It may seem surprising that we often find models that fit
like this: big changes in Background and Outcome over Period are
not accompanied by a change in the association between Background
and Outcome.
However, some simple processes of allocation can give rise to
near constant odds ratios.
Assume outcomes are decided according to a threshold
measurement: anyone over a certain score gets the good outcome.
Increases in the supply of the outcome cause a lowering of the
threshold.
Assume we have two subpopulations with different
distributions of the score variable: let's say same standard
deviation but different mean.
The odds ratio is defined by the intersection of the
distributions and the threshold.
What happens to the OR as we increase the supply?
This depends on the distribution: if it is a logistic
distribution (very like the normal distribution with slightly
fatter tails, but easier to deal with mathematically), the odds
ratio is constant as long as the difference in means is constant.
If the distribution is normal, the odds ratio changes quite
slowly: a large change in the supply/threshold may have little
effect on the odds ratio.
The odds ratio will be most close to equal the closer the
threshold is to the means of the distributions: it has a shallow
U-shaped curve.
In summary, two points:
It's not necessarily surprising when odds ratios are
relatively stable despite big marginal change.
Odds ratios may directly relate to simple processes
generating the data: they are not necessarily statistical
abstractions.
