• Bias modelling;
  • Hierarchical models;
  • Meta-analysis;
  • Mixed effects models;
  • Multiple survey data;
  • Small area estimation;
  • Smoking prevalence

Summary.  Combining information from multiple surveys can improve the quality of small area estimates. Customary approaches, such as the multiple-frame and statistical matching methods, require individual level data, whereas in practice often only multiple aggregate estimates are available. Commercial surveys usually produce such estimates without clear description of the methodology that is used. In this context, bias modelling is crucial, and we propose a series of Bayesian hierarchical models which allow for additive biases. Some of these models can also be fitted in a classical context, by using a mixed effects framework. We apply these methods to obtain estimates of smoking prevalence in local authorities across the east of England from seven surveys. All the surveys provide smoking prevalence estimates and confidence intervals at the local authority level, but they vary by time, sample size and transparency of methodology. Our models adjust for the biases in commercial surveys but incorporate information from all the sources to provide more accurate and precise estimates.