As a critical mental health disorder, depression has severe effects on both human physical and mental well-being. Recent developments in EEG-based depression analysis have shown promise in improving depression detection accuracies. However, EEG features often contain redundant, irrelevant, and noisy information. Additionally, real-world EEG data acquisition frequently faces challenges, such as data loss from electrode detachment and heavy noise interference. To tackle the challenges, we propose a novel feature selection approach for robust depression analysis, called Incomplete Depression Feature Selection with Missing EEG Channels (IDFS-MEC). IDFS-MEC integrates missing-channel indicator information and adaptive channel weighting learning into orthogonal regression to lessen the effects of incomplete channels on model construction, and then utilizes global redundancy minimization learning to reduce redundant information among selected feature subsets. Extensive experiments conducted on MODMA and PRED-d003 datasets reveal that the EEG feature subsets chosen by IDFS-MEC have superior performance than 10 popular feature selection methods among 3-, 64-, and 128-channel settings.