Odds Ratios and what they mean

• 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).