Joseph W. Sakshaug, Arkadiusz Wiśniowski, Diego Andres Perez Ruiz and Annelies G. Blom
Carefully designed probability-based sample surveys can be prohibitively expensive to conduct. As such, many survey organizations have shifted away from using expensive probability samples in favor of less expensive, but possibly less accurate, nonprobability web samples. However, their lower costs and abundant availability make them a potentially useful supplement to traditional probability-based samples. We examine this notion by proposing a method of supplementing small probability samples with nonprobability samples using Bayesian inference. We consider two semi-conjugate informative prior distributions for linear regression coefficients based on nonprobability samples, one accounting for the distance between maximum likelihood coefficients derived from parallel probability and non-probability samples, and the second depending on the variability and size of the nonprobability sample. The method is evaluated in comparison with a reference prior through simulations and a real-data application involving multiple probability and nonprobability surveys fielded simultaneously using the same questionnaire. We show that the method reduces the variance and mean-squared error (MSE) of coefficient estimates and model-based predictions relative to probability-only samples. Using actual and assumed cost data we also show that the method can yield substantial cost savings (up to 55%) for a fixed MSE.
Arkadiusz Wiśniowski, Jakub Bijak, Solveig Christiansen, Jonathan J. Forster, Nico Keilman, James Raymer and Peter W.F. Smith
In this article, we first discuss the need to augment reported flows of international migration in Europe with additional knowledge gained from experts on measurement, quality and coverage. Second, we present our method for eliciting this information. Third, we describe how this information is converted into prior distributions for subsequent use in a Bayesian model for estimating migration flows amongst countries in the European Union (EU) and European Free Trade Association (EFTA). The article concludes with an assessment of the importance of expert information and a discussion of lessons learned from the elicitation process.