Principal Stratification. Often when comparing treatment effects on outcome, other surrogate out- comes and post-treatment characteristics Sobs are measured and available for analysis. Xxxxxxxxx and Xxxxx (2002) suggest that the estimating the effect of treatment within a group of individuals who have the same measure of Sobs will result in a causal estimand. They formally define principal stratifi- cation in two steps. First, the basic principal stratification P 0 is the partition of units i = 1,2,...,n, such that within any P 0 all units will have the same vector Si(), Si(). Si(0) and Si(1) are potential outcomes of Si that vