I would like to add another perspective to that in Shrier et al [1] on the application of principal stratification [2] to all-or-none versus partial compliance. With all-or-none compliance in randomized trials [3] or all-or-none changes in availability in the paired availability design [4, 5], the assumptions needed for identifiability of the principal stratification model are often reasonable. Consider a binary outcome and suppose participants are randomized to either group 0 (assigned T0) or group 1 (assigned T1). The principal strata partition participants into pairs defined by (treatment received if in group 0, treatment received if in group 1), namely (T0,T0), (T0,T1), (T1,T0), and (T1,T1). Assuming rational decision making so no (T1,T0), persons receiving T0 in group 0 are a mixture of (T0,T0) and (T0,T1); persons receiving T1 in group 0 are (T1,T1); persons receiving T0 in group 1 are (T0,T0); and persons receiving T1 in group 1 are a mixture of (T0,T1) and (T1,T1). With the additional assumption that probabilities of outcome not depend on group for (T0, T0) and (T1, T1), one can estimate the effect of receipt of T0 versus T1 (treatment efficacy) in the principal stratum (T0,T1) for compliers, a result which can be elucidated graphically [5, 6]. In essence, this estimation peels away the irrelevant (T0,T0) and (T1,T1) strata [7] to obtain a more informative estimate of treatment efficacy (if the assumptions hold) than an intent-to-treat estimate. This identifiability of treatment efficacy in principal stratification under all-or-none compliance extends to missing binary outcomes [7, 8] and survival outcomes [9].
In contrast an application of principal stratification to partial compliance requires strong assumptions for identifiability. Consider treatment received initially and later yielding 16 principal strata, each of the form (initial treatment if randomized to group 0: later treatment if randomized to group 0, initial treatment if randomized to group1: later treatment if randomized to group 1). Strong assumptions are needed to estimate treatment efficacy via the stratum of most interest, namely (T0:T0,T1:T1) [10]. However, under reasonable assumptions, the study can still yield an approximate estimate of treatment efficacy. The rational decision-making assumption implies no (T0:T1, T0:T0), (T1:T0,T0:T0), (T1:T0,T0:T1), (T1:T1,T0:T0), (T1:T1,T0:T1), or (T1:T1,T1:T0). Persons receiving T0:T0 in group 0 are (T0:T0,T0:T0) and persons receiving T1:T1 in group 1 are (T1:T1,T1:T1). We call the remaining persons partial compliers as they are a mixture of (T0:T0,T0:T1), (T0:T0,T1:T0), (T0:T0, T1:T1), (T0:T1,T0:T1), (T0:T1,T1:T0), (T0:T1,T1:T1), (T1:T0, T1:T0), and (T1:T0,T1:T1). With the additional assumption that probabilities of outcome not depend on group for (T0:T0,T0:T0) and (T1:T1,T1:T1), the estimate of treatment efficacy when applying the usual principal stratification analysis under all-or-none compliance is an approximate estimate of treatment efficacy (with mixtures of initial and late treatments) among partial compliers. This approximate estimate of treatment efficacy in partial compilers, while not ideal, is still more informative than an intent-to-treat estimate with respect to treatment efficacy (if the assumptions hold) because it peels away strata (T0:T0,T0:T0) and (T1:T1,T1:T1). Another approach to estimating treatment efficacy under partial compliance involves a third randomization group and a modified principal stratification model [10].
Acknowledgments
This research was supported by the National Institutes of Health. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
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