Bayesian Ordinal Regression for Crop Development and Disease Assessment

Zhanglong Cao, Rose Megirian, Matthew Nguyen, Adam Sparks

CBADA - CCDM & EECMS, Curtin University

26/11/2025

Background

Experiments

Objective: evaluating deep sowing of oats in WA production systems

  • Design: split-plot design at two sites

  • Sowing depths at 30, 80 and 120 mm

  • Variety-herbicide combinations: 10 levels

trial layout

Growth stage data

  • The growth scale data is measured on an ordinal scale following the Zadoks growth scale measurement methods.

  • Ten samples were taken from each plot four weeks after sowing.

Zadoks growth scale

The Zadok’s growth scale is based on ten principal cereal growth stages (Zadoks, Chang, and Konzak (1974)):

0 - Germination

1 - Seeding growth

2 - Tillering

3 - Stem elongation

4 - Booting

5 - Awn emergence

6 - Flowering (anthesis)

7 - Milk development

8 - Dough development

9 - Ripening

Zadoks growth scale

The Zadok’s growth scale is not chronological time points but a series of ordinal stages.

0 - Germination

00 - Dry seed

01 - Start of water absorption

03 - Seed fully swollen

05 - First root emerged from seed

07 - Coleoptile emerged from seed

09 - First green leaf just at tip of coleoptile

1 - Seedling growth

10 - First leaf through coleoptile

11 - First leaf emerged

12 - Two leaves emerged

13 - Three leaves emerged

14 - Four leaves emerged

15 - Five leaves emerged

16 - Six leaves emerged

17 - Seven leaves emerged

18 - Eight leaves emerged

19 - Nine or more leaves emerged

2 - Tillering

20 - Main stem only

21 - Main stem and one tiller

22 - Main stem and two tillers

23 - Main stem and three tillers

24 - Main stem and four tillers

25 - Main stem and five tillers

26 - Main stem and six tillers

27 - Main stem and seven tillers

28 - Main stem and eight tillers

29 - Main stem and nine or more tillers

Initial data exploration

Growth stage distribution by depth - Merredin

Growth stage distribution by depth - Wickepin

The spatial effect

Spatial patterns - Merredin

Spatial patterns - Wickepin

  • Field trials have spatial gradients

  • Better estimates → Better decisions

Modelling challenge

  1. Ordinal (not continuous) response

  2. Hierarchical design (plots, sites)

    • Multiple plants per plot (pseudo-replication)

  3. Complex interactions

    • Treatment effects differ by variety
    • Effects differ by growth stage

  4. Spatial correlation

    • Field gradients affect growth

  5. Biological interpretability

    • Results must inform agronomic decisions

Bayesian approach

rstan

  • Powerful brms package for Bayesian multilevel models using rstan (Stan Development Team (2025), Bürkner (2017)).

  • Flexibility: supports a wide range of distributions and link functions, incorporating personalised features, such as spatial effects.

  • Advanced features: handles missing data, multilevel structures, and more.

  • Visualisation: offers extensive plotting capabilities for model diagnostics and results.

Choosing the right ordinal model

Source: Bürkner and Vuorre (2019)

Why sequential model

Sequential model logic

Plant at stage \(k\) asks: “Do I advance to \(k+1\)?”

\[P(\text{advance to } k+1 | \text{currently at } k)\]

For Z13 → Z21 transition:

“Does this plant develop its first tiller?”

Treatment effects = changes in advancement probability!

This directly answers: “Does shallow sowing ACCELERATE development?” ⏱️

Why not cumulative?

Cumulative models: \(P(Y \leq k)\)

“Probability of being at or below stage k”

Problem: Doesn’t capture the sequential progression through development!

  • Can’t isolate specific transitions
  • Harder to interpret biologically
  • Doesn’t match how plants actually develop

Bayesian hierarchical model

\[ \begin{split} &Y_i \in \{Z00, Z01, \ldots, Z24\} \\ &\eta_{k,i} = \alpha_k + \boldsymbol\beta_k^\top \mathbf{x}_i + u_{\text{Plot}[i]} + u_{\text{Depth}[i]|\text{Plot}[i]} + f(\mathbf{s}_i) \\ &P(Y_i \geq k \mid Y_i \geq k-1) = \text{logit}^{-1}(\eta_{k,i}) \\ &u_{\text{Plot}} \sim N(0, \sigma_p^2) \\ &u_{\text{Depth|Plot}} \sim N(0, \sigma_d^2) \\ &f \sim \text{GP}(\mathbf{0}, \Sigma_s) \quad \text{(optional)}. \end{split} \]

  • \(\alpha_k, \beta_k\) are category-specific (vary by growth stage)

  • Captures how sowing depth affects different developmental transitions differently

  • Hierarchical structure accounts for plot and treatment effects

Bayesian workflow

workflow

(source: Gelman et al. (2020))

No-U-Turn sampler (NUTS)

(source:https://github.com/chi-feng)

Leave-one-out (LOO) cross validation

In Bayesian statistics, the expected \(\log\) LOO predictive density (ELPD) is used to measure the predictive accuracy : \[ \mbox{elpd} = \sum_{i=1}^{n}\log p(y_i\mid y_{-i}), \] where \(p(y_i \mid y_{-i}) = \int p(y_i \mid \theta)p(\theta \mid y_{-i})d\theta\) is the LOO predictive density with the \(i\)-th observation omitted from the data set (Vehtari, Gelman, and Gabry (2017)).

