With more patients receiving external beam radiation therapy with protons, it becomes increasingly important to refine the clinical understanding of the relative biological effectiveness (RBE) for dose delivered during treatment. Treatment planning systems used in clinics typically implement a constant RBE of 1.1 for proton fields irrespective of their highly heterogeneous linear energy transfer (LET). Quality assurance tools that can measure beam characteristics and quantify or be indicative of biological outcomes become necessary in the transition towards more sophisticated RBE weighted treatment planning and for verification of the Monte Carlo and analytical based models they use. In this study the RBE for the CHO-K1 cell line in a passively delivered clinical proton spread out Bragg peak (SOBP) is determined both in vitro and using a silicon-on-insulator (SOI) microdosimetry method paired with the modified microdosimetric kinetic model. The RBE along the central axis of a SOBP with 2 Gy delivered at the middle of the treatment field was found to vary between 1.11-1.98 and the RBE for 10% cell survival between 1.07-1.58 with a 250 kVp x-ray reference radiation and between 1.19-2.34 and 0.95-1.41, respectively, for a Co60 reference. Good agreement was found between RBE values calculated from the SOI-microdosimetry-MKM approach and in vitro. A strong correlation between proton lineal energy and RBE was observed particularly in the distal end and falloff of the SOBP.