Abstract: We provide marginal coordinate tests based on two competing Principal Hessian Directions (PHD) methods. Predictor contributions to central mean subspace can be effectively identified by our proposed testing procedures. PHD-based tests avoid choosing the number of slices, which is a well-known shortcoming of similar tests based on Sliced Inverse Regression (SIR) or Sliced Average Variance Estimation (SAVE). The asymptotic distributions of our test statistics under the null hypothesis are provided and the effectiveness of the new tests is illustrated by simulations. \newcommand\fundinfoThe first and corresponding authors were supported by National Social Science Foundation (08CTJ001).