Among the various methods adopted to compare health inequality, Makdissi and Yazbeck (J Health Econ 34:84–95, 2014) developed positional stochastic dominance conditions to identify an ordering. To reach a conclusion, their rules require that the (generalized) health concentration curve of the dominant distribution lie above that of the dominated one. However, it is frequently observed in practice that these curves intersect. Our paper proposes new criteria to cope with this problem by allowing a relatively small violation of the condition proposed by Makdissi and Yazbeck (2014). We characterize our conditions by linking them with some ethical constraints of the weight functions. We further use individual data for Côte d’Ivoire and Guinea from the Demographic and Health Survey to demonstrate the usefulness of our newly-proposed method.