Should we take measurements at an intermediate design point?

It is well known that, for estimating a linear treatment effect with constant variance, the optimal design divides the units equally between the two extremes of the design space. If the dose-response relation may be nonlinear, however, intermediate measurements may be useful in order to estimate the effects of partial treatments. We consider the decision of whether to gather data at an intermediate design point: do the gains from learning about nonlinearity outweigh the loss in efficiency in estimating the linear effect? Under reasonable assumptions about nonlinearity, we find that, unless sample size is very large, the design with no interior measurements is best, because with moderate total sample sizes, any nonlinearity in the dose-response will be difficult to detect. We discuss in the context of a simplified version of the problem that motivated this work-a study of pest-control treatments intended to reduce asthma symptoms in children.