In program evaluation, it is often desired to distinguish the effect of a treatment on individuals who receive the full dosage of that treatment from the effects on those who receive the partial dosage(s) of that and potentially other treatment conditions. We compare the identification, estimation, and policy interpretation of several estimators that aim to address this issue in a multi-armed longitudinal randomized control trial (RCT) with repeated non-compliers (i.e., individuals who switch back and forth between different treatment conditions over the course of the study).  For example, in a study with K possible treatment assignments implemented over T observation periods, there are K^T possible treatment histories (or exposure patterns) and this paper is concerned with the estimation of separate impact estimates for each treatment history. Previously, the complier-average causal effect (CACE) and the local average treatment effect (LATE) frameworks have been used to estimate impacts on compliers with only two possible assignment outcomes where sample units can switch from treatment to control (no-shows) or from control to treatment (cross-overs) only once throughout the study (Bloom, 1984; Angrist, Imbens, and Rubin, 1996; Yau and Little 2001; Jo 2002; and Schochet and Chiang 2011). We extend these frameworks to handle situations where non-compliers may switch between treatment conditions multiple times throughout a longitudinal study. Specifically, we derive the assumptions required to extend and identify the CACE/LATE model to estimate separate impacts for each exposure pattern yielded by such repeated non-compliers. We also contrast these assumptions to those required in an alternative Heckman-based approach that involves modeling the choice of participants to switch between treatment conditions (Heckman 1979). 

Our extension of CACE/LATE is presented along with simulation results and its application to the longitudinal evaluation of TRIAD, a model that employs guidelines to scale up successful interventions using research-based components—instruction, formative assessments, and professional development—that are all based on empirically-supported (children’s) learning trajectories. The goal of the TRIAD model is to avoid the dilution and pollution that usually plague efforts to achieve broad success (Clements & Sarama, 2011; Sarama, Clements, Wolfe, & Spitler, 2012)  In this evaluation, schools in low-income areas were randomly assigned to no treatment, treatment in the Pre-K classrooms only, and treatment in the Pre-K classroom plus follow-through in grades K and 1. Treatment in pre-K entailed implementation of the Building Blocks early mathematics curriculum (Clements & Sarama, 2013), while treatment in the K and 1^st grade years consisted of teacher professional development focused on effective mathematics teaching and learning.  Students were followed through grade 4.  As is common in low-income areas, many students switched schools, and thus treatment conditions, several times throughout the study.   In addition, complications arise from some students who switched to schools outside the study but were still followed, some students could not be tracked, and some students were held back a year.  The proposed paper demonstrates how we estimate the long-term effect of each of the distinct treatments caused by these complications and discusses the plausibility of the underlying identifying assumptions.