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Confidence intervals of ratios Risk ratio, odds ratio, and rate ratioRatios for summarising relationships Risk ratio confidence interval Odds ratio confidence interval Confidence interval of incidence rate ratioRatios for summarising relationshipsWe introduced the use of risk ratios, odds ratios and incidence rate ratios as measures of association for binary variables in
The confidence interval of a ratio provides a measure of the reliability of the estimate of the ratio. It is common practice to also use the confidence interval as a surrogate statistical test. This is unwise  a significance test (such as Pearson's chi square test or Fisher's exact test) and a confidence interval around a ratio should instead be considered as complementary. The test is to formally assess a null hypothesis and the interval gives an indication of reliability of the estimate. As with the confidence intervals we have met before, large sample normal approximation intervals are those most commonly used. We give below computational details for calculating these intervals. But such intervals are only valid for large samples and, even then, may be misleading if some proportions or odds are very small.
Confidence interval of risk ratioLarge sample normal approximationA transformation is required for risk ratios to be approximately normal. The most appropriate transformation is the natural logarithm of the risk ratio. An approximate estimate of the standard error of the log risk ratio (lnRR) is given by:
The 95% Wald confidence interval of the risk ratio is then given by:
If the frequencies are suitably large (none less than 5), and the risk ratio not too extreme, the errors can be accepted as 'approximately' normal. Exact methods and other approximationsExact intervals sensu stricto do not exist for the risk ratio  other than by Monte Carlo. However Thomas & Gart (1977) suggest an "exact" type method based on fixed marginals in the 2 × 2 table. R provides a bootstrap interval which gives a somewhat wider interval than the simple formulae. There are a number of approximate methods which are suitable for application to small samples which are available in various software packages. One approach is to invert a single twosided test using the Wilson score statistic  we term the resulting interval the score method interval. The methods of Koopman
Confidence interval of odds ratioLarge sample normal approximationAgain a transformation is required for the odds ratio to be approximated by a normal distribution. The most appropriate transformation is the natural logarithm of the odds ratio (lnOR). An approximate estimate of the standard error of the log odds ratio is given by the square root of the sum of the reciprocal of the cell frequencies:
If these cell frequencies are suitably large (none less than 5), and the odds ratio not too extreme, the errors can be accepted as 'approximately' normal. The 95% Wald confidence interval of the odds ratio is then given by:
If the frequencies are suitably large (none less than 5), and the odds ratio not too extreme, the errors can be accepted as 'approximately' normal. Exact methods and other approximationsThere is an exact confidence interval for the odds ratio based on the nonnull hypergeometric model  which we term the conditional exact interval. The oddsratio function, provided by the 'epitools' package for R, gives 'exact' midP confidence intervals, and Fisher exact intervals. The most commonly used method (that used by StatXact for example) was suggested by Cornfield. It consists of inverting two separate onesided tests  an approach termed the tail method. This interval is attached not to the conventional odds ratio but to the conditional maximum likelihood estimate of the odds ratio. This differs slightly from the conventional sample odds ratio. Both the interval and the maximum likelihood estimate of the odds ratio have to be obtained iteratively. Because the conditional exact odds ratio is much more discrete than the unconditional odds ratio, Cornfield intervals can be extremely conservative. Hence there are strong theoretical reasons for applying midP criteria to the testinversion when obtaining this interval. In this case the medianunbiased odds ratio is used instead of the conditional odds ratio. The 'epitools' package for R gives this interval as the midp exact interval. Cornfield, and later Fisher, proposed a largesample approximation to Cornfield's exact interval for odds ratios  which we term the Cornfield approximate interval. It is much easier to evaluate than the null hypergeometric probability function and provides the equivalent of the midp exact interval.
Confidence interval of incidence rate ratioLarge sample normal approximationAgain a transformation is required for it to be approximated by a normal distribution. An approximate standard error of the log rate ratio is given by:
The 95% confidence limits of the rate ratio are then given by:
Exact methods and other approximationsAn exact (and midP exact) interval for a rate ratio can be obtained by treating the total number of cases as fixed  so computation of expected values and their variance is done conditional on the observed case margin total. This is a similar approach to that used for estimating an exact confidence interval for the conditional odds ratio. The midP exact interval is given by the 'epitools' package for R. Another large sample approximate confidence interval of the incidence rate ratio (IR) can be calculated based on the Poisson distribution (see Woodward (2004)). First the probability that an event occurs in (say) population 2 is calculated as e_{2} / (e_{1}+e_{2}). The upper and lower confidence limits for this proportion () are then obtained using the normal approximation, namely ±1.96√{[(1)] / (e_{1}+e_{2})} The lower and upper limits for limit for IR are then given by:
