Gene-set analysis provides insight into which functional and biological properties of genes are aetiologically relevant for a particular phenotype. But genes have multiple properties, and these properties are often correlated across genes. This can cause confounding in a gene-set analysis, because one property may be statistically associated even if biologically irrelevant to the phenotype, by being correlated with gene properties that are relevant. To address this issue we present a novel conditional and interaction gene-set analysis approach, which attains considerable functional refinement of its conclusions compared to traditional gene-set analysis. We applied our approach to blood pressure phenotypes in the UK Biobank data (N = 360,243), the results of which we report here. We confirm and further refine several associations with multiple processes involved in heart and blood vessel formation but also identify novel interactions, among others with cardiovascular tissues involved in regulatory pathways of blood pressure homoeostasis.