Date Published: February 22, 2019
Publisher: Public Library of Science
Author(s): Maria Bekker-Nielsen Dunbar, Thomas J. R. Finnie, Barney Sloane, Ian M. Hall, Jeffrey Shaman.
Employing historical records we are able to estimate the risk of premature death during the second plague pandemic, and identify the Black Death and pestis secunda epidemics. We show a novel method of calculating Bayesian credible intervals for a ratio of beta distributed random variables and use this to quantify uncertainty of relative risk estimates for these two epidemics which we consider in a 2 × 2 contingency table framework.
We present a means of calculating two forms of credible interval for ratios of binomial proportions which we use with the relative risk, a commonly used measure in medical and epidemiological research. Kawasaki and Miyaoka  have previously examined credible intervals for a difference between binomial proportions. We extend this work by considering ratios, such that our work can be considered an addition to the current scientific evidence on binomial proportions. Our approach is similar to that of Nurminen and Mutanen’s examination of Bayesian counterparts to frequentist tests . They consider the posterior cumulative function for null hypothesis testing while we considered the ratio as being a random variable. Furthermore we extended Pham-Gia’s work on the density of such a variable by calculating the distribution on the basis of that density. Beyond this work, we prove the unimodality of this density and consider priors beyond the pairing Nurminen and Mutanen call “rectangular”, which is α = β = θ = ϕ = 1 following our notation, though we include it in our examples, which further considers non-equal priors. This means, unlike other instances found in the literature, we allow for non-symmetrical beta distributions hence it is possible to skew the priors towards probabilities considered more likely.
We see that the shape of the beta distribution changes based on the parameter values. If both shape parameters are the same we obtain a symmetric distribution. B(0, 0) corresponds to the situation where we have no prior beliefs. B(1, 1) is the uniform prior as all options are equiprobable. Both B(0, 0) and B(1, 1) are improper priors. We see that when the shape parameters are between 0 and 1 we observe a U-shaped distribution. When they are larger than 1, they flip with respect to the uniform prior curve. If we consider shape parameters that are not equal, we observe asymmetric distributions, B(1, 2) being the most extreme case in our plot. B(2, 1) will act in a similar way but point in the opposite direction, producing a linear curve with slope 2. The asymmetry is not as obvious for the B(3, 4) case but there is a slight shift towards the left of the plot as a result of having more belief in the members of the unaffected group, causing the distribution to flatten out. Fig 1 illustrates the versatility of the beta distribution as it covers many different distribution shapes.
We considered the ratio of two beta distributions and calculated a closed form expression of the posterior distribution of such a ratio. To consider the uncertainty of relative risks, we considered two types of credible interval, the quantile interval and the highest posterior density. We saw that in the situation considered they gave similar results and did not contain 1 indicating that risk of death due to plague was significantly different to non-plague death in the periods we considered. We saw a reduced risk of dying from plague during the pestis secunda outbreak comapred to the risk of dying from plague during the Black Death outbreak.