Absfracr: We study regression designs, with an eye to maximizing the minimum power of the standard test for Lack of Fit. The minimum is taken over a broad class of departures from the assumed multiple linear regression model. We show that the uniform design is maximin. This design attains its optimality by maximizing the minimum bias in the regression estimate of 0’. It is thus surprising that this same design has an optimality property relative to the estimation of (r2 it minimizes the maximum bias, in a closely related class of departures from linearity.