We understand the definiton of the odds ratio, and how it is one
way of parameterising association: ORs plus margins give you the
whole table.
What of interpreting odds ratios in context? What does an OR
mean?
In a simple table: white victim panel of the death penalty
data:
Defendent's
Death Penalty
race
Yes
No
White
53
414
Black
11
37
(Source: Agresti p. 54ff.)
The odds ratio for whites versus blacks getting versus not
getting the death penalty is (53/414)/(11/37) = 0.43. That is,
the odds of a white person convicted of killing a white person
getting the death penalty are only 43% of those of a black
person.
How do we map this arithmetic concept onto social-science or
everyday concepts? Consider the general case of an OR describing
the relationship between two types of people and two outcomes,
one of which is preferable, e.g., people from two rich versus poor
families, completing or not completing second level-education.
Here we have two tables with the same odds ratios but
different margins:
Education
Yes
No
Period 1
OR=4.2
Rich
12
188
Poor
3
198
Period 2
OR=4.2
Rich
103
97
Poor
40
160
What has happened between the two periods? The provision of
education has increased enormously, and both rich and poor people
have taken advantage of it. However, the rich group have much
more of the extra places, so that their odds of completing have
risen by as much as have the odds for poor people.
So, if we're concerned with social justice, is this progress
or not? What does it mean in social-justice terms that the OR
stays constant?
At one level there is unmistakable progress: more education
all round. But the relative effect of family background
has remained constant.
The odds ratio measures what might be
considered equality of opportunity: as it deviates from one there
is inequality, and if its size remains the same the inequality of
opportunity remains the same, even though the `quantity' of
opportunity changes.
This is an artificial example, but it is not unrealistic. In
terms of loglinear models, it is the case that arises when a
model like B*O + B*P + O*P fits well (Background,
Outcome, Period) suggesting that there is no 3-way B*O*P
interaction.
That is, that the B*O interaction is not
significantly different at the different levels of P.
That is, the odds-ratios that the B*O interaction term
represents are constant despite the changes in the distribution
of Outcome (O*P) and changes in the distribution of
Background (B*P).
