Microphone arrays based on the minimum variance distortionless response (MVDR) beamformer are among the most popular for speech enhancement applications. The original MVDR is excessively sensitive to source location and microphone gains. Previous research has made MVDR practical by successfully increasing the robustness of MVDR to source location, and MVDR-based microphone arrays are already commercially available. Nevertheless, MVDR performance is still weak in cases where microphone gain variations are too large e.g., for circular arrays of directional microphones. In this paper we propose an improved MVDR beamformer which takes into account the effect of sensors (e.g. microphones) with arbitrary, potentially directional responses. Specifically, we form estimates of the relative magnitude responses of the sensors based on the data received at the array and include those in the original formulation of the MVDR beamforming problem. Experimental results on real-world audio data show an average 2.4 dB improvement over conventional MVDR beamforming, which does not account for the magnitude responses of the sensors.