Cross-over designs allow observation of an outcome variable to more than one treatment for an individual. Individual effects of one treatment with respect to another can be observed. Such designs have been proposed to evaluate how much the effect of a treatment varies across individuals in a population. But the observed effects include contributions of treatment and period, and these contributors are not easily separated at the level of the individual. This is illustrated first using a two-period two treatment cross-over design. Then additional details are presented in the context of a three-period two treatment cross-over design involving six sequences. The assumptions that are needed to separate individual effects of treatment from individual effects of the period at which outcomes are observed are specified using a potential outcomes framework. The computed variance of observed individual treatment effects is biased for a true variance, but the bias can be estimated with assumptions. Relatively large samples are required for precise estimates. Simulated data from a hypothetical blood pressure study are used to illustrate concepts, and a simulation evaluates properties of proposed estimators.

causal inference, clinical trial, counterfactual, potential outcomes, subject-treatment interaction.