Hoof Trimming Schedule — DOE Use Case

Summary

This experiment investigates hoof trimming schedule. Fractional factorial screening of trim interval, heel height, toe angle, weight distribution, and exercise hours for lameness prevention and hoof health.

The design varies 5 factors: trim weeks (weeks), ranging from 6 to 16, heel height mm (mm), ranging from 20 to 40, toe angle deg (deg), ranging from 45 to 55, balance pct (%), ranging from 45 to 55, and exercise hrs (hrs), ranging from 2 to 8. The goal is to optimize 2 responses: lameness score (pts) (minimize) and hoof health (pts) (maximize). Fixed conditions held constant across all runs include animal = dairy_cow, surface = concrete_rubber.

A fractional factorial design reduces the number of runs from 32 to 8 by deliberately confounding higher-order interactions. This is ideal for screening — identifying which of the 5 factors matter most before investing in a full study.

Key Findings

For lameness score, the most influential factors were exercise hrs (38.3%), trim weeks (25.2%), toe angle deg (19.6%). The best observed value was 1.2 (at trim weeks = 16, heel height mm = 20, toe angle deg = 45).

For hoof health, the most influential factors were exercise hrs (47.3%), toe angle deg (18.7%), trim weeks (14.3%). The best observed value was 7.0 (at trim weeks = 16, heel height mm = 20, toe angle deg = 45).

Recommended Next Steps

Follow up with a response surface design (CCD or Box-Behnken) on the top 3–4 factors to model curvature and find the true optimum.
Consider whether any fixed factors should be varied in a future study.
The screening results can guide factor reduction — drop factors contributing less than 5% and re-run with a smaller, more focused design.

Experimental Setup

Factors

Factor	Low	High	Unit
`trim_weeks`	6	16	weeks
`heel_height_mm`	20	40	mm
`toe_angle_deg`	45	55	deg
`balance_pct`	45	55	%
`exercise_hrs`	2	8	hrs

Fixed: animal = dairy_cow, surface = concrete_rubber

Responses

Response	Direction	Unit
`lameness_score`	↓ minimize	pts
`hoof_health`	↑ maximize	pts

Configuration

use_cases/297_hoof_trimming/config.json

{
  "metadata": {
    "name": "Hoof Trimming Schedule",
    "description": "Fractional factorial screening of trim interval, heel height, toe angle, weight distribution, and exercise hours for lameness prevention and hoof health"
  },
  "factors": [
    {
      "name": "trim_weeks",
      "levels": [
        "6",
        "16"
      ],
      "type": "continuous",
      "unit": "weeks"
    },
    {
      "name": "heel_height_mm",
      "levels": [
        "20",
        "40"
      ],
      "type": "continuous",
      "unit": "mm"
    },
    {
      "name": "toe_angle_deg",
      "levels": [
        "45",
        "55"
      ],
      "type": "continuous",
      "unit": "deg"
    },
    {
      "name": "balance_pct",
      "levels": [
        "45",
        "55"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "exercise_hrs",
      "levels": [
        "2",
        "8"
      ],
      "type": "continuous",
      "unit": "hrs"
    }
  ],
  "fixed_factors": {
    "animal": "dairy_cow",
    "surface": "concrete_rubber"
  },
  "responses": [
    {
      "name": "lameness_score",
      "optimize": "minimize",
      "unit": "pts"
    },
    {
      "name": "hoof_health",
      "optimize": "maximize",
      "unit": "pts"
    }
  ],
  "settings": {
    "operation": "fractional_factorial",
    "test_script": "use_cases/297_hoof_trimming/sim.sh"
  }
}

Experimental Matrix

The Fractional Factorial Design produces 8 runs. Each row is one experiment with specific factor settings.

Run	`trim_weeks`	`heel_height_mm`	`toe_angle_deg`	`balance_pct`	`exercise_hrs`
1	6	40	55	45	2
2	16	20	45	45	2
3	16	40	45	55	2
4	16	40	55	55	8
5	6	40	45	45	8
6	16	20	55	45	8
7	6	20	45	55	8
8	6	20	55	55	2

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/297_hoof_trimming/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/297_hoof_trimming/config.json \
    --output use_cases/297_hoof_trimming/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/297_hoof_trimming/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/297_hoof_trimming/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/297_hoof_trimming/config.json

6

Multi-objective optimization

With 2 competing responses, use --multi to find the best compromise via Derringer–Suich desirability.

