# Is a very large odds ratio from a contingency table an issue?

I computed an odds ratio from a contingency table (exposure/disease). My odds ratio ended up being 3329.044 which is ridiculously large. The reason it is large is because the prevalence of the disease in the unexposed group is very small and the number of individuals in the unexposed group is very large. My question is how would I interpret such a large odds ratio? Would I just say it suggests a very large association between the exposure and disease? Is there an alternative measure I should use to quantify the association? The 2x2 is below..

Disease    No Disease
Exposed       73          78
Unexposed     33        117383
• How was the sampling performed?
– LSC
Jan 29, 2020 at 18:34

Norm Breslow argued that we should prefer the odds ratio as an association measure precisely because it's possible to have an odds ratio of exactly $$\infty$$. Some exposures in reality have deterministic relationship with the outcomes, but relative risks have theoretical upper bounds that depend on the "design" (margins of the contingency table).

If the design of this study does not employ outcome dependent sampling, the most conservative measure of association (both statistically and practically) comes from reporting the risk difference. So much so some journals require the RD to be reported as an effect so as to preclude any exaggeration of effects (such as interpreting an odds ratio as a risk ratio).

The model is easily estimated along with 95% CIs in R:

x <- c(0,1,0,1)
y <- c(0,0,1,1)
w <- c(117383, 78, 33, 73)

fit <- glm( y ~ x, weights=w, family=binomial(link='identity'))

And we find the risk of disease is 48.3% greater among exposed participants compared to unexposed (95% CI: 40.4% - 56.3%).

If the study does actually use outcome-dependent sampling, you will need to correct the design by weights to obtain correct estimates and 95% CIs. Only the odds ratio is invariant to the design in this sense.

• this is very informative, thanks! Jan 29, 2020 at 21:26

It looks like those who were exposed have a WAY higher risk of disease. The disease risk little under 50% in the exposed group and is .02% in the unexposed group. That is consistent with a massive odds ratio. What's the problem?

I guess you could use the absolute risk difference, or the risk ratio, as alternative measures but you're going to find, no matter what you do, that this exposure is strongly related to disease risk.