Design and Spatial Analysis of On-Farm Strip Trials: From Statistical Principles to Methodological Innovation and Global Impact

Zhanglong Cao
Julia Easton, Suman Rakshit, Barbara Kachigunda, Mark Gibberd

7/10/25

Introduction

Region/country	Farms	Fields	Field size (ha)	Primary focus
Nebraska, USA	4,000	150	2-150	Broadacre cropping
Montana, USA	800	65	8-65	Precision experiments
UK/N. Europe	300	200-1,000	5-40	Yield enhancement
France (DEPHY)	3,000	65	1-5	Varied systems
Australia (WA)	50-70	Variable	15-500	Broadacre cropping
China (STB)	5,000	10-30	0.1-10	Smallholder systems
Argentina	35	Variable	5-150	Nutrition optimisation
Canada	15	12,000	15-300	Broadacre systems

Statistical analysis for OFE - GWR

The underlying template model for GWR is given by

y (s_{i}) = β_{0} + \sum_{j = 1}^{k} β_{j} x_{j} (s_{i}) + ε_{i},

$\begin{equation}\label{eq:basicgwr} y(s_i) = \beta_0 + \sum_{j=1}^{k} \beta_j x_j(s_i) + \varepsilon_i, \end{equation}$ where

β = (β_{0}, β_{1}, \dots, β_{k})^{⊤}

$\beta = (\beta_0, \beta_{1}, \ldots, \beta_{k})^{\top}$ are model parameters corresponding to the

k

$k$ treatment levels and

ε_{i} \sim N (0, τ^{2})

$\varepsilon_{i} \sim N(0,\tau^2)$ ,

i = 1, \dots, n

$i=1,\ldots,n$ , are independent and identically distributed error terms at

n

$n$ grid points.

For a query location $s$ , the local log-likelihood is given by

\begin{array}{rcl} \log L (s; β) = - \frac{1}{2 τ^{2}} \sum_{i = 1}^{n} K (s, s_{i}) {(y (s_{i}) - β_{0} - \sum_{j = 1}^{k} β_{j} x_{j} (s_{i}))}^{2} \end{array}

$\begin{eqnarray}\label{eq:locallog} \log L(s; \beta) = - \frac{1}{2\tau^2}\sum_{i=1}^{n}K(s,s_i)\left( y(s_i) -\beta_0 - \sum_{j=1}^{k}\beta_j x_j(s_i) \right)^2 \end{eqnarray}$ where

K (\cdot, \cdot)

$K(\cdot,\cdot)$ is a given kernel function, such as Gaussian, exponential, bi-square or tri-cube Gollini et al. (2015).

Statistical analysis for OFE - GWR

Core concept: GWR is a locally weighted regression method that operates by assigning different weights to observations based on their distance from each query location on the field (Páez et al. 2002).

Simple mathematical framework:

y (location) = β_{0} (location) + β_{1} (location) \times treatment (location) + error

$y(\text{location}) = \beta_0(\text{location}) + \beta_1(\text{location}) \times \text{treatment}(\text{location}) + \text{error}$ where coefficients $\beta$ vary across the field

“Instead of one average effect across the entire field, GWR estimates unique treatment effects for every location”

Statistical analysis for OFE - LMM

Suppose the entire field is partitioned into $p$ pseudo-environments (PE), and the $j$ -th PE consists of $n_j$ grid points, such that the total number of grid points in the experiment is $n=\sum_{j=1}^p n_j$ . Let $y_j$ be the $n_{j} \times 1$ vector of observed responses corresponding to the $j$ -th PE. The LMM for the combined vector of data $Y = \lbrack y^\top_{1}, \ldots, y^\top_{p}\rbrack^{\top}$ across all PEs is given by

Y = X τ + Z u + e,

$\begin{equation}\label{eq:modelmatrix} Y = X \tau + Zu + e, \end{equation}$ where

τ

$\tau$ and

u

$u$ are

t \times 1

$t\times 1$ and

b \times 1

$b \times 1$ vectors of fixed and random effects, respectively. The matrices

X

$X$ and

Z

$Z$ are

n \times t

$n\times t$ and

n \times b

$n\times b$ design matrices corresponding to the fixed and random effects, respectively. The

n \times 1

$n\times 1$ vector

e

$e$ provides the combined residual effects from all PEs.

Statistical analysis for OFE - LMM

Core concept: Linear Mixed Models treat large strip OFEs similar to Multi-Environment Trials (MET), where PE represent different zones within the field.

