In a previous post I presented a possible solution to the extraction of standard errors and hazard ratios from publications in which only rounded approximations to the risk ratios and the associated 95% confidence interval are considered.
This solution, is subject to a criticism that I will now discuss : it suffers from an internal contradiction. Specifically, while one can use two of the three pieces of data (the actual risk ratio and the two limits of the 95% CI) simultaneously, one will arrive at different and conflicting probability estimates which is clearly not what one wants!.
To show the inconsistency, we need to see that the three pieces of data are algebraic manipulations of two random variables which are interrelated via the following equations .
When one tries to use standard probability calculus (or rudimentary Monte Carlo) to derive the distributions of from or or even one will arrive at joint (and marginal) distributions with different support and shapes. This is illustrated in the figure below, in which yellow is the area of the joint probability density function (PDF) that assumes non-zero values, black is the area in which the PDF is zero and white is the white space that makes the figures look nice 🙂
What this little exercise tells us, is that one cannot hope to arrive at an (unconditional) probability model that is internally consistent. Even though this is possible when only pairs of are considered, the three solutions do not agree with each other. Hence, one would either need to define one’s model directly on the basis of or or use the specifications discussed in the previous post in a differential fashion.
Stay tuned, more to follow 🙂
EDIT (February 9th)
The three panel image creates the (false) visual impression that the three joint PDF have the same support on the x-axis, which is definitely NOT the case. The following image corrects this impression by plotting the PDFs over the same rectangular area in the plane.