Ordinal Logistic
🎯 Purpose
This QuickRef covers ordinal logistic regression — a model used when the target variable is ordered but not continuous. Commonly used in rating scales, satisfaction levels, or risk categories.
📦 1. When to Use¶
| Condition | Use Ordinal Logistic? |
|---|---|
| Target has ≥3 ordered categories | ✅ Yes |
| Values have clear ranking (e.g. low < med < high) | ✅ Yes |
| Target is unordered | ❌ Use multinomial logistic |
| Continuous regression needed | ❌ Use linear model |
🧮 2. Model Logic (Proportional Odds)¶
The model assumes a single set of coefficients across multiple threshold logits:
$$ \log \left( \frac{P(Y \leq j)}{P(Y > j)} \right) = \theta_j - X \cdot \beta $$
θ_j= intercept (cutoff) for category jβ= shared coefficient vector
⚙️ 3. Fitting the Model¶
# Python (mord package)
from mord import LogisticIT
model = LogisticIT().fit(X, y)
# R (MASS package)
polr(y ~ x1 + x2, data = df, method = "logistic")
✔️ Encode target labels as ordered integers (0, 1, 2, ...)
📊 4. Output Interpretation¶
| Output | Meaning |
|---|---|
| Coef (β) | Effect on odds of being in higher category |
| Intercepts (θ_j) | Logit cutoffs between class levels |
exp(coef) |
Proportional odds ratio per feature |
✔️ “A 1-unit increase in X increases odds of being in a higher category by OR.”
🧪 5. Assumptions¶
| Assumption | Notes |
|---|---|
| Proportional Odds | Effect of X is consistent across class splits |
| Linearity in logit | X must relate linearly to cumulative logit |
📉 If violated: Consider adjacent-category models or partial proportional odds models
✅ Modeling Checklist¶
- [ ] Target verified as ordered (e.g. ordinal categories or numeric codes)
- [ ] Model fit with ordinal-compatible library (mord, polr, etc.)
- [ ] Intercepts and β interpreted with respect to category ordering
- [ ] Proportional odds assumption considered or tested
💡 Tip¶
“Ordinal logistic doesn’t ask which class — it asks how far up the scale you’re likely to go.”