Phase screen techniques are commonly used to model scintillation of radio signals passing through a disturbed ionosphere, but observational phase or in situ density from both archival and real-time sources is often sampled at rates well below the resolution desired for input to such models. Previous phase screen resolution criteria do not address the computation of the amplitude scintillation index (S4), which is widely used in scintillation monitoring systems, for practical applications that rely on coarsely sampled phase data. We investigate the accuracy of S4 estimates from sparsely populated one-dimensional phase screen models through systematic removal of samples from high-resolution discrete phase screen models in novel numerical experiments. We also provide analytic approximations for the weak-scatter case and study the effects of antialias filtering. This subsampling analysis shows that 2--5 samples per Fresnel radius (rF) are usually sufficient to compute S4 to within 90% of its true value, depending on the form of the phase spectral density function (SDF) and the strength of scattering. We make initial application of the subsampling techniques to data-based phase screens for studying equatorial ionospheric scintillation with generally satisfactory results. Finally, we outline the use of the techniques developed in this paper to other practical problems, including SDF determination and computing irradiance patterns.