Recent observational and theoretical studies indicate that the damping of solar coronal loop oscillations depends on the oscillation amplitude. We consider two mechanisms: linear resonant absorption and a nonlinear damping model. We confront theoretical predictions from these models with observed data in the plane of observables defined by the damping ratio and the oscillation amplitude. The structure of the Bayesian evidence in this plane displays a clear separation between the regions where each model is more plausible relative to the other. There is qualitative agreement between the regions of high marginal likelihood and Bayes factor for the nonlinear damping model and the arrangement of observed data. A quantitative application to 101 loop oscillation cases observed with Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA) results in the marginal likelihood for the nonlinear model being larger in the majority of them. The cases with conclusive evidence for the nonlinear damping model outnumber considerably those in favor of linear resonant absorption.