Reflection factor R as a function of the strength of the soft excess q (in the case of distant reflection). Top panel: linear regression performed using the linmix_err procedure is represented by the blue line, and 99.9% CI is given by the blue contour. The red line represents the positive correlation expected if the soft excess is due to ionized reflection (result of relxilllp_ion simulations, see Sect. 4.2). Density contours of the simulations are plotted in red (solid line: 68%; dashed line: 90%). Bottom panel: blue points represent our 79 objects showing a soft excess. They are mostly contained (75%) in the blue box (R< 2.0 and q< 0.75). Red triangles are relxilllp_ion simulation results.