Total $(B)$ $(\beta)$ $N$ Where:

$(AB)$: Number of units possessing both A and B.
$(A\beta)$: Number of units possessing A but not B.
$(\alpha B)$: Number of units possessing B but not A.
$(\alpha \beta)$: Number of units possessing neither A nor B.
$(A) = (AB) + (A\beta)$: Total number of units possessing A.
$(B) = (AB) + (\alpha B)$: Total number of units possessing B.
$N = (A) + (\alpha) = (B) + (\beta) = (AB) + (A\beta) + (\alpha B) + (\alpha \beta)$: Total number of observations.

3.2.2 Measures of Association

Yule's Coefficient of Association ($Q$):
- Description: A measure of association for $2 \times 2$ contingency tables. It is based on the ratio of cross-products of cell frequencies.
- Formula: $$Q = \frac{(AB)(\alpha\beta) - (A\beta)(\alpha B)}{(AB)(\alpha\beta) + (A\beta)(\alpha B)}$$
- Range: $-1 \le Q \le +1$.
  - $Q = +1$: Perfect positive association (A and B always occur together).
  - $Q = -1$: Perfect negative association (A and B never occur together).
  - $Q = 0$: No association (attributes are independent).
- Features: Useful for binary attributes. It's not affected by the sample size or the marginal totals, making it relatively robust. However, it can give a strong association even if frequencies in some cells are very small.
Coefficient of Colligation ($K$):
- Description: Another measure of association, often considered more stable than $Q$ when some cell frequencies are zero.
- Formula: $$K = \frac{\sqrt{(AB)(\alpha\beta)} - \sqrt{(A\beta)(\alpha B)}}{\sqrt{(AB)(\alpha\beta)} + \sqrt{(A\beta)(\alpha B)}}$$
- Relationship to Q: $Q = \frac{2K}{1+K^2}$
- Range: $-1 \le K \le +1$. Interpretation is similar to $Q$.
Chi-square Test for Independence: While not a measure of association, the Chi-square test (discussed in Section 9.1.3) is used to statistically determine if an association exists between two categorical variables.

Definition: A set of observed frequencies (class frequencies) is said to be consistent if they do not violate any fundamental laws of probability or logic, meaning that no theoretical frequency should be negative.
Conditions for Consistency (for two attributes A and B):
- All class frequencies of order zero and one must be non-negative:
  - $(A) \ge 0$, $(B) \ge 0$, $(\alpha) \ge 0$, $(\beta) \ge 0$
- All class frequencies of higher orders must be non-negative:
  - $(AB) \ge 0$, $(A\beta) \ge 0$, $(\alpha B) \ge 0$, $(\alpha \beta) \ge 0$
- In addition, the following fundamental inequalities must hold:
  - $(AB) \le (A)$ and $(AB) \le (B)$
  - $(\alpha B) \le (\alpha)$ and $(\alpha B) \le (B)$
  - $(A\beta) \le (A)$ and $(A\beta) \le (\beta)$
  - $(\alpha \beta) \le (\alpha)$ an