From transportation networks to complex infrastructures, and to social and economic networks, a large variety of systems can be described in terms of multiplex networks formed by a set of nodes interacting through different network layers. Network robustness, as one of the most successful application areas of complex networks, has also attracted great interest in both theoretical and empirical researches. However, the vast majority of existing researches mainly focus on the robustness of single-layer networks an interdependent networks, how multiplex networks respond to potential attack is still short of further exploration. Here we study the robustness of multiplex networks under two attack strategies: layer node-based random attack and layer node-based targeted attack. A theoretical analysis framework is proposed to calculate the critical threshold and the size of giant component of multiplex networks when a fraction of layer nodes are removed randomly or intentionally. Via numerous simulations, it is unveiled that the theoretical method can accurately predict the threshold and the size of giant component, irrespective of attack strategies. Moreover, we also compare the robustness of multiplex networks under multiplex node-based attack and layer node-based attack, and find that layer node-based attack makes multiplex networks more vulnerable, regardless of average degree and underlying topology. Our finding may shed new light on the protection of multiplex networks.