Through a series of investigations of the quantitative structure of monolayer surface alloy phases of Cu(1 0 0)/Au, Cu(1 0 0)/Mn, Cu(1 1 1)/Sb and Ni(1 1 1)/Pb a number of aspects of the methodology of medium energy ion scattering (MEIS) have been investigated. These include the form of reliability-factors (R-factors) used to provide an objective test of the fit of experimental blocking curves and those simulated for different model structures, and the criteria used to define the precision and uniqueness of such models. The role of absolute calibration of the scattered ion yields and the appropriate choice of scattering geometry are also discussed in the context of these specific studies. The quantitative results for these surface alloy structures also cast some light on the effective atomic radii in such structures and highlight the general trend for phases with low surface corrugation or rumpling