The paper is concerned with randomized experiments with one treatment. Two randomization schemes are considered: randomized pairs and unrestricted randomization. If effective at all, the treatment is supposed to affect the conditional distribution of the “experimental” variableYgiven another variableX, called “predictor”. The distribution ofXis not affected by the treatment. Using the general theory published elsewhere, the paper deduces the locally asymptotically optimal test of the hypothesis that the treatment has no effect. Apart from the usual difficulties connected with asymptotic tests (how large mustNbe?), the theory is easily applicable in many “live” cases even though the conditional distribution ofYgivenXmay contain nuisance parameters and be of unusual form.