Goal: Compare tillage methods (conventional vs. no-till) and nitrogen fertilizer rates on burley tobacco yield
Design: Split-plot with tillage as the main-plot factor and fertilizer as the subplot factor
Main plots: Tillage (CT, NT) - applied to whole plots because tillage requires large equipment
Subplots: Fertilizer (0, 140, 280 kg N/ha) - can vary within a main plot
Replicates: 4 blocks (randomized complete block design at the main-plot level)
Split-Plot Design Diagram
Figure 1: Schematic of a split-plot design with 2 tillage levels and 3 fertilizer rates across 4 blocks.
Key Properties of Split-Plot Designs
Two levels of randomization:
Main-plot factor (tillage) is randomized across whole plots within each block
Subplot factor (fertilizer) is randomized within each main plot
Two error terms:
Main-plot error: variation among main plots within blocks (used to test tillage)
Subplot error: variation among subplots within main plots (used to test fertilizer and the interaction)
Why split-plot? Some factors (e.g., tillage) are hard or impractical to apply at a small scale
The Model
In R, we fit the split-plot model using lmer from the lme4 package:
# Yield ~ fixed effects + random effects# Tillage * Fertilizer = Tillage + Fertilizer + Tillage:Fertilizer# (1 | Block/WU) = random intercept for Block and whole-plot unit nested in Blockmodel <-lmer(Yield ~ Tillage * Fertilizer + (1| Block/WU), data = data)
Term
Meaning
Tillage * Fertilizer
Fixed effects: main effects and interaction
(1 | Block)
Random effect for blocks (replicates)
(1 | Block:WU)
Random effect for whole-plot units (main-plot error)
Residual
Subplot error
Fitting the Model in R
# Source the simulation and plotting code from the repo.# This script (projects/examplesplitplot.R) creates:# - data: a data.frame with Tillage, Fertilizer, Replicate, Yield, WU, Block# - model: an lmer fit (Yield ~ Tillage * Fertilizer + (1 | Block/WU))# - p: a ggplot object showing the simulated split-plot datasource("../projects/examplesplitplot.R")# Display the ANOVA tableanova(model)
Analysis of Variance Table
npar Sum Sq Mean Sq F value
Tillage 1 723.8 723.8 27.0436
Fertilizer 2 12843.6 6421.8 239.9364
Tillage:Fertilizer 2 38.1 19.1 0.7121
Model Summary
summary(model)
Linear mixed model fit by REML ['lmerMod']
Formula: Yield ~ Tillage * Fertilizer + (1 | Block/WU)
Data: data
REML criterion at convergence: 118.6
Scaled residuals:
Min 1Q Median 3Q Max
-1.7507 -0.5957 -0.0413 0.4511 1.8075
Random effects:
Groups Name Variance Std.Dev.
WU:Block (Intercept) 0.000e+00 0.000e+00
Block (Intercept) 3.282e-17 5.729e-09
Residual 2.676e+01 5.173e+00
Number of obs: 24, groups: WU:Block, 8; Block, 4
Fixed effects:
Estimate Std. Error t value
(Intercept) 61.253 2.587 23.680
TillageNT -13.575 3.658 -3.711
Fertilizer140 25.807 3.658 7.055
Fertilizer280 55.778 3.658 15.248
TillageNT:Fertilizer140 6.007 5.173 1.161
TillageNT:Fertilizer280 1.768 5.173 0.342
Correlation of Fixed Effects:
(Intr) TllgNT Frt140 Frt280 TNT:F1
TillageNT -0.707
Fertilzr140 -0.707 0.500
Fertilzr280 -0.707 0.500 0.500
TllgNT:F140 0.500 -0.707 -0.707 -0.354
TllgNT:F280 0.500 -0.707 -0.354 -0.707 0.500
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')
Diagnostic Plot: Residuals vs Fitted
Figure 2: Residuals vs fitted values for the split-plot model. Look for patterns indicating model misspecification.
Interpreting the Results
Main-plot effect (Tillage): Tested against the whole-plot error. A significant tillage effect means the two tillage methods produce different average yields.
Subplot effect (Fertilizer): Tested against the subplot (residual) error. A significant fertilizer effect means nitrogen rate influences yield.
Interaction (Tillage x Fertilizer): Also tested against subplot error. A significant interaction means the fertilizer effect depends on which tillage method is used.
Check: The whole-plot error variance should be non-negligible. If it is near zero, the split-plot structure may not be needed.
Practical Tips
Balance: Ensure equal replication of all treatment combinations within each block.
Blocking: Use blocks to control for known sources of variation (e.g., field position, soil type).
Diagnostics: Always check residual plots for non-constant variance or non-normality.
Random vs. fixed: The whole-plot grouping factor must be random. If you treat it as fixed, you underestimate the main-plot error.
Software:lme4::lmer and nlme::lme both handle split-plot models. Use car::Anova() or anova() for Type III tests.
Next Steps
Student activity (10 min):
Write a split-plot experiment proposal for your thesis or research area.
Identify your main-plot factor and subplot factor.
Describe how you would analyze the data (e.g., write out the lmer call).
