Pearson Correlation Coefficient (r)
What it measures
The strength and direction of the linear relationship between a soil nutrient and yield.
Formula
n = count of paired values
sumX = sum(nutrient values)
sumY = sum(yield values)
sumXY = sum(nutrient × yield)
sumX² = sum(nutrient²)
sumY² = sum(yield²)
r = (n × sumXY - sumX × sumY) /
sqrt((n × sumX² - sumX²) × (n × sumY² - sumY²))
Significance Levels
| |r| Value | Significance |
|---|
| > 0.7 | High (strong relationship) |
| 0.4-0.7 | Medium (moderate relationship) |
| 0.2-0.4 | Low (weak relationship) |
| < 0.2 | None (no meaningful relationship) |
Correlation ≠ Causation: A high correlation between P and yield doesn't prove P is limiting yield. The correlation could be driven by other factors (soil type, drainage, management) that happen to correlate with P.
Yield Correlation Table
Scatter Plot
Normalized (Field-Relative) Correlation
What it measures
Correlation after removing between-field differences, showing only within-field relationships.
Method
fieldMean = mean(all samples in that field)
normalizedValue = (value / fieldMean) × 100
Why normalize: Raw correlations include field-to-field differences. A high-yielding field might have high P simply because it's a better field overall. Normalization isolates whether P variation within a field affects yield, which is more actionable for variable-rate decisions.
Yield Correlation (normalized mode)
95% Confidence Interval
What it shows
The range where we expect 95% of individual predictions to fall, shown as a band around the regression line.
Formula
tValue = 1.96
MSE = sum(residuals²) / (n - 2)
SE = sqrt(MSE)
SE_y = SE × sqrt(1 + 1/n + (x - meanX)² / SS_x)
upper = predicted + tValue × SE_y
lower = predicted - tValue × SE_y
Scatter Plot
Yield by Nutrient Level (Bucket Analysis)
What it measures
Average yield at different nutrient levels (Low, Medium, High) based on agronomic thresholds.
Method
Low = value < critical threshold
Medium = critical ≤ value < optimal max
High = value ≥ optimal max
avgYield_low = mean(yield of all Low samples)
avgYield_medium = mean(yield of all Medium samples)
avgYield_high = mean(yield of all High samples)
yieldDiff = avgYield_high - avgYield_low
Interpretation
| Pattern | What it suggests |
|---|
| Low < Medium < High | Classic response - nutrient is limiting in low areas |
| Low ≈ Medium ≈ High | No yield response to this nutrient (not limiting) |
| High < Low | Possible toxicity, imbalance, or confounding factor |
Sample size matters: Buckets with fewer than 10 samples may not be reliable. Look for consistent patterns across multiple years before making management changes.
Yield by Nutrient Level Tab
Breakpoint Analysis
What it measures
The critical nutrient threshold where yield response changes - below = yield penalty, above = diminishing returns.
Algorithm (Binning with Bootstrap)
for each candidate threshold t:
below = samples where nutrient < t
above = samples where nutrient ≥ t
if below.count < minPerSide OR above.count < minPerSide: skip
penalty = mean(yield_above) - mean(yield_below)
if penalty > bestPenalty AND penalty ≥ MIN_PENALTY:
bestBreakpoint = t
for i = 1 to 50:
subset = random 80% of samples
bootBreakpoint = run algorithm on subset
if |bootBreakpoint - bestBreakpoint| ≤ tolerance:
nearCount++
stabilityPct = nearCount / 50 × 100
Minimum Penalty Thresholds
| Crop | Min Penalty | Why |
|---|
| Corn | 5 bu/ac | Smaller differences not economically significant |
| Soybeans | 2 bu/ac | Lower yield baseline, smaller absolute differences |
Confidence Levels
| Stability % | Confidence | Meaning |
|---|
| ≥ 70% | High | Breakpoint found consistently across resamples |
| 50-69% | Medium | Breakpoint is likely but variable |
| 30-49% | Medium-Low | Weak signal, needs more data |
| < 30% | Low | No reliable breakpoint detected |
Data-driven thresholds: Unlike fixed textbook thresholds, breakpoint analysis finds YOUR threshold based on YOUR data. This accounts for soil type, climate, hybrid/variety, and management that make your operation unique.
Breakpoint Analysis Tab
Multivariate Regression (MVR)
What it measures
The combined effect of multiple soil nutrients on yield, accounting for the influence of each variable while controlling for others.
Algorithm (Ordinary Least Squares)
Yield = β₀ + β₁×P + β₂×K + β₃×OM + β₄×pH + ... + ε
β = (X'X)⁻¹ × X'Y
X = matrix of nutrient values (with intercept column of 1s)
Y = vector of yield values
β = vector of coefficients (slopes for each nutrient)
Key Statistics
| Statistic | What it tells you |
|---|
| R² (R-squared) | % of yield variation explained by the model (higher = better fit) |
| Adjusted R² | R² adjusted for number of variables (penalizes overfitting) |
| Coefficient (β) | Expected yield change per 1-unit increase in nutrient |
| p-value | Probability coefficient is zero (< 0.05 = statistically significant) |
| VIF | Variance Inflation Factor - detects collinearity (> 5 = concern) |
Collinearity Check (VIF)
Xj = α₀ + α₁×X₁ + ... + αₖ×Xₖ
VIF_j = 1 / (1 - R²_j)
Why multivariate? Single-nutrient correlations can be misleading. For example, P and K might both correlate with yield simply because high-fertility fields have both. MVR isolates each nutrient's unique contribution, controlling for the others.
Multivariate Regression Tab
Hinge-MVR (Segmented Regression)
What it measures
A two-segment linear model that captures different yield responses below vs. above a breakpoint. Also called "piecewise regression" or "bent-stick model."
Algorithm
lowPart = max(0, t - x)
highPart = max(0, x - t)
Yield = β₀ + β₁×lowPart + β₂×highPart + β₃×cov₁ + ... + ε
β₁ = yield change per unit BELOW breakpoint (deficiency response)
β₂ = yield change per unit ABOVE breakpoint (luxury response)
Visual Interpretation
╱ β₂ slope (above)
╱
∙ ← Breakpoint (t)
╱
╱ β₁ slope (below)
───────────────┴──────────────────
Low Nutrient High
What the coefficients mean
| Scenario | β₁ (below) | β₂ (above) | Interpretation |
|---|
| Classic deficiency | Large positive | Small/zero | Strong response below threshold, plateau above |
| Linear response | ≈ equal | ≈ equal | No breakpoint needed, use simple regression |
| Toxicity | Small | Negative | Yield decreases at high levels |
When to use: Hinge-MVR is most useful when breakpoint analysis finds a stable threshold. It quantifies HOW MUCH yield responds on each side, while controlling for other nutrients.
Breakpoint Analysis Tab (toggle)