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Confidence intervals of ratios Risk ratio, odds ratio, and rate ratio

Worked example I

Our first example uses results from a randomized trial on the effect of vitamin E supplementation on the incidence of macular degeneration.

Effect of Vitamin E on the incidence of macular degeneration
TypeTreatmentNo. positiveNo negative% positiveRisk ratio (RR)
EarlyTreated353748.561.05
Placebo343848.131.00

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 interval to estimate the standard error:

Using
SE(lnRR)   =     = √0.05315 = 0.23054
1 - 1 + 1 - 1
3540934418

95% CI (lnRR)   =    ln (1.05)  ± (1.96 0.23054)
    =   0.40307 - 0.50065

95% CI (RR)   =    0.67 - 1.65

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 (0.670 - 1.653).

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).

Worked example II

Our second example uses a result from a cross-sectional 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
BreedNo. positiveNo negative% positiveOdds ratio
OR
Risk ratio
RR
Manx1175.5515.68614.87
Colony3 8000.37(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:

Using
SE(lnOR)   =     = √1.393407 = 1.180427
1+1+1+1
1173800

95% CI (lnOR)   =    ln (15.686)  ± (1.96 1.180427)
    =   0.439133 - 5.066405

95% CI (OR)   =    1.551 - 158.60

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.

Using