Goodman-Kruskal Gamma & Kendall’s Tau-b
Measuring association strength and direction in ordinal data
Why Not Just Chi-Square?
Chi-square tests IF variables are related, but ignores order. Gamma and Tau-b use the natural ordering of categories to tell you how strong and in what direction the association is.
Core Concept: Concordant, Discordant & Tied Pairs
For any two observations (Person A vs Person B):
| Type | Condition | Meaning |
|---|---|---|
| Concordant (C) | Both variables go same direction | Consistent pattern |
| Discordant (D) | Variables go opposite directions | Inconsistent pattern |
| Tied | At least one variable is equal | Can’t determine direction |
CONCORDANT: X: A > B and Y: A > B (or both lower)
DISCORDANT: X: A > B and Y: A < B (or vice versa)
TIED: X: A = B (or Y: A = B)
Goodman-Kruskal Gamma (γ)
Key Idea
Out of all pairs where we CAN determine direction (ignoring ties), what proportion are concordant?
Formula
γ = (C - D) / (C + D)
Ties are completely ignored.
Interpretation
| γ | Meaning |
|---|---|
| +1 | Perfect positive association |
| 0 | No association |
| −1 | Perfect negative association |
PRE interpretation: |γ| = how much knowing X reduces your errors in predicting the order of Y.
Kendall’s Tau-b (τ_b)
Key Idea
Like gamma, but penalises for ties — gives a more conservative estimate.
Formula
τ_b = (C - D) / √[(C + D + T_x)(C + D + T_y)]
Where T_x = pairs tied on X only, T_y = pairs tied on Y only.
Key property
|τ_b| ≤ |γ| (tau is always ≤ gamma in magnitude)
Gamma vs Tau-b
| Feature | Gamma (γ) | Tau-b (τ_b) |
|---|---|---|
| Treats ties | Ignores | Penalises |
| Magnitude | Larger | Smaller |
| Best when | Many ties, want simple interpretation | Want conservative/correlation-like measure |
Rule of thumb: If gamma and tau-b are very different, there are many ties in your data.
Python
from scipy.stats import kendalltau
# Convert contingency table to long format first
tau, p_value = kendalltau(rater_A, rater_B)
print(f"Kendall's τ_b = {tau:.4f}, p = {p_value:.4f}")For gamma, count C and D manually from the table:
gamma = (C - D) / (C + D)When to Use Each Measure
| Goal | Use |
|---|---|
| Test IF association exists | Chi-square |
| Measure direction + strength (ordinal, simple) | Gamma |
| Measure direction + strength (conservative) | Tau-b |
Use chi-square and gamma/tau-b together — they answer different questions.
What to Report
“A chi-square test indicated a significant association (χ² = 89.3, p < 0.001). Ordinal measures confirmed strong positive concordance: γ = 0.85, τ_b = 0.75.”