Statistical data-driven process monitoring is critical for efficient operations of industrial processes. However, deviations from normal regions in the process data may or may not lead to poor quality of products. This paper proposes a new combined index for detecting output-relevant faults, which affect the output data, and studies the output-relevant fault detectability based on total projection to latent structures (T-PLS). Given actual fault direction, fault-free data can be reconstructed and output-relevant part of fault magnitude can be estimated. Two new methods are derived to extract output-relevant fault subspace from faulty data. A simulation example and a case study on the Tennessee Eastman process are used to show the effectiveness of the proposed methods.