Bürkner, Gabry, and Vehtari (2021) proposed approximated LOO CV, which uses only a single model fit and calculating the pointwise \(\log\) predictive density as a fast approximation to the exact LOO CV (Cao et al. (2022)).

Results: Merredin Data

Model comparison

Winner: Sequential for Better ELPD + biologically interpretable!

  1. Cumulative (proportional odds)
    • Models: \(P(Y \leq k)\)
    • Interpretation: “At or below stage \(k\)”
  2. Sequential (continuation ratio) ✅
    • Models: \(P(Y \leq k | Y \leq k-1)\)
    • Interpretation: “Advance to next stage”
  3. Adjacent-category
    • Models: \(P(Y = k \mid Y \in \{k, k+1\})\)
    • Interpretation: “Adjacent stage ratios”

Model diagnostics

Posterior predictive check

Posterior predictive check

Also refer to Cao et al. (2022) for detailed diagnostics.

Results

Variety-specific response by sowing depth (Merredin); ribbons show 95% credible intervals.

Results

What this shows:

  • Clear threshold between seed emerge (Z01) and leaf development (Z11). This explains why sowing depth affects it most strongly!

Site consistency

Treatment effects comparison across Merredin and Wickepin

Extension: disease severity

The same framework works for disease assessment!

  • Powdery mildew (0-9 scale)

  • Fusarium head blight severity

  • Leaf rust infection scores

  • Root health ratings

Why sequential model?

Disease progression is also sequential:

Healthy -> Mild -> Moderate -> Severe -> Very Severe

“Does treatment prevent advancement to next severity level?”

Same logic as growth stages! 🌱→🦠

AI integration

Current workflow:

  1. Manual field assessment
  2. Record growth stage visually
  3. Enter data manually
  4. Analyse
  5. Generate recommendations

Limitations:

  • Labour intensive

  • Subjective scoring

  • Limited temporal resolution

  • Can’t scale to whole fields

2026 vision: automated system

  1. Drone imagery of entire field (done!)
  2. Computer vision classifies growth stage/diesea score (in progress)
  3. Real-time Bayesian updating (in progress)
  4. Automatic recommendations to farmer (plan)

Advantages:

✅ Continuous monitoring

✅ Objective classification

✅ Whole-field coverage

✅ Early intervention possible

Bayesian advantages

First time buying:

  • Prior: “Nice houses cost ~$500k” (market knowledge)
  • Data: Visit 10 houses
  • Posterior: “Actually $550-600k” (updated belief)

Second time buying:

  • Prior: $550-600k (your posterior becomes the prior!)
  • Data: Visit more houses
  • Posterior: “Trending toward $650k” (continuously refined)

Take-Home Messages

  1. Growth stages are ordinal - respect the structure! Don’t use linear models.

  2. Sequential models match biology - plants progress through stages, model should too.

  3. Bayesian inference answers real questions - not just p-values, actual probabilities.

  4. Spatial correlation matters - field trials need spatial structure in models.

  5. Framework is general - works for growth stages, disease severity, any other ordinal outcome.

  6. Ready for precision agriculture - principled statistics + AI = future of agronomy.

Acknowledgments

People:

  • Rose Megirian
  • Matthew Nguyen
  • Professor Adam Sparks
  • All Growers

Funding & Support:

Bürkner, Paul-Christian. 2017. “Brms: An r Package for Bayesian Multilevel Models Using Stan.” Journal of Statistical Software 80 (1): 1–28. https://doi.org/10.18637/jss.v080.i01.
Bürkner, Paul-Christian, Jonah Gabry, and Aki Vehtari. 2021. “Efficient Leave-One-Out Cross-Validation for Bayesian Non-Factorized Normal and Student-t Models.” Comput Stat 36 (2): 1243–61. https://doi.org/10.1007/s00180-020-01045-4.
Bürkner, Paul-Christian, and Matti Vuorre. 2019. “Ordinal Regression Models in Psychology: A Tutorial.” Advances in Methods and Practices in Psychological Science 2 (1): 77–101.
Cao, Zhanglong, Katia Stefanova, Mark Gibberd, and Suman Rakshit. 2022. “Bayesian Inference of Spatially Correlated Random Parameters for on-Farm Experiment.” Field Crops Research 281: 108477.
Gelman, Andrew, Aki Vehtari, Daniel Simpson, Charles C Margossian, Bob Carpenter, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, and Martin Modrák. 2020. “Bayesian Workflow.” arXiv Preprint arXiv:2011.01808.
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Zadoks, J. C., T. T. Chang, and C. F. Konzak. 1974. “A Decimal Code for the Growth Stages of Cereals.” Weed Research 14 (6): 415–21. https://doi.org/10.1111/j.1365-3180.1974.tb01084.x.