Worked example II
Our second example uses a result from a crosssectional survey on the prevalence of dystocia in cats. We previously looked at this work in relation to the confidence intervals attached to the prevalence estimates. We will calculate both the odds ratio (as used by the authors) and the risk ratio with their accompanying intervals.
Prevalence of dystocia in cats in relation to breed 
Breed  No. positive  No negative  % positive  Odds ratio OR  Risk ratio RR 
Manx  1  17  5.55  15.686  14.87 
Colony  3  800  0.37  (1.00)  (1.00)

In this case one of the sample sizes is small and one of the proportions is small. Hence the Wald interval calculated below may be unreliable, so we would do better to also calculate a conditional exact interval using the epitools oddsratio function for R:
For the odds ratio in R we obtain the same for the Wald interval (OR = 15.69, 95% CI 1.55 to 158.60), but the conditional exact interval overlaps 1 (OR = 15.48, 95% CI 0.28 to 204.67), as does the (more reliable) midP interval (OR = 16.77, 95% CI 0.56 to 153.09). Hence it is now highly questionable whether we have actually demonstrated that there is any difference between breeds.
For the risk ratio we obtained a risk ratio of 14.87 with a Wald interval of 1.62 to 136.2, the same as those given by the epitool package riskratio function for the normal approximation (Wald) confidence interval. Using the same Rfunction the Wald normal approximation, with small sample adjustment, gave a risk ratio of 11.17 with an interval 1.22 to 102.25. The exact midP value however was 0.0876, somewhat above the conventional 0.05 level.
Using