Simple mathematical framework:

Yield = Zone Effects + Zone × Treatment + Spatial Correlation + Error

$\text{Yield} = \text{Zone Effects} + \text{Zone × Treatment} + \text{Spatial Correlation} + \text{Error}$

Why pseudo-environments matter:

Fields are not uniform - different zones behave differently
One-size-fits-all recommendations often fail
Zone-specific management improves overall performance

Practical implementation:

Clustering approach using elevation, soil, or other covariates
Systematic partitioning when covariates unavailable
3-6 zones typically optimal for most fields
Statistical modeling within each zone

Simulation study 2 - LMM

We simulate baseline yield using unconditional Gaussian geo-statistical simulation based on a first-order stationary random field model (Evans et al. (2020)).

z (s) = μ + ε (s)

$z(s) = \mu + \varepsilon(s)$ where

z (s)

$z(s)$ is the simulated yield at location

s = (x, y)

$s = (x, y)$ , fertiliser trials are simulated using different designs and is added to the baseline yield:

y (s) = z (s) + β \times N (s)

$y(s) = z(s) + \beta \times N(s)$

Simulation workflow:

Simulate baseline yield surfaces using spatial random fields.
Overlay treatment effects using different layouts (strip, stacked).
Fit a range of LMMs with/without spatial terms, using both averaged and full data.
Assess estimation accuracy (relative absolute difference, RAD) and statistical power for each scenario.
Compare performance across all combinations of design, model, and data type.

Result 2 - LMM

The statistical powers of strip trials

Key findings are

Critical Design Thresholds

Minimum 3 replications required for 95%+ power in most scenarios

Strip length matters: Longer strips ( $\ge$ 50m) show consistent power improvement

Spatial models (M12, M22) enhance power by 5-15% over non-spatial models

Full data analysis consistently outperforms averaged data approaches

Design-specific performance

Randomised designs: Superior power for hypothesis testing (0.85-1.0 typical)

Systematic designs: Adequate power (0.70-0.95) with operational advantages

Length effect: 110m strips achieve 98%+ power with 2 reps vs 70% for 10m strips

Model selection: Spatial correlation models critical for realistic field conditions

Result 2 - LMM

The statistical powers of stacked trials

Key findings are

Critical performance patterns:

M21 (Full non-spatial): Consistently achieves 100% power across all scenarios
M23 models: Generally poor performance (0.1-0.7 power) - avoid in practice
Length sensitivity: Dramatic improvement from 10m (0.0-0.7) to 110m (0.25-1.0) trials
Replication effects: Optimal performance at 10-12 reps, diminishing returns beyond 6 reps

Model selection insights:

Averaged data models: Highly variable power (0.0-1.0) - unreliable for inference
Full data analysis: More consistent and higher power overall
Spatial vs non-spatial: Non-spatial models (M21) surprisingly outperform spatial models (M22, M23)
Design comparison: Randomised slightly better than systematic, but differences minimal

Practical recommendations: For stacked trials, prioritize M21 models with full data analysis and ensure adequate trial length ( $\ge$ 50m) for reliable statistical inference.

Research Objective	Primary Analysis	Recommended Design	Key Considerations
Variable rate maps	GWR, spatial models	Systematic	Bandwidth selection critical
Treatment comparison	ANOVA, LMM	Randomised	Block design preferred
Yield response mapping	Spatial interpolation	Systematic	Consider treatment spacing
Economic analysis	Mean comparisons	Either	Depends on spatial components

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Design and Spatial Analysis of On-Farm Strip Trials: From Statistical Principles to Methodological Innovation and Global Impact Zhanglong Cao Julia Easton, Suman Rakshit, Barbara Kachigunda, Mark Gibberd 7/10/25

Design and Spatial Analysis of On-Farm Strip Trials: From Statistical Principles to Methodological Innovation and Global Impact
Slide 2
Introduction
About myself
What is on-farm experimentation?
Examples of worldwide OFE initiatives1
Examples of worldwide OFE initiatives1
Primary motivators across global networks
Key differences from traditional research
Benefits: real-world validation and innovation
Benefits: real-world validation and innovation
Benefits: real-world validation and innovation
Research focus
Key objectives
Experimental design principles of OFE
Illustration of OFE trial designs
Historical context: the Flawborough chequerboard experiment
Why our focus is on strip trials?
Methodological evolution
Methodological evolution
A typical strip trial design
Design selection guidelines
Visual examples
Visual examples
Stacked replicated design
Stacked replicated design
Design selection guidelines by AAGI (CBADA)
Data collection
Spatial statistics for OFE analysis
Theoretical framework (Cressie 2013)
Statistical analysis for OFE - GWR
Statistical analysis for OFE - GWR
Statistical analysis for OFE - GWR
Statistical analysis for OFE - GWR
Statistical analysis for OFE - GWR
Statistical analysis for OFE - GWR
Statistical analysis for OFE - LMM
Statistical analysis for OFE - LMM
Statistical analysis for OFE - LMM
Statistical analysis for OFE - LMM
Method selection framework
Research findings
Simulation study 1 - GWR
Result 1 - GWR
Result 1 - GWR
Simulation study 2 - LMM
Simulation study 2 - LMM
Result 2 - LMM
Result 2 - LMM
Result 2 - LMM
Result 2 - LMM
Result 2 - LMM
Result 2 - LMM
Result 2 - LMM
Design and analysis tool selection table
National standard and research impact
Curtin University’s leadership and impact in OFE
Curtin University’s (CBADA) leadership
Global impact
Key takeaways & future directions
Acknowledgements
Acknowledgements
Thank you!
Reference