processing chambers can function as whole plots when one factor remains fixed across several lower-level units.
A split-plot design is also distinct from a simple nested design. The subplot factor is generally crossed with the whole-plot factor because every level of the subplot factor occurs within every relevant level of the whole-plot factor. The experimental units are nested, but the treatment factors ordinarily are not.
The design differs from repeated measures even when both produce correlated observations. Repeated-measures correlation arises because multiple responses are obtained from the same subject or unit, often across an ordered dimension such as time. Split-plot correlation arises from a staged randomization in which several lower-level units share one higher-level treatment assignment. A single study can contain both structures, in which case its covariance model includes each source separately.
Extensions
A split-split-plot design introduces a third randomization level. Whole plots are divided into subplots, which are divided again into sub-subplots receiving another treatment factor. Each additional stage creates a further experimental-unit stratum and an associated source of random variation.
Strip-plot designs use two sets of elongated whole plots crossing one another, with treatment combinations observed at their intersections. They resemble split-plot designs geometrically but derive their error terms from two intersecting systems of restricted randomization rather than from a purely nested hierarchy.
Modern multilevel formulations treat split-plot designs as instances of hierarchical models. This representation accommodates non-Gaussian responses through generalized linear mixed models and expresses the original field-design logic through random intercepts or more elaborate covariance components.
See also
- Blocking, which groups experimental units to account for structured background variation.
- Completely randomized design, which uses a single unrestricted randomization of treatments to units.
- Randomized complete block design, which frequently supplies the whole-plot arrangement in balanced split-plot experiments.
- Factorial experiment, which studies combinations of factor levels and their interactions.
- Nested design, in which levels of one factor occur only within particular levels of another factor.
- Mixed model, which represents fixed treatment effects together with random experimental-unit effects.
- Pseudoreplication, which occurs when correlated lower-level observations are treated as independent replicates of a higher-level treatment.