ASSOCIATION: Bivariate Ordinal/Nominal Distribution
Theta (): = Di/T_{2 }
Where: T_{2} = Sum of product of nominal totals multiplied
by each other nominal total, taking them 2 at a time.
For instance, if the totals are 3, 4, 7, 2: T_{2} =
3x4 + 3x7 + 3x2 + 4x7 + 4x2 + 7x2 = 89.
Di = | Ai - Bi |
Where: Ai = For any pair of nominal categories, take the frequency
in each cell of the first category and multiply it by the sum
of the frequencies in the cells to the right of it in the
second nominal category. Sum these products.
Bi = For any pair of nominal categories, take the frequency
in each cell of the first category and multiply it by the sum
of the frequencies in the cells to the left of it in the
second nominal category. Sum these products.
*It is crucial that
table is set up so that nominal categories are on the side and
ordinal categories are across the top.
Example:
Number of job offers | ||||||
Major | 20-24 | 18-19 | 17 | 16 | 13-15 | |
Criminal Justice | 10 | 10 | 2 | 1 | 1 | 24 |
Public Administration | 1 | 2 | 10 | 10 | 1 | 24 |
Child Development | 1 | 1 | 2 | 2 | 10 | 16 |
12 | 13 | 14 | 13 | 12 | 64 |
Q: What is the association between major and job offers?
A: CJ = criminal justice
PA = public administration
CD = child development
Di _{(CJ,PA)} = ^{Ai}^{ }| [10(2+10+10+1)
+ 10(10+10+1) + 2(10+1) + 1(1)]
- ^{Bi}[1(1+2+10+10) + 1(1+2+10) + 2(1+2) + 10(1)]
| = 411
Di _{(CJ,CD)} = ^{Ai }| [10(1+2+2+10) +
10(2+2+10) + 2(2+10) + 1(10)]
- ^{Bi}[1(1+1+2+2) + 1(1+1+2) + 2(1+1) + 10(1)]
| = 300
Di _{(PA,CD)} = ^{Ai} | [1(1+2+2+10) +
2(2+2+10) + 10(2+10) + 10(10)]
- ^{Bi}[1(1+1+2+2) + 10(1+1+2) + 10(1+1) + 2(1)]
| = 195
Theta (): = (411 +300 +195) / [24(24)
+ 24(16) + 24(16)] = .67
or Change to percent: "In % of the comparisons made,
(nominal variable) showed systematic differences in
(ordinal variable).
From example above:
.67 = "A moderately large association between major and number
of job offers." Or
"In 67% of the comparisons made, persons with different
majors showed systematic differences in their number of job offers.