Binary Association Coefficient

In statistics and data science, quantifying the relationship between two categorical variables is a fundamental task. When both variables are binary (taking only values 0 or 1), the Binary Association Coefficient provides an elegant measure of their correlation strength and direction.

This coefficient is mathematically equivalent to the Pearson correlation coefficient when applied specifically to two binary variables. It computes a value between -1 and +1:

• A value of +1 indicates perfect positive association: when one variable is 1, the other is also 1 (and similarly for 0)
• A value of -1 indicates perfect negative association: when one variable is 1, the other is 0 (and vice versa)
• A value of 0 indicates no linear association: knowing one variable tells you nothing about the other

The Four-Cell Contingency Table:

Given two binary sequences x and y of equal length, we first construct a 2×2 contingency table counting co-occurrences:

	y = 1	y = 0
x = 1	a (both 1)	b (x=1, y=0)
x = 0	c (x=0, y=1)	d (both 0)

Where:

• a = count of positions where both x and y are 1
• b = count of positions where x is 1 and y is 0
• c = count of positions where x is 0 and y is 1
• d = count of positions where both x and y are 0

The Association Coefficient Formula:

$$\phi = \frac{ad - bc}{\sqrt{(a+b)(c+d)(a+c)(b+d)}}$$

This formula computes the normalized difference between the diagonal products of the contingency table. The denominator normalizes the result to the [-1, +1] range.

Your Task: Implement a function that takes two binary sequences (lists of 0s and 1s) and returns their binary association coefficient, rounded to 4 decimal places. Both sequences will always have the same length.