We study the performance of shape-constrained methods for evaluating immune response profiles from early-phase vaccine trials. The motivating problem for this work involves quantifying and comparing the IgG binding immune responses to the first and second variable loops (V1V2 region) arising in HVTN 097 and HVTN 100 HIV vaccine trials. We consider unimodal and log-concave shape-constrained methods to compare the immune profiles of the two vaccines, which is reasonable because the data support that the underlying densities of the immune responses could have these shapes. To this end, we develop novel shape-constrained tests of stochastic dominance and shape-constrained plug-in estimators of the Hellinger distance between two densities. Our techniques are either tuning parameter free, or rely on only one tuning parameter, but their performance is either better (the tests of stochastic dominance) or comparable with the nonparametric methods (the estimators of Hellinger distance). The minimal dependence on tuning parameters is especially desirable in clinical contexts where analyses must be prespecified and reproducible. Our methods are supported by theoretical results and simulation studies.