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Supplementary Material for “Sample Size Formulae for Two-Stage Randomized Trials with Censored Data”

In the simple two-stage sequential multiple assignment randomized trial considered in the paper, nonresponders to both first-stage treatments are rerandomized to one of two secondstage treatments, while responders are not rerandomized. In practice, all kinds of variants of two-stage designs exist. For example, in a two-stage sequential multiple assignment randomized trial that is currently underway for drug addicted pregnant women conducted by Hendree Jones at the RTI International, Research Triangle Park, North Carolina, USA, both responders and and nonresponders to the first stage of treatment are rerandomized. At first, all subjects are randomized to either a traditional reinforcement based treatment or a reduced reinforcement based treatment with reduced intensity/scope. Response to the initial treatment is assessed at week 2. The criterion for nonresponse involved adherence to treatment as well as positive urine tests. Early non-responders are rerandomized to receive either the same treatment or a greater intensity/scope of their initial treatment, while responders are randomized to either the same intensity of treatment or a decreased intensity or scope of the initial treatment. An interesting outcome in this trial is time until dropout from counseling. In another trial concerning the treatment of children with autism conducted by Connie Kasari at the University of California at Los Angels, only nonresponders to one of the two first-stage treatments are rerandomized. In this trial, all subjects are initially randomized to either joint attention/joint engagement supplemented with an individualized augmentative/alternative communication system, or joint attention/joint engagement and spoken communication intervention, both treatments lasting for 12 weeks. At the end of the 12 week period, response is assessed, using measures such as the number of words and the number of communicative functions used spontaneously during parent-child interaction. All responders stay on the initial treatment; nonresponders to joint attention/joint engagement supplemented with an individualized augmentative/alternative communication system