I am wondering if there is an example of a two-by-two contingency table dataset where an Odds Ratio estimate would work and a Risk Ratio would fail, or vice versa? In additional are there cases where a Risk Difference would fail as well? Thanks.
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Depends on what you mean by "fail". You have a problems to calculate (i.e. the likelihood is maximized on the boundary of the parameter space $(0, \infty)$) the maximum likelihood estimates of the odds and risk ratio in a single 2 by 2 table, if any cell has a zero entry. In contrast the risk difference can still be calculated. Of course, there are a lot of possible approaches for still getting an odds or risk ratio estimate in these circumstances (e.g. Bayesian approaches, continuity corrections, median unbiased estimates etc.).
Once you go to stratified tables or use the estimate from one table for a new table, you run into situations that the odds ratio handles better than the risk ratio or risk difference. E.g. $$\begin{array} . & Yes & No \\ A & 3 & 9 \\ B & 1 & 12 \end{array}$$ The odds ratio for A versus B is 4, the risk ratio 3.25 and the risk difference about 0.17. Then you have this second table $$\begin{array} . & Yes & No \\ A & 48 & 1 \\ B & 12 & 1 \end{array}$$ The odds ratio for A versus B is also 4. However, if you assumed a risk ratio of 3.25 on top of the proportion of 0.923 for group B, then you would get a proportion of 3.0 for group A. This is of course nonsense, because a proportion needs to be in [0, 1,]. Similarly, if you were to add a risk difference of 0.17 to 0.923 you get a proportion for group A that is >1.
This illustrates how risk ratios and risk differences can fail in the sense that they cannot be constant across extremely different proportions for group B. At least mathematically the odds ratio does not have that problem, but of course that does not mean that the odds ratio would truly stay constant across different group B proportions in any real practical problem. You do not notice that kind of issue in 2 by 2 tables.
