Analyzing Within-Subject Experiments: Identification, Testing, and Sensitivity

Abstract

Recent work encourages political scientists to move from post-only toward within-subject designs for improved precision from repeated measurements. We formalize a potential-outcomes framework for two-period within-subject designs that allows for unequal allocation and heterogeneous treatment and carryover effects. We characterize the pooled estimator and evaluate the carryover test used to justify pooling. We find: first, pooling identifies the average treatment effect only when the gap in the average carryover effects is zero across the two treatment sequences. The unit-clustered standard error for the pooled estimator is identical to its design-based counterpart. Second, under mild conditions, the carryover test has strictly less power than the average-treatment-effect test with post-only data. The resulting two-step procedure, which pools only after a nonrejected test, produces confidence intervals that typically undercover. When the gap is zero, undercoverage occurs if and only if pooling is more efficient than post-only analysis, precisely when the within-subject design is worthwhile. When the gap is nonzero, undercoverage is typical unless the gap or sample size is large. Third, we derive a sensitivity analysis and find published conclusions robust to plausible carryover gaps. We therefore endorse within-subject designs but recommend justifying a zero carryover gap substantively and reporting sensitivity to departures.