The precision of biological assays is increased if each subject can be tested with several different doses of the materials under assay, so that the estimate of relative potency is based upon intra-subject comparison of responses. In some assay techniques, doses can be given simultaneously at different sites on a subject. In others, a time-sequence of doses is inevitable, and any response may be affected by the previous doses or responses of the same subject. Five models are elaborated, these involving combinations of error correlation, residual dose effects, and autoregressive effects of previous responses to represent mathematically the expected and observed responses. Cross-over designs, involving a balancing of dose sequences for different subjects, permit estimation of the important parameters of the models by analysis of variance and least-squares techniques. Three designs, each applicable to a 4-point parallel line assay, are discussed in detail. These are the twin cross-over, in which only two doses are tested on each subject, and the orthogonal square and serially balanced single-square designs in which each subject receives in turn all four doses. The construction of the analysis of variance, tests of assay validity and the formation of estimates of relative potency are described for each model. Despite their different logical bases, the model involving correlation and additive residual dose effects and that for a simple autoregressive scheme always lead to essentially the same statistical analysis; this analysis seems likely to be fairly insensitive to small deviations from the strict specifications of the models, and its use is an insurance against the possibility that one of the more complicated models is applicable. A final section describes the randomization necessary in the selection of a particular design.