Table 1 Outcomes measures, analytical perspective and statistical methods

Yield of TB screening:

The fraction of active TB cases detected among the number of asylum seekers actively screened for TB, expressed as cases per 100,000 persons.

Number needed to screen (NNS):

The number of asylum seekers that need to undergo screening in order to diagnose one person with active TB [13], calculated as the inverse of yield.

Bayesian perspective:

Bayesian estimation and inference generally differ from frequentist methods that are still mostly seen in clinical and public health research in treating parameters as random variables (as opposed to constants in frequentist methods) [24]. The learning process in Bayesian methods works by modifying initial probability statements about parameters (prior distributions) before observing the data to updated or posterior knowledge that combines both previous knowledge and the data at hand. It allows hypotheses to be assessed by using a collection of parameter samples from their posterior distribution. A main advantage of Bayesian methods is the probabilistic (more common sense) interpretation of the confidence interval, here termed credible interval (CrI) on parameters. Key ingredients of a Bayesian statistical model are the likelihood function, reflecting information about the parameters contained in the data, and the prior distribution, quantifying what is known about the parameters before observing data. The prior distribution and likelihood can be combined to the posterior distribution, which represents total knowledge about the parameters after the data has been observed in the following sense [38]:

posterior ∞ prior ×  likelihood,

where ∝ means “is proportional to”. When considering the occurrence of a TB case in the screening process as “success” in n independent trials, the prevalence may be modeled to be binomially distributed. We exploit here that the conjugate prior for the binomial distribution is the beta distribution (Additional file 1).

Determination of prior distributions using WHO data:

WHO reports indirect estimates with a 95% CrI, except for Gambia and Pakistan where estimates are based on population-based surveys. This information was used to derive a Beta(p;q) prior distribution for the TB prevalence (Additional file 1: Figure S2). According to the method of moments [39] the shape parameters p and q of the beta distribution Beta(p;q) were estimated on the basis of the WHO data as follows:

\( \overset{\wedge }{p}\kern0.5em =\kern0.5em \left(\frac{1\kern0.5em -\kern0.5em \overset{\_}{p}}{\overset{\_\kern0.5em }{\sigma^2}}\kern0.5em -\kern0.5em \frac{1}{\overset{\_}{p}}\right)\kern0.5em \overset{\_}{p^2} \) and \( \hat{q}\kern0.5em =\kern0.5em \hat{p}\kern0.5em \left(\frac{1}{\overset{\_}{p}}\kern0.5em -\kern0.5em 1\right) \),

where \( \overline{p} \) is the mean prevalence (averaged over the years 1990 to 2014) for each country and \( \overline{\sigma} \) is the mean standard deviation. The mean standard deviation \( \overline{\sigma} \) is computed on the basis of the lower and upper bounds of the 95% credible intervals given by the WHO data for each country and year. It is assumed that:

width (95 % credible interval)  ≈  2 × 1.96 σ, and therefore \( \frac{\mathrm{mean}\ \left(\mathrm{width}\kern0.5em \left(95\%\mathrm{credible}\ \mathrm{interval}\right)\right)}{2\times 1.96}\kern0.5em \approx \kern0.5em \overset{\_}{\sigma } \).

This gives a Beta(\( \widehat{p} \); \( \widehat{q} \)) distribution for each country.

Modelling country-specific probabilities of the NNS to lie above a given cut-off value

The probability that the NNS for a certain country is above a given cut-off value t can be calculated from the derived posterior distributions of the prevalence in each country as follows: First, we use the inverse of the expected prevalence as NNS, and then we calculate the cumulative distribution function of the prevalence at 1/t.

\( {\displaystyle \begin{array}{l}P\kern0.5em \left[ NNS\kern0.5em \ge \kern0.5em t\right]\kern0.5em =\kern0.5em P\kern0.5em \left[\frac{1}{\mathrm{expected}\kern0.5em \mathrm{prevalence}}\kern0.5em =\kern0.5em NNS\kern0.5em \ge t\right]\\ {}\kern5.12em =\kern0.5em P\kern0.5em \left[\mathrm{expected}\kern0.5em \mathrm{prevalence}\kern0.5em \le \kern0.5em \frac{1}{t}\right]\\ {}\kern5.12em =\kern0.5em F\kern0.5em \left(\frac{1}{t}\right)\end{array}} \)