An Innovation to Test Treatment X Pretest Interactions within Difference-in-Differences

Abstract

We introduce a way to test Treatment X Pretest interactions within difference-in-differences (DID). Mathematically adding a Treatment X Pretest interaction to DID transforms the treatment estimate to an ANCOVA-type estimate, which differs from DID's estimate and is often biased against at-risk cases. Dual-centered ANCOVA duplicates DID's treatment estimate and can test whether that estimate varies by pretest scores. To illustrate, we test a Treatment X Pretest interaction for the effects of therapy for depression using the Fragile Families and Child Wellbeing longitudinal dataset. After centering posttest and pretest outcome data on pretest group means, DID and ANCOVA estimates are both equivalent to the original DID treatment estimate. ANCOVA of these dual-centered data can then test a Treatment X Pretest interaction.