Drug screening studies typically involve assaying the sensitivity of a range of cancer cell lines across an array of anti-cancer therapeutics. Alongside these sensitivity measurements high dimensional molecular characterizations of the cell lines are typically available, including gene expression, copy number variation and genomic mutations. We propose a sparse multitask regression model which learns discriminative latent characteristics that predict drug sensitivity and are associated with specific molecular features. We use ideas from Bayesian nonparametrics to automatically infer the appropriate number of these latent characteristics. The resulting analysis couples high predictive performance with interpretability since each latent characteristic involves a typically small set of drugs, cell lines and genomic features. Our model uncovers a number of drug-gene sensitivity associations missed by single gene analyses. We functionally validate one such novel association: that increased expression of the cell-cycle regulator C/EBPxcexb4 decreases sensitivity to the histone deacetylase (HDAC) inhibitor panobinostat.