Abstract: Electromagnetic ion cyclotron (EMIC) waves have been proposed to be an important mechanism in causing relativistic electron precipitation (REP).  However, not all EMIC waves are associated with REP.  Knowledge of the factors that influence the efficiency of EMIC waves to scatter relativistic electrons in observations is still limited. In our study, we perform 6 years of analysis from 2013 to 2018, with relativistic electron precipitation (REP) observed by POES and EMIC wave observations from Van Allen Probes. The coincidence occurrence rate between EMIC waves and relativistic electron precipitation events is about 34%. Additionally, the coincidence occurrence rate is about 10% higher than the random coincidence occurrence rate, indicating that EMIC waves and relativistic electrons can be statistically related, but the link is weaker than expected. H⁺ band EMIC waves have been regarded as less important than He⁺ band EMIC waves for the precipitation of relativistic electrons. We demonstrate that the proportion of H⁺ band EMIC wave events that are associated with REP is slightly higher than for He⁺ band EMIC wave activity. An even greater proportion of EMIC waves are accompanied by REP events when H⁺ band and He⁺ band EMIC waves occur simultaneously.  A parametric study further showed that the coincidence occurrence rate of EMIC wave events and REP events increases with respect to increased background plasma density, with increases in the ratio of plasma frequency to local gyrofrequency, increasing EMIC wave power and when the wave frequency approaches the gyrofrequency. The dependence on background electron density is stronger than the dependence on the ratio of plasma frequency to gyrofrequency. The coincidence occurrence rate decreases as the magnetic field increases between 120-270 nT, consistent with a previous study. These results are consistent with the quasi-linear interaction between EMIC waves and relativistic electrons. Next, a rapid loss event of relativistic radiation belt electrons at low L_ values (2.4-3.2) during a strong geomagnetic storm on 22 June 2015 is investigated along with a local H⁺ band electromagnetic ion cyclotron (EMIC) wave-driven loss mechanism. Both the particle and wave data are obtained from the Van Allen Probes. Duskside H⁺ band EMIC waves were observed during a rapid decrease of relativistic electrons with energy above 5.2 MeV occurring outside the plasmasphere during extreme magnetopause compression. Lower He⁺ composition and enriched O⁺ composition is found compared to typical values assumed in other studies of cyclotron resonant scattering of relativistic electrons by EMIC waves. Quantitative analysis demonstrates that even with the existence of He⁺ band EMIC waves, it is the H⁺ band EMIC waves that are likely to cause the depletion of relativistic electrons with energy above 5.2 MeV at low L values for the event studied.