Terminal

$ doe optimize --config use_cases/297_hoof_trimming/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/297_hoof_trimming/config.json \
    --output use_cases/297_hoof_trimming/results/report.html

Features Exercised

Feature	Value
Design type	`fractional_factorial`
Factor types	`continuous` (all 5)
Arg style	`double-dash`
Responses	2 (lameness_score ↓, hoof_health ↑)
Total runs	8

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: lameness_score

Top factors: exercise_hrs (38.3%), trim_weeks (25.2%), toe_angle_deg (19.6%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
trim_weeks	1	0.9112	0.9112	1.682	0.2513
heel_height_mm	1	0.2813	0.2813	0.519	0.5035
toe_angle_deg	1	0.5512	0.5512	1.017	0.3594
balance_pct	1	0.0112	0.0112	0.021	0.8911
exercise_hrs	1	2.1012	2.1012	3.878	0.1060
trim_weeks*heel_height_mm	1	0.0113	0.0113	0.021	0.8911
trim_weeks*toe_angle_deg	1	2.1013	2.1013	3.878	0.1060
trim_weeks*balance_pct	1	0.2813	0.2813	0.519	0.5035
trim_weeks*exercise_hrs	1	0.5513	0.5513	1.017	0.3594
heel_height_mm*toe_angle_deg	1	0.0113	0.0113	0.021	0.8911
heel_height_mm*balance_pct	1	0.9112	0.9112	1.682	0.2513
heel_height_mm*exercise_hrs	1	0.3613	0.3613	0.667	0.4513
toe_angle_deg*balance_pct	1	0.3613	0.3613	0.667	0.4513
toe_angle_deg*exercise_hrs	1	0.9113	0.9113	1.682	0.2513
balance_pct*exercise_hrs	1	0.0113	0.0113	0.021	0.8911
Error	(Lenth	PSE)	5	2.7094	0.5419
Total	7	4.2288	0.6041

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for lameness_score

Half-Normal Plot of Effects

Model Diagnostics

Response: hoof_health

Top factors: exercise_hrs (47.3%), toe_angle_deg (18.7%), trim_weeks (14.3%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
trim_weeks	1	0.2113	0.2113	0.667	0.4513
heel_height_mm	1	0.2113	0.2113	0.667	0.4513
toe_angle_deg	1	0.3612	0.3612	1.140	0.3345
balance_pct	1	0.0312	0.0312	0.099	0.7662
exercise_hrs	1	2.3113	2.3113	7.294	0.0427
trim_weeks*heel_height_mm	1	0.0312	0.0312	0.099	0.7662
trim_weeks*toe_angle_deg	1	2.3113	2.3113	7.294	0.0427
trim_weeks*balance_pct	1	0.2112	0.2112	0.667	0.4513
trim_weeks*exercise_hrs	1	0.3613	0.3613	1.140	0.3345
heel_height_mm*toe_angle_deg	1	0.2812	0.2812	0.888	0.3894
heel_height_mm*balance_pct	1	0.2112	0.2112	0.667	0.4513
heel_height_mm*exercise_hrs	1	0.4513	0.4513	1.424	0.2863
toe_angle_deg*balance_pct	1	0.4513	0.4513	1.424	0.2863
toe_angle_deg*exercise_hrs	1	0.2113	0.2113	0.667	0.4513
balance_pct*exercise_hrs	1	0.2812	0.2812	0.888	0.3894
Error	(Lenth	PSE)	5	1.5844	0.3169
Total	7	3.8588	0.5513

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Half-Normal Plot of Effects

Model Diagnostics

Response Surface Plots

3D surfaces fitted with quadratic RSM. Red dots are observed data points.

hoof health balance pct vs exercise hrs

hoof health heel height mm vs balance pct

hoof health heel height mm vs exercise hrs

hoof health heel height mm vs toe angle deg

hoof health toe angle deg vs balance pct

hoof health toe angle deg vs exercise hrs

hoof health trim weeks vs balance pct

hoof health trim weeks vs exercise hrs

hoof health trim weeks vs heel height mm

hoof health trim weeks vs toe angle deg

lameness score balance pct vs exercise hrs

lameness score heel height mm vs balance pct

lameness score heel height mm vs exercise hrs

lameness score heel height mm vs toe angle deg

lameness score toe angle deg vs balance pct

lameness score toe angle deg vs exercise hrs

lameness score trim weeks vs balance pct

lameness score trim weeks vs exercise hrs

lameness score trim weeks vs heel height mm

lameness score trim weeks vs toe angle deg

Multi-Objective Optimization

When responses compete, Derringer–Suich desirability finds the best compromise. Each response is scaled to a 0–1 desirability, then combined via a weighted geometric mean.

