Unit V · Lecture 39 of 40 · B.Sc. Data Science, 1st Year, Semester II
Y = (√ad − √bc) / (√ad + √bc)
Where a, b, c, d are the four cells of the 2×2 table:
a = (AB), b = (Aβ), c = (αB), d = (αβ)
"Colligation" means "binding together" — Y measures how strongly the two attributes are "bound" to each other.
Range: −1 ≤ Y ≤ +1
Y = +1: Perfect positive association (√bc = 0, i.e., b=0 or c=0)
Y = −1: Perfect negative association (√ad = 0, i.e., a=0 or d=0)
Y = 0: No association (ad = bc, i.e., independence)
Symmetric: Y(A,B) = Y(B,A)
Unit-free: Pure number, no units
Yule introduced Y as a measure of the "binding" or "tying together" of two attributes. It uses the geometric mean approach (√ad and √bc) rather than the direct products (ad and bc) used in Q.
The square root transformation makes Y less sensitive to extreme values in any single cell and provides a more balanced measure of association.
| Feature | Yule's Q | Yule's Y |
|---|---|---|
| Formula | (ad−bc)/(ad+bc) | (√ad−√bc)/(√ad+√bc) |
| Range | −1 to +1 | −1 to +1 |
| Based on | Products ad and bc | Geometric means √ad and √bc |
| Q = 0 iff | ad = bc | Same (ad = bc) |
| Sensitivity | More sensitive to large cell values | Less sensitive (square root dampens extremes) |
| Relationship | |Y| ≤ |Q| always (Y is always closer to 0 than Q) | |
| When Q = ±1 | Y = ±1 (same) | Y = ±1 (same) |
| When Q = 0 | Y = 0 (same) | Y = 0 (same) |
| Preferred by | More commonly used in exams | Some textbooks prefer Y for robustness |
Key inequality: |Y| ≤ |Q| for all data. Y is always closer to zero than Q.
When are they equal? Q = Y only when Q = Y = 0, Q = Y = +1, or Q = Y = −1.
Conversion formula: Q = 2Y / (1 + Y²) and Y can be found from Q by solving this quadratic.
Using the same data as Lecture 38, Example 1:
| Disease (B) | No Disease (β) | |
|---|---|---|
| Smoker (A) | 120 | 80 |
| Non-smoker (α) | 60 | 240 |
a = 120, b = 80, c = 60, d = 240
√ad = √(120×240) = √28800 = 169.71
√bc = √(80×60) = √4800 = 69.28
Y = (√ad − √bc) / (√ad + √bc)
= (169.71 − 69.28) / (169.71 + 69.28)
= 100.43 / 238.99
= 0.4202
Q = 0.714 | Y = 0.420
|Y| = 0.420 < |Q| = 0.714 ✓
Both indicate positive association, but Y is more conservative (closer to zero). Q suggests "strong" association while Y suggests "moderate" — the truth depends on context.
Compute Q and Y for: a=90, b=10, c=10, d=90.
Q = (90×90 − 10×10) / (90×90 + 10×10)
= (8100 − 100) / (8100 + 100) = 8000/8200 = 0.9756
Y = (√8100 − √100) / (√8100 + √100)
= (90 − 10) / (90 + 10) = 80/100 = 0.80
Q = 0.976 | Y = 0.800
Q is very close to 1, suggesting near-perfect association. Y is more moderate at 0.80. The square root in Y dampens the effect of the large diagonal products (ad = 8100).
Enter the four cell frequencies to compute both Q and Y:
(A) = (AB) + (Aβ)
(B) = (AB) + (αB)
N = (A) + (α) = (B) + (β)
(α) = N − (A), (β) = N − (B)
max(0, (A)+(B)−N) ≤ (AB) ≤ min((A),(B))
All cross-class frequencies ≥ 0
(AB) = (A)(B)/N
Positive: (AB) > (A)(B)/N
Negative: (AB) < (A)(B)/N
χ² = N(ad−bc)²/[(a+b)(c+d)(a+c)(b+d)]
φ = √(χ²/N)
C = √(χ²/(χ²+N))
Q = (ad−bc)/(ad+bc)
Y = (√ad−√bc)/(√ad+√bc)
|Y| ≤ |Q| always
Q = 2Y/(1+Y²)
Q1: An attribute is:
Q2: Given N=200, (A)=120, (B)=150. The minimum value of (AB) for consistency is:
Q3: A and B are independent when:
Q4: If (AB) > (A)(B)/N, the attributes are:
Q5: The φ coefficient equals 0 when:
Q6: For a=50, b=20, c=30, d=100, Yule's Q = ?
Q7: Yule's Y uses:
Q8: The relationship between Q and Y is:
Q9: Pearson's C can never reach:
Q10: Q = +1 occurs when:
You have completed all 5 units of BDS113 Introduction to Statistics. Best of luck with your exams!
Unit V · Lecture 39 · BDS113 Introduction to Statistics · B.Sc. Data Science, Semester II