Raking mortality rates across cause, population group and geography with uncertainty quantification
The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) is the single largest and most detailed scientific effort ever conducted to quantify levels and trends in health. This global health model to estimate mortality rates and other health metrics is run at different scales, leading to large data sets of results for a global region and its different sub-regions, or for a cause of death and different sub-causes for example. These models do not necessarily lead to consistent data tables where, for instance, the sum of the number of deaths for each of the sub-regions is equal to the number of deaths for the global region. Raking is widely used in survey inference and global health models to adjust the observations in contingency tables to given marginals, in the latter case reconciling estimates between models with different granularities. The results of global health models usually associate to the point estimates an uncertainty, such as standard deviations or confidence intervals. In this paper, we propose an uncertainty propagation approach that obtains, at the cost of a single solve, nearly the same uncertainty estimates as computationally intensive Monte Carlo techniques that pass thousands of observed and marginal samples through the entire raking process. We introduce a convex optimization approach that provides a unified framework to raking extensions such as uncertainty propagation, raking with differential weights, raking with different loss functions in order to ensure that bounds on estimates are respected, verifying the feasibility of the constraints, raking to margins either as hard constraints or as aggregate observations, and handling missing data.