Aims. We perform a statistical study of the relations between the properties of solar energetic electron (SEE) events measured by the MESSENGER mission from 2010 to 2015 and the parameters of the respective parent solar activity phenomena in order to identify the potential correlations between them. During the time of analysis, the MESSENGER heliocentric distance varied between 0.31 and 0.47 au. Methods. We used a published list of 61 SEE events measured by MESSENGER, which includes information on the near-relativistic electron peak intensities, the peak-intensity energy spectral indices, and the measured X-ray peak intensity of the flares related to the SEE events. Taking advantage of multi-viewpoint remote-sensing observations, we reconstructed, whenever possible, the associated coronal mass ejections (CMEs) and shock waves; and we determined the three-dimensional (3D) properties (location, speed, and width) of the CMEs and the maximum speed of the 3D CME-driven shocks in the corona. We used different methods (Spearman, Pearson, and a Bayesian approach, namely the Kelly method to linear regression) to estimate the correlation coefficients between the flare intensity, maximum speed at the apex of the CME-driven shock, CME speed at the apex, and CME width with the electron peak intensities and with the energy spectral indices. In this statistical study, we considered and addressed the limitations of the particle instrument on board MESSENGER (elevated background intensity level, anti-Sun pointing). Results. There is an asymmetry to the east in the range of connection angles (CAs) for which the SEE events present the highest peak intensities, where the CA is the longitudinal separation between the footpoint of the magnetic field connecting to the spacecraft and the flare location. Based on this asymmetry, we define a subsample of well-connected events as when −65° ≤ CA ≤ +33°. For the well-connected sample, we find moderate to strong correlations between the near-relativistic electron peak intensity and the 3D CME-driven shock maximum speed at the apex (Spearman: cc = 0.53 ± 0.05; Pearson: cc = 0.65 ± 0.04; Kelly: cc = 0.87 ± 0.20), the flare peak intensity (Spearman: cc = 0.63 ± 0.03; Pearson: cc = 0.59 ± 0.03; Kelly: cc = 0.74 ± 0.30), and the 3D CME speed at the apex (Spearman: cc = 0.50 ± 0.04; Pearson: cc = 0.46 ± 0.03; Kelly: cc = 0.60 ± 0.39). When including poorly connected events (full sample), the relations between the peak intensities and the solar-activity phenomena are blurred, showing lower correlation coefficients. Conclusions. Based on the comparison of the correlation coefficients presented in this study using near 0.4 au data, (1) both flare and shock-related processes may contribute to the acceleration of near relativistic electrons in large SEE events, in agreement with previous studies based on near 1 au data; and (2) the maximum speed of the CME-driven shock is a better parameter to investigate particle-acceleration-related mechanisms than the average CME speed, as suggested by the stronger correlation with the SEE peak intensities.