Assumptions for unbiasedness
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Ensuring the mITT estimator in This assumption implies there are only two principal strata in the trial, ‘never initiators’ and ‘always initiators’; hence, if we observe a participant’s initiation status in one treatment arm, we can infer their initiation status had they been allocated to the alternative treatment, and identify the principal stratum to which they belong (i.e. if we observe Then, the mITT estimator in The mITT principal stratum estimator under the assumption of no ‘intervention initiators’ and no ‘control initiators’.mITT: modified intention-to-treat.The estimator compares the shaded cells in the intervention column against the shaded cells in the control column. Grey shading with a strikethrough indicates that these strata do not exist.This assumption can be partly assessed, on the basis that if it is true, then the proportion of non-initiators should, on average, be the same across randomised groups. Hence, if there are large discrepancies in the proportion of non-initiators between groups, this may provide evidence that this assumption has been violated. However, even if this proportion is the same across groups, this is not a guarantee the assumption is true; hence, this assumption cannot be fully tested, and therefore must rely primarily on contextual information around its underlying plausibility.We demonstrate the unbiasedness of the mITT estimator under the assumptions given above as follows. First, under consistency assumption
can be written as
Then, under the assumption that there are no ‘intervention initiators’ and no ‘control initiators’, 
which is the principal stratum estimand in We note that this estimator is unbiased regardless of what happens to ‘never initiators’ (e.g. regardless of whether they receive no treatment at all or receive a non-trial treatment instead).