  Clinicians and researchers alike are increasingly interested in how best to
personalize interventions. A dynamic treatment regimen (DTR) is a sequence of
pre-specified decision rules which can be used to guide the delivery of a
sequence of treatments or interventions that are tailored to the changing needs
of the individual. The sequential multiple-assignment randomized trial (SMART)
is a research tool which allows for the construction of effective DTRs. We
derive easy-to-use formulae for computing the total sample size for three
common two-stage SMART designs in which the primary aim is to compare mean
end-of-study outcomes for two embedded DTRs which recommend different
first-stage treatments. The formulae are derived in the context of a regression
model which leverages information from a longitudinal outcome collected over
the entire study. We show that the sample size formula for a SMART can be
written as the product of the sample size formula for a standard two-arm
randomized trial, a deflation factor that accounts for the increased
statistical efficiency resulting from a longitudinal analysis, and an inflation
factor that accounts for the design of a SMART. The SMART design inflation
factor is typically a function of the anticipated probability of response to
first-stage treatment. We review modeling and estimation for DTR effect
analyses using a longitudinal outcome from a SMART, as well as the estimation
of standard errors. We also present estimators for the covariance matrix for a
variety of common working correlation structures. Methods are motivated using
the ENGAGE study, a SMART aimed at developing a DTR for increasing motivation
to attend treatments among alcohol- and cocaine-dependent patients.
