Open and Distance Learning

The odds-ratio and the 2 x 2 table

The odds-ratio is a measure that describes the degree of association in a 2 x 2 table. If a joint probability distribution is given in a 2 x 2 table

Y

X

	1	2
1	p11	p12	p1.
2	p21	p22	p2.
	p.1	p.2

within row 1 the odds is W ₁ = p₁₂ / p₁₁, and within row 2 the odds is W ₂ = p₂₂ / p₂₁ (similarly we could start from column 1 and column 2). The odds-ratio is

If independence is true, in an r x c table, p_ij = p_i. p_.j and, in a 2 × 2 table,

For demonstration, let us consider row 1 in the 2 x 2 table
(similarly we could start from row 2, or column 1 or column 2).
When independence is true,
p₁₁ = p_1. p_.1, and p₁₂ = p_1. p_.2, that is, p_1. = p₁₁ / p_.1, and p_1. = p₁₂ / p_.2.
Hence,

from which it follows that p₁₁ p_.2 = p₁₂ p_.1, where p_.2 = p₁₂ + p₂₂, and p_.1 = p₁₁ + p₂₁. Thus the result is p₁₁ p₂₂ = p₁₂ p₂₁. Hence, if independence is true,

p₁₁ p₂₂p₁₂ p₂₁
= 1.

In most applications the population probabilities p_ij are unknown and hence is q.
For sample cell frequencies {n_ij} a sample analogue of q is

^
q
  =

The log transformation is

log( ^
q
  ) = log(n₁₁) + log(n₂₂) - log(n₁₂) - log(n₂₁) .

If independence is true, ^
q
  = 1, hence log( ^
q
  ) = 0 .

If log( ^
q
  ) > 0 , then there is a positive covariation of the two cross-classified variables.

If log ( ^
q
  ) < 0 , then there is a negative covariation.

.