Overall Desirability

D = 0.9545

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`lameness_score`	1.5	0.9545	1.20 0.9545 1.20 pts	↓
`hoof_health`	2.0	0.9545	7.00 0.9545 7.00 pts	↑

Recommended Settings

Factor	Value
`trim_weeks`	6 weeks
`heel_height_mm`	20 mm
`toe_angle_deg`	55 deg
`balance_pct`	55 %
`exercise_hrs`	2 hrs

Source: from observed run #5

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`hoof_health`	7.00	7.00	+0.00

Top 3 Runs by Desirability

Run	D	Factor Settings
#1	0.7747	trim_weeks=16, heel_height_mm=20, toe_angle_deg=45, balance_pct=45, exercise_hrs=2
#7	0.4937	trim_weeks=6, heel_height_mm=40, toe_angle_deg=55, balance_pct=45, exercise_hrs=2

Model Quality

Response	R²	Type
`hoof_health`	0.4525	linear

Full Multi-Objective Output

doe optimize --multi
============================================================
MULTI-OBJECTIVE OPTIMIZATION
Method: Derringer-Suich Desirability Function
============================================================

Overall desirability: D = 0.9545

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
lameness_score               1.5       0.9545        1.20 pts   ↓
hoof_health                  2.0       0.9545        7.00 pts   ↑

Recommended settings:
  trim_weeks = 6 weeks
  heel_height_mm = 20 mm
  toe_angle_deg = 55 deg
  balance_pct = 55 %
  exercise_hrs = 2 hrs
  (from observed run #5)

Trade-off summary:
  lameness_score: 1.20 (best observed: 1.20, sacrifice: +0.00)
  hoof_health: 7.00 (best observed: 7.00, sacrifice: +0.00)

Model quality:
  lameness_score: R² = 0.8410 (linear)
  hoof_health: R² = 0.4525 (linear)

Top 3 observed runs by overall desirability:
  1. Run #5 (D=0.9545): trim_weeks=6, heel_height_mm=20, toe_angle_deg=55, balance_pct=55, exercise_hrs=2
  2. Run #1 (D=0.7747): trim_weeks=16, heel_height_mm=20, toe_angle_deg=45, balance_pct=45, exercise_hrs=2
  3. Run #7 (D=0.4937): trim_weeks=6, heel_height_mm=40, toe_angle_deg=55, balance_pct=45, exercise_hrs=2

Full Analysis Output

doe analyze
=== Main Effects: lameness_score ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
exercise_hrs             1.0250       0.2748            38.3%
trim_weeks               0.6750       0.2748            25.2%
toe_angle_deg            0.5250       0.2748            19.6%
heel_height_mm          -0.3750       0.2748            14.0%
balance_pct             -0.0750       0.2748             2.8%

=== ANOVA Table: lameness_score ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
trim_weeks                   1       0.9112       0.9112      1.682     0.2513
heel_height_mm               1       0.2813       0.2813      0.519     0.5035
toe_angle_deg                1       0.5512       0.5512      1.017     0.3594
balance_pct                  1       0.0112       0.0112      0.021     0.8911
exercise_hrs                 1       2.1012       2.1012      3.878     0.1060
trim_weeks*heel_height_mm    1       0.0113       0.0113      0.021     0.8911
trim_weeks*toe_angle_deg     1       2.1013       2.1013      3.878     0.1060
trim_weeks*balance_pct       1       0.2813       0.2813      0.519     0.5035
trim_weeks*exercise_hrs      1       0.5513       0.5513      1.017     0.3594
heel_height_mm*toe_angle_deg    1       0.0113       0.0113      0.021     0.8911
heel_height_mm*balance_pct    1       0.9112       0.9112      1.682     0.2513
heel_height_mm*exercise_hrs    1       0.3613       0.3613      0.667     0.4513
toe_angle_deg*balance_pct    1       0.3613       0.3613      0.667     0.4513
toe_angle_deg*exercise_hrs    1       0.9113       0.9113      1.682     0.2513
balance_pct*exercise_hrs     1       0.0113       0.0113      0.021     0.8911
Error (Lenth PSE)            5       2.7094       0.5419
Total                        7       4.2288       0.6041
  Note: Error estimated using Lenth's pseudo-standard-error (unreplicated design)

=== Interaction Effects: lameness_score ===
Factor A             Factor B              Interaction   % Contribution
------------------------------------------------------------------------
trim_weeks           toe_angle_deg             -1.0250            23.6%
heel_height_mm       balance_pct               -0.6750            15.5%
toe_angle_deg        exercise_hrs              -0.6750            15.5%
trim_weeks           exercise_hrs              -0.5250            12.1%
heel_height_mm       exercise_hrs               0.4250             9.8%
toe_angle_deg        balance_pct                0.4250             9.8%
trim_weeks           balance_pct                0.3750             8.6%
heel_height_mm       toe_angle_deg             -0.0750             1.7%
balance_pct          exercise_hrs              -0.0750             1.7%
trim_weeks           heel_height_mm             0.0750             1.7%

=== Summary Statistics: lameness_score ===

trim_weeks:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  16                  4     2.3750     0.9535     1.2000     3.2000
  6                   4     3.0500     0.4435     2.4000     3.4000

heel_height_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  20                  4     2.9000     0.6000     2.0000     3.2000
  40                  4     2.5250     0.9777     1.2000     3.4000

toe_angle_deg:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  45                  4     2.4500     1.0376     1.2000     3.4000
  55                  4     2.9750     0.3862     2.4000     3.2000

balance_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  45                  4     2.7500     0.6608     2.0000     3.4000
  55                  4     2.6750     0.9845     1.2000     3.2000

exercise_hrs:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4     2.2000     0.8327     1.2000     3.2000
  8                   4     3.2250     0.1258     3.1000     3.4000

=== Main Effects: hoof_health ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
exercise_hrs            -1.0750       0.2625            47.3%
toe_angle_deg           -0.4250       0.2625            18.7%
trim_weeks              -0.3250       0.2625            14.3%
heel_height_mm           0.3250       0.2625            14.3%
balance_pct             -0.1250       0.2625             5.5%

=== ANOVA Table: hoof_health ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
trim_weeks                   1       0.2113       0.2113      0.667     0.4513
heel_height_mm               1       0.2113       0.2113      0.667     0.4513
toe_angle_deg                1       0.3612       0.3612      1.140     0.3345
balance_pct                  1       0.0312       0.0312      0.099     0.7662
exercise_hrs                 1       2.3113       2.3113      7.294     0.0427
trim_weeks*heel_height_mm    1       0.0312       0.0312      0.099     0.7662
trim_weeks*toe_angle_deg     1       2.3113       2.3113      7.294     0.0427
trim_weeks*balance_pct       1       0.2112       0.2112      0.667     0.4513
trim_weeks*exercise_hrs      1       0.3613       0.3613      1.140     0.3345
heel_height_mm*toe_angle_deg    1       0.2812       0.2812      0.888     0.3894
heel_height_mm*balance_pct    1       0.2112       0.2112      0.667     0.4513
heel_height_mm*exercise_hrs    1       0.4513       0.4513      1.424     0.2863
toe_angle_deg*balance_pct    1       0.4513       0.4513      1.424     0.2863
toe_angle_deg*exercise_hrs    1       0.2113       0.2113      0.667     0.4513
balance_pct*exercise_hrs     1       0.2812       0.2812      0.888     0.3894
Error (Lenth PSE)            5       1.5844       0.3169
Total                        7       3.8588       0.5513
  Note: Error estimated using Lenth's pseudo-standard-error (unreplicated design)

=== Interaction Effects: hoof_health ===
Factor A             Factor B              Interaction   % Contribution
------------------------------------------------------------------------
trim_weeks           toe_angle_deg              1.0750            25.0%
heel_height_mm       exercise_hrs               0.4750            11.0%
toe_angle_deg        balance_pct                0.4750            11.0%
trim_weeks           exercise_hrs               0.4250             9.9%
heel_height_mm       toe_angle_deg             -0.3750             8.7%
balance_pct          exercise_hrs              -0.3750             8.7%
trim_weeks           balance_pct               -0.3250             7.6%
heel_height_mm       balance_pct                0.3250             7.6%
toe_angle_deg        exercise_hrs               0.3250             7.6%
trim_weeks           heel_height_mm             0.1250             2.9%

=== Summary Statistics: hoof_health ===

trim_weeks:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  16                  4     6.2000     0.8756     5.3000     7.0000
  6                   4     5.8750     0.6702     4.9000     6.4000

heel_height_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  20                  4     5.8750     0.9323     4.9000     6.9000
  40                  4     6.2000     0.5888     5.6000     7.0000

toe_angle_deg:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  45                  4     6.2500     0.9678     4.9000     7.0000
  55                  4     5.8250     0.4787     5.3000     6.4000

balance_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  45                  4     6.1000     0.6583     5.3000     6.9000
  55                  4     5.9750     0.9179     4.9000     7.0000

exercise_hrs:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4     6.5750     0.4646     6.0000     7.0000
  8                   4     5.5000     0.5477     4.9000     6.2000

Optimization Recommendations

doe optimize
=== Optimization: lameness_score ===
Direction: minimize

Best observed run: #5
  trim_weeks = 16
  heel_height_mm = 20
  toe_angle_deg = 45
  balance_pct = 45
  exercise_hrs = 2
  Value: 1.2

RSM Model (linear, R² = 0.6589, Adj R² = -0.1939):
  Coefficients:
    intercept                      +2.7125
    trim_weeks                     -0.5125
    heel_height_mm                 +0.0375
    toe_angle_deg                  +0.1625
    balance_pct                    -0.0375
    exercise_hrs                   +0.2375

  Predicted optimum (from linear model, at observed points):
    trim_weeks = 6
    heel_height_mm = 40
    toe_angle_deg = 45
    balance_pct = 45
    exercise_hrs = 8
    Predicted value: 3.3750

  Surface optimum (via L-BFGS-B, linear model):
    trim_weeks = 16
    heel_height_mm = 20
    toe_angle_deg = 45
    balance_pct = 55
    exercise_hrs = 2
    Predicted value: 1.7250

  Model quality: Moderate fit — use predictions directionally, not precisely.

Factor importance:
  1. trim_weeks  (effect: 1.0, contribution: 51.9%)
  2. exercise_hrs  (effect: 0.5, contribution: 24.1%)
  3. toe_angle_deg  (effect: 0.3, contribution: 16.5%)
  4. balance_pct  (effect: -0.1, contribution: 3.8%)
  5. heel_height_mm  (effect: 0.1, contribution: 3.8%)

=== Optimization: hoof_health ===
Direction: maximize

Best observed run: #5
  trim_weeks = 16
  heel_height_mm = 20
  toe_angle_deg = 45
  balance_pct = 45
  exercise_hrs = 2
  Value: 7.0

RSM Model (linear, R² = 0.5458, Adj R² = -0.5896):
  Coefficients:
    intercept                      +6.0375
    trim_weeks                     +0.2625
    heel_height_mm                 -0.0375
    toe_angle_deg                  -0.3625
    balance_pct                    +0.1875
    exercise_hrs                   +0.1625

  Predicted optimum (from linear model, at observed points):
    trim_weeks = 16
    heel_height_mm = 40
    toe_angle_deg = 45
    balance_pct = 55
    exercise_hrs = 2
    Predicted value: 6.6500

  Surface optimum (via L-BFGS-B, linear model):
    trim_weeks = 16
    heel_height_mm = 20
    toe_angle_deg = 45
    balance_pct = 55
    exercise_hrs = 8
    Predicted value: 7.0500

  Model quality: Moderate fit — use predictions directionally, not precisely.

Factor importance:
  1. toe_angle_deg  (effect: -0.7, contribution: 35.8%)
  2. trim_weeks  (effect: -0.5, contribution: 25.9%)
  3. balance_pct  (effect: 0.4, contribution: 18.5%)
  4. exercise_hrs  (effect: 0.3, contribution: 16.0%)
  5. heel_height_mm  (effect: -0.1, contribution: 3.7%)

Run	`trim_weeks`	`heel_height_mm`	`toe_angle_deg`	`balance_pct`	`exercise_hrs`
1	6	40	55	45	2
2	16	20	45	45	2
3	16	40	45	55	2
4	16	40	55	55	8
5	6	40	45	45	8
6	16	20	55	45	8
7	6	20	45	55	8
8	6	20	55	55	2

Run	`trim_weeks`	`heel_height_mm`	`toe_angle_deg`	`balance_pct`	`exercise_hrs`
1	6	40	55	45	2
2	16	20	45	45	2
3	16	40	45	55	2
4	16	40	55	55	8
5	6	40	45	45	8
6	16	20	55	45	8
7	6	20	45	55	8
8	6	20	55	55	2

Run	`trim_weeks`	`heel_height_mm`	`toe_angle_deg`	`balance_pct`	`exercise_hrs`
1	6	40	55	45	2
2	16	20	45	45	2
3	16	40	45	55	2
4	16	40	55	55	8
5	6	40	45	45	8
6	16	20	55	45	8
7	6	20	45	55	8
8	6	20	55	55	2