A transcendental meromorphic function is called a Speiser function if it has only finitely many singular values (i.e., critical and asymptotic values). We show how to construct such functions with any given Hausdorff dimension of escaping set by using quasiconformal mappings. This strengthens an earlier result of Bergweiler and Kotus. Our construction also shows that the escaping sets of quasiconformally equivalent meromorphic functions may be different. This work is joint with Magnus Aspenberg.