Rates of species diversification vary widely across the tree of life and there is considerable interest in identifying organismal traits that correlate with rates of speciation and extinction. However, it has been challenging to develop methodological frameworks for testing hypotheses about trait-dependent diversification that are robust to phylogenetic pseudoreplication and to directionally biased rates of character change. We describe a semi-parametric test for trait-dependent diversification that explicitly requires replicated associations between character states and diversification rates to detect effects. To use the method, diversification rates are reconstructed across a phylogenetic tree with no consideration of character states. A test statistic is then computed to measure the association between species-level traits and the corresponding diversification rate estimates at the tips of the tree. The empirical value of the test statistic is compared to a null distribution that is generated by structured permutations of evolutionary rates across the phylogeny. The test is applicable to binary discrete characters as well as continuous-valued traits and can accommodate extremely sparse sampling of character states at the tips of the tree. We apply the test to several empirical data sets and demonstrate that the method has acceptable Type I error rates.