vignettes/mathematical_details.Rmd
mathematical_details.RmdWe start by defining the following terms:
| Parameter | Definition |
|---|---|
| \(p\) | Prevalence at the domain level. This is usually the main thing we want to estimate |
| \(r\) | Intra-cluster correlation coefficient (between 0 and 1) |
| \(\alpha\), \(\beta\) |
Shape parameters of beta prior on site-level prevalence. Defined based
on other model parameters as follows: \(\alpha = p(1/r - 1)\) \(\beta = (1 - p)(1/r - 1)\) |
| \(c\) | Number of sites |
| \(x_i\) | Prevalence in site \(i\) |
| \(n_i\) | Sample size in site \(i\) |
| \(k_i\) | Observed pfhrp2/3 deletions in site \(i\) |
We assume a multilevel model in which \(p\) is the overall prevalence at the domain level, but the individual site-level prevalence varies around this value according to a beta distribution. The parameters of this beta distribution are chosen such that \(r\) represents the level of intra-cluster correlation.
Data, in the form of counts of observed pfhrp2/3 deletions, are assumed to be binomial within each site. The expected value of this distribution is the site-level prevalence. This results in counts following a beta-binomial distribution, with the following probability distribution:
\(Pr(k_i | n_i, p, r) = {n_i \choose k_i}\frac{B(k_i + \alpha, n_i - k_i + \beta)}{B(\alpha, \beta)}\)
where \(B(x, y)\) is the beta function. The overall likelihood is the product of this distribution over all sites:
\(Pr(\mathbf{k} | \mathbf{n}, p, r) = \prod_{i=1}^c Pr(k_i | n_i, p, r)\)
We assume beta priors on the two free parameters, \(p\) and \(r\). By default, the shape parameters of the prior on \(p\) are both equal to 1, meaning this simplifies to the uniform distribution. In contrast, based on historical data we assume shape parameters of 1 and 9 for the prior on \(r\), giving a mean of 0.1 and entertaining values in the plausible range [0, 0.3]:
The joint posterior probability is obtained (up to a constant of proportionality) by multiplying the likelihood by both priors:
\(Pr(p, r | \mathbf{k}, \mathbf{n}) \propto Pr(\mathbf{k} | \mathbf{n}, p, r)Pr(p)Pr(r)\)
Finally, the marginal posterior distribution of either \(p\) or \(r\) can be obtained by integrating over the other free parameter:
\(Pr(p | \mathbf{k}, \mathbf{n}) \propto \int_0^1 Pr(p, r | \mathbf{k}, \mathbf{n}) dr\)
\(Pr(r | \mathbf{k}, \mathbf{n}) \propto \int_0^1 Pr(p, r | \mathbf{k}, \mathbf{n}) dp\)
This integration is performed inside DRpower using an adaptive quadrature-based method. The marginal posterior distributions can be returned in full, or can be summarised using various measures of central tendency and/or credible intervals (CrI). There are multiple methods of calculating CrIs - we use the high density region (HDI) by default as this avoids the possibility of the maximum a posteriori (MAP) estimate being outside the CrI, which is possible by the more commonly used equal-tailed interval (ETI).