The shift alternative model has been the canonical alternative hypothesis since the early days of statistics. This holds true both in parametric and nonparametric testing. In this talk, we depart from shift alternatives, into *mixture* alternatives. We trace back the history of mixture alternatives, dating back to Good's 1979 Contamination Alternatives. We then present some modern applications where mixture alternatives are more appropriate than shift alternatives; examples include personalized medicine, and medical imaging. Finally, we present several candidate test statistics to detect such alternatives. In particular, we show that when detecting mixtures, Wilcoxon's signed-rank test may (surprisingly) be more powerful than a t-test, even under a Gaussian null.