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Confidence intervals of ratios Risk ratio, odds ratio, and rate ratio
Worked example I
The risk ratio is an appropriate summary measure to use to assess the association between treatment and outcome. Since sample sizes are reasonably large we attach a Wald normal approximation
The statistic and confidence interval as calculated above are the same as those given by the riskratio function, of epitools package for R, for the normal approximation (Wald) confidence interval: Risk ratio = 1.052
The interval widely overlaps 1.0 suggesting that vitamin E has no significant effect on the incidence of macular degeneration. This conclusion is supported by the non-significant P-value from a Pearson's chi square test (0.826).
Our second example uses a result from a cross-sectional survey on the prevalence of dystocia in
In this case one of the sample sizes is small and one of the proportions is small. Hence the Wald interval calculated
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) mid-P 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 R-function 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 mid-P value however was 0.0876, somewhat above the conventional 0.05 level.