Split Plot Design

A Split Plot Design is a type of experimental design used when one or more factors are difficult or expensive to randomize at the level of individual experimental units. It is widely used in agricultural field trials, industrial experiments, and manufacturing studies.

The design partitions experimental units into two levels:

• Whole plots — the larger units to which the hard-to-change factor (the whole-plot factor) is applied
• Subplots — subdivisions of whole plots, to which the easy-to-change factor (the subplot factor) is applied

Key idea: Whole-plot factors are randomized among whole plots; subplot factors are randomized within each whole plot.

When to Use a Split Plot Design

Use a split plot design when:

1. One factor is difficult, costly, or impractical to change for every run (e.g., oven temperature, tillage method, irrigation system).
2. You want to study both a hard-to-change factor and an easy-to-change factor in the same experiment.
3. You can accept lower precision for the whole-plot factor in exchange for higher precision on the subplot factor.

Typical applications:

Field	Whole-Plot Factor	Sub-Plot Factor
Agriculture	Irrigation method	Fertilizer type
Food Science	Oven temperature	Packaging type
Manufacturing	Machine setting	Raw material batch
Textile	Dyeing temperature	Fabric type
Agronomy	Tillage method	Crop variety

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Structure of a Split Plot Design

Components

Experiment
├── Block 1
│   ├── Whole Plot A  →  Subplot: level 1, level 2, level 3
│   └── Whole Plot B  →  Subplot: level 1, level 2, level 3
├── Block 2
│   ├── Whole Plot A  →  Subplot: level 1, level 2, level 3
│   └── Whole Plot B  →  Subplot: level 1, level 2, level 3
└── Block 3
    ├── Whole Plot A  →  Subplot: level 1, level 2, level 3
    └── Whole Plot B  →  Subplot: level 1, level 2, level 3

Notation

Symbol	Meaning
$r$	Number of blocks (replications)
$a$	Number of whole-plot factor levels
$b$	Number of subplot factor levels
$N = r \times a \times b$	Total number of observations

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Statistical Model

The linear model for a split plot design (with blocks) is:

\[Y_{ijk} = \mu + \rho_i + \alpha_j + \delta_{ij} + \beta_k + (\alpha\beta)_{jk} + \varepsilon_{ijk}\]

Where:

Term	Description
$\mu$	Overall mean
$\rho_i$	Effect of the $i$-th block ($i = 1, \ldots, r$)
$\alpha_j$	Effect of the $j$-th whole-plot factor level ($j = 1, \ldots, a$)
$\delta_{ij}$	Whole-plot error (block × whole-plot interaction)
$\beta_k$	Effect of the $k$-th subplot factor level ($k = 1, \ldots, b$)
$(\alpha\beta)_{jk}$	Interaction between whole-plot and subplot factors
$\varepsilon_{ijk}$	Subplot error (residual)

Note: There are two error terms in a split plot design — one for testing whole-plot effects and one for testing subplot effects and interactions.

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ANOVA Table

Source of Variation	df	MS	F-ratio
Whole-Plot Stratum
Blocks	$r - 1$	$MS_{Block}$	—
Whole-plot factor (A)	$a - 1$	$MS_A$	$MS_A / MS_{WPE}$
Whole-plot error	$(r-1)(a-1)$	$MS_{WPE}$	—
Subplot Stratum
Subplot factor (B)	$b - 1$	$MS_B$	$MS_B / MS_{SPE}$
A × B interaction	$(a-1)(b-1)$	$MS_{AB}$	$MS_{AB} / MS_{SPE}$
Subplot error	$a(r-1)(b-1)$	$MS_{SPE}$	—
Total	$rab - 1$

Key Rules

• Test whole-plot factor A using the whole-plot error as denominator.
• Test subplot factor B and A×B interaction using the subplot error as denominator.
• The subplot error is typically smaller than the whole-plot error, giving more power to detect subplot and interaction effects.

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Worked Example

Experiment: Effect of irrigation method (Factor A: furrow, drip) and nitrogen level (Factor B: low, medium, high) on wheat yield, conducted in 3 blocks.

Parameters

\[r = 3,\quad a = 2,\quad b = 3,\quad N = 3 \times 2 \times 3 = 18\]

Randomisation Plan

Block 1:
  WP-1 (Furrow):  N-Low  |  N-Med  |  N-High
  WP-2 (Drip):    N-Med  |  N-High |  N-Low

Block 2:
  WP-1 (Drip):    N-High |  N-Low  |  N-Med
  WP-2 (Furrow):  N-Low  |  N-High |  N-Med

Block 3:
  WP-1 (Furrow):  N-Med  |  N-Low  |  N-High
  WP-2 (Drip):    N-High |  N-Med  |  N-Low

Within each whole plot, the order of subplot treatments is randomised independently.

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Assumptions

For valid inference, the following assumptions must hold:

1. Normality — Residuals at both whole-plot and subplot levels are approximately normally distributed.
2. Homogeneity of variance — Variances are equal within strata.
3. Independence — Whole plots are independent of each other.
4. Correct error term — Whole-plot factor is tested against whole-plot error, not subplot error.

Violating assumption 4 (a common mistake when using standard one-error ANOVA software) leads to anti-conservative tests for whole-plot factors.

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Analysis in R

library(nlme)
library(lme4)

# Load data
data <- read.csv("split_plot_data.csv")

# Fit split plot model using lme (nlme package)
model <- lme(yield ~ irrigation * nitrogen,
             random = ~1 | block/whole_plot,
             data = data,
             method = "REML")

# ANOVA table
anova(model)

# Summary
summary(model)

# Pairwise comparisons (emmeans)
library(emmeans)
emmeans(model, pairwise ~ nitrogen | irrigation)

Using aov() (base R)

# Traditional aov approach
model_aov <- aov(yield ~ irrigation * nitrogen +
                   Error(block/irrigation),
                 data = data)

summary(model_aov)

Tip: Prefer lme() or lmer() for unbalanced data or when variance components are of interest.

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Analysis in SAS

proc mixed data=split_plot;
  class block irrigation nitrogen;
  model yield = irrigation nitrogen irrigation*nitrogen / ddfm=satterth;
  random block block*irrigation;
  lsmeans irrigation*nitrogen / pdiff adjust=tukey;
run;

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Advantages and Disadvantages

Advantages ✓

• Accommodates hard-to-change factors without an impractically large number of factor-level changes.
• Provides high precision for subplot factor and interaction comparisons.
• Reduces experimental effort compared to a completely randomised design with the same number of factor combinations.
• Naturally suits many agricultural and industrial scenarios.

Disadvantages ✗

• Lower precision for the whole-plot factor compared to the subplot factor.
• More complex analysis — two error terms required.
• Unbalanced data (missing observations) complicates analysis.
• Software that ignores the split-plot structure produces incorrect F-tests for whole-plot effects.

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Extensions

Extension	Description
Split-Split Plot	A third factor added as sub-subplots within subplots
Strip Plot (Criss-Cross)	Both factors applied in perpendicular strips; no nesting
Repeated Measures	Time as the subplot factor; observations correlated within subject
Unbalanced Split Plot	Handled with mixed model (REML) when cells are missing
Split Plot in RCBD	Each block contains one replicate of all whole-plot treatments

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References

1. Montgomery, D.C. (2017). Design and Analysis of Experiments (9th ed.). Wiley.
2. Cochran, W.G. & Cox, G.M. (1957). Experimental Designs (2nd ed.). Wiley.
3. Littell, R.C., Milliken, G.A., Stroup, W.W., Wolfinger, R.D., & Schabenberger, O. (2006). SAS for Mixed Models (2nd ed.). SAS Institute.
4. Pinheiro, J.C. & Bates, D.M. (2000). Mixed-Effects Models in S and S-PLUS. Springer.
5. Stroup, W.W. (2012). Generalized Linear Mixed Models. CRC Press.

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