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Modeling of X-ray pulse profiles from millisecond pulsars offers a promising method of inferring the mass-to-radius ratios of neutron stars. Recent observations with Neutron Star Interior Composition Explorer (NICER) resulted in measurements of radii for three neutron stars using this technique. In this paper, we explore correlations between model parameters and the degree to which individual parameters can be inferred from pulse profiles, using an analytic model that allows for efficient and interpretable exploration. We introduce a new set of model parameters that reduce the most prominent correlations and allow for efficient sampling of posteriors. We then demonstrate that the degree of beaming of radiation emerging from the neutron star surface has a large impact on the uncertainties in the inferred model parameters. Finally, we show that the uncertainties in the model parameters for neutron stars for which the polar cap temperature falls outside of the NICER energy range are significantly degraded.