A Bayesian approach to synthesize estimates of the size of hidden populations: the Anchored Multiplier.

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The multiplier method is one of the most frequently used population size estimation (PSE) methods for key populations, yet estimates from this method are often inconsistent with each other, other PSE methods and local knowledge. We developed a novel Bayesian approach, the 'Anchored Multiplier', which synthesizes estimates from multipliers coupled to an a priori estimate to arrive at a single consensus estimate and credible range. Data for size estimation were collected from three cross-sectional bio-behavioural surveillance studies of people who inject drugs (PWID) in San Francisco, CA, USA (2005, 2009 and 2012). We demonstrate the application of the Anchored Multiplier and a Variance Adjusted-Anchored Multiplier using PSE produced by multipliers in the three surveys and the literature for the USA. Size estimates were compared with estimates from other available PSE methods. Using the Anchored Multiplier, we estimated the PWID population made up 2.41% [95% credible interval (CI): 1.9-2.85] of the adult population in 2005, 2.1% (95% CI: 1.8-2.48) in 2009 and 2.3% (95% CI: 2.03-2.61) in 2012. The Variance Adjusted-Anchored Multiplier calculated similar point estimates, with wider 95% credible intervals. Credible intervals from both approaches were substantially narrower than from other standard PSE methods and, unlike other methods, indicated that the prevalence of PWID was stable over time. The Anchored Multiplier is a promising new approach to size estimation, which generates a single estimate to inform programmatic strategies to counter the HIV epidemic, and provides a robust denominator to quantify the burden of disease for key populations.