Two-tailed approximate confidence intervals for the ratio of proportions

Abstract

Various approximate methods have been proposed for obtaining a two-tailed confidence interval for the ratio R of two proportions (independent samples). This paper evaluates 73 different methods (64 of which are new methods or modifications of older methods) and concludes that: (1) none of the classic methods (including the well-known score method) is acceptable since they are too liberal; (2), the best of the classic methods is the one based on logarithmic transformation (after increasing the data by 0.5), but it is only valid for large samples and moderate values of R; (3) the best methods among the 73 methods is based on an approximation to the score method (after adding 0.5 to all the data), with the added advantage of obtaining the interval by a simple method (i.e. solving a second degree equation); and (4) an option that is simpler than the previous one, and which is almost as effective for moderate values of R, consists of applying the classic Wald method (after adding a quantity to the data which is usually $z_{\alpha /2}^{2}/4$).

Appendix: Approximate value of the maximum likelihood estimator

Martín Andrés et al. (2011) demonstrated that under the null hypothesis H:L=λ, with L=β ₁ p ₁+β ₂ p ₂, the maximum likelihood estimators p _iE of the p _i are in the solution for equations $n_{i}( \bar{p}_{i} - p_{i\mathrm{E}} )/\beta_{i}p_{i\mathrm{E}}q_{i\mathrm{E}}=C (\forall i)$, when C is a constant that remains to be determined. This indicates that for i=1 one must have $n_{1}\beta_{1}\bar{p}_{1} - n_{1}\beta_{1}p_{1\mathrm{E}} = C\{ \beta_{1}^{2}p_{1\mathrm{E}} - \beta_{1}^{2}p_{1\mathrm{E}}^{2} \}$, whilst for i=2 it will be $n_{2}\beta_{2}\bar{p}_{2} - n_{2}\beta_{2}p_{2\mathrm{E}} = C\{ \beta_{2}^{2}p_{2\mathrm{E}} - \beta_{2}^{2}p_{2\mathrm{E}}^{2} \}$. As β ₂ p _2E=λ−β ₁ p _1E, by substituting in the second equality above it can be deduced that $n_{2}\beta_{2}\bar{p}_{2} - n_{2}\lambda + n_{2}\beta_{1}p_{1\mathrm{E}} = C\{ \lambda( \beta_{2} - \lambda) - \beta_{1}^{2}p_{1\mathrm{E}}^{2} + \beta_{1}p_{1\mathrm{E}}( 2\lambda - \beta_{2} ) \}$, so that by subtracting this equality from the one in the case i=1 and working out p _iE one gets $p_{1\mathrm{E}} = \{ ( n_{2}\beta_{2}\bar{p}_{2} - n_{1}\beta_{1}\bar{p}_{1} - n_{2}\lambda) - \lambda( \beta_{2} - \lambda)C \}/ \{ - n\beta_{1} + \beta_{1}C( 2\lambda - \beta_{1} - \beta_{2} ) \}$. By doing the division, and disregarding all terms lower than or equal to −1, one can deduce that $p_{1\mathrm{E}} \approx p_{1\mathrm{A}} = ( n_{1}\beta_{1}\bar{p}_{1} - n_{2}\beta_{2}\bar{p}_{2} + n_{2}\lambda) / ( n\beta_{1} )$, and given that $p_{2\mathrm{E}} = ( \lambda - \beta_{1}p_{1\mathrm{E}} )/\beta_{2},p_{2\mathrm{E}} \approx p_{2\mathrm{A}} = ( - n_{1}\beta_{1}\bar{p}_{1} + n_{2}\beta_{2}\bar{p}_{2} + n_{1}\lambda) / ( n\beta_{2} )$ where p _2A=(λ−β ₁ p _1A)/β ₂.

When the interest is on the hypothesis H:d=δ for the difference of proportions d=p ₂−p ₁, then β ₁=−1,β ₂=+1,λ=δ and the previous expression indicates that p _1A=(a ₁−n ₁ δ)/n and p _2A=(a ₁+n ₂ δ)/n: the classic conditioned estimators of Dunnett and Gent (1977).

When the interest lies in the hypothesis H:R=ρ for the ratio of proportions R=p ₂/p ₁ then, as the previous hypothesis is equivalent to H:L=p ₂−ρp ₁=0, one will have β ₁=−ρ,β ₂=+1 and λ=0. Substituting these values in the two previous expressions for the p _iA we obtain the estimators for expression (12), which have been corrected in order to avoid p _iA being greater than 1.