The improvement obtained by including site terms in ground-motion predictive relations is often small. Does it mean that local effects play a marginal role in ground shaking? Or could different geophysical site parameters or more complicated functional forms provide better results? These hypotheses are investigated for 399 peak ground accelerations (PGA) observed for strong earthquakes (5≤Mw≤7.6) in Europe and the Middle East in comparison with the ground-motion relation by Ambraseys et al. (2005). An analysis of the residuals log10(observed)-log10(predicted) reveals that site effects could contribute up to 33% of the overall standard deviation σ and that a large part of this component is potentially removable by site classification: the theoretical optimal classification (that making the best trade-off between high accuracy and low number of classes) reduces σ by 27% using six classes. Similar reductions have been obtained for response spectral acceleration (SA) at 0.2, 1, and 2 sec. For PGA, using a nonparametric approach, it is found that functions of the shear-wave velocity in the upper 30 m (VS30) reduce σ less then 6% (less than 1% using the National Earthquake Hazards Reduction Program [NEHRP] and the Eurocode 8 [EC8] classifications as well as linear regression on VS30). The reduction could reach 10% if a further dependency on the PGA predicted for rock is introduced. The large difference between the potential improvement by 33% and the practical results using VS30 evidence the utility and need of considering other parameters as well.