Propagation of random variables through computer codes of many inputs is primarily limited by computational expense. The use of sparse grids mitigates these costs somewhat; here we show how Sobol indices can be used to perform dimension adaptivity to mitigate them further. The method is compared to conventional adaptation schemes on sparse grids (Gerstner and Griebel, Computing 71(1), 65-87, 2003), and seen to perform comparably,without requiring the expense associated with a look-ahead error estimate. It is demonstrated for an expensive computer model of contaminant flow over a barrier.