After a very long leave of absence I return to the issue of extracting the effect estimate () and standard error () from reported and (rounded to a fixed number of decimal points) relative risk (), limits of 95% confidence intervals ( and ) and p-value () figures found in scientific publications. This solution is a Bayesian one, requiring nothing more than a straightforward application of the Bayes theorem for the posterior distribution of the
A straightforward application of Bayes theorem for the quantities given the :
However , because the rounding operation only depends on the value to be rounded (thus dropping conditioning on all other variables but ). Since rounding to digits is a deterministic operation, admitting only one value () for a specific , the conditional probability is essentially a point mass function at given by the Kronecker function.
When viewed as a parametric function of , this conditional probability will be non-zero in the interval and zero otherwise.
Further application of the Bayes theorem yields:
The last factor in the right hand side of the formula is simply the (suitably transformed) joint probability function obtained from the pair as discussed in the previous post . More specifically, it is the probability of obtained by taking the logarithm of and the ratio of conditioning on the pair .
On the other hand, p-value calculations are based on the ratio , allowing us to drop conditioning of on . Applying the same line of reasoning as in the case of the conditional probability is mutatis mutandis equal to one when is in the interval and zero otherwise.
To compute the Bayesian estimates of we can resort to a rejection type of algorithm:
- Simulate a pair of values from the joint density
- Transform to
- Reject if is not in the interval
- Reject if the corresponding p-value is not in the interval
Steps 1,2 ensure that the are compatible with the limits of the confidence interval, while steps 3,4 guarantee compatibility with the reported relative risk and p-value respectively. The non-rejected Monte Carlo samples can be summarized by various statistics (e.g. the mean or the median) to yield point estimates for meta-analytic applications.
- True Values:
- Rounding (to 2 figures): ,
- Result of steps 1,2 are shown in the figure below (the blue cross corresponds to the true values of :
- Rejecting the pairs that are not compatible with the RR point estimate (step 3) yields the following two dimensional density:
- Applying the criterion in step 4 does not reject further samples in this particular example so the corresponding image is virtually indistinguishable from the previous one:\
Point estimates for the two log-relative risk and the standard error are: 0.4043738 and 0.1131314 respectively. These are in close agreement the true values of 0.403876 and 0.11345 !