Ocean Island Basalts (OIBs) are generated by mantle plumes, with their geochemistry controlled by a combination of source composition, temperature, and thickness of overlying lithosphere. For example, OIBs erupting onto thicker, older oceanic lithosphere are expected to exhibit signatures indicative of higher average melting pressures. Here, we quantitatively investigate this relationship using a global dataset of Neogene and younger OIB compositions. Local lithospheric thicknesses are estimated using theoretical plate-cooling models and Bayes factors are applied to identify trends. Our findings provide compelling evidence for a correlation between OIB geochemistry and lithospheric thickness, with some variables SiO2,  Al2O3, FeO, Lu, Yb and λ2) showing linear trends that can be attributed to increasing average melting pressure, whereas others (λ0 and λ1, CaO) require a bi-linear fit with a change in gradient at ~55 km. Observed variations in highly incompatible elements are consistent with melt fractions that decrease with increasing lithospheric thickness, as expected. Nevertheless, at thicknesses beyond ~55 km, the implied melt fraction does not decrease as rapidly as suggested by theoretical expectations. This observation is robust across different lithospheric thickness estimates, including those derived from seismic constraints. We interpret this result as weak plumes failing to effectively thin overlying lithosphere and/or producing insufficient melt to erupt at the surface, in combination with a 'memory effect' of incomplete homogenisation of melts during their ascent. This view is supported by independent estimates of plume buoyancy flux, indicating that OIB magmatism on older lithosphere may be biased towards hotter plumes.