One effective way to estimate the impact of contingencies is to utilize linear distribution sensitivity factors, such as injection shift factor and line outage distribution factor. Compared to other impact estimation approaches, estimating the line flows with sensitivity factors is computationally less demanding, as a linearized DC power flow model is utilized. However, the accuracy of the power flow model is highly dependent on the received system information. Hence, wrong or missing system information can yield inaccurate results. Phasor Measurement Units provide measurements that can be used to estimate sensitivity factors such that the impact of wrong system model information can be minimized. This paper introduces a new methodology, based on the alternating direction method of multipliers, to leverage PMU data for estimating sensitivity factors. The developed methodology is particularly applicable to near real-time conditions, where the speed of estimation is of essence. The performance of the developed method is compared with the traditional estimation methods for multiple testbeds.