A general recursive strategy is introduced to test nonparametrically for an increasing dose-response relationship when a downturn in response at high doses is possible. A specific implementation involving a test of A. R. Jonckheere [ibid. 41, 133-145 (1954; Zbl 0058.353)] and T. J. Terpstra [Indagationes Math. 14, 327-333 (1952; Zbl 0046.363)] is considered, and its size and power properties are investigated via Monte Carlo methods. Data from mutation research illustrate the phenomenon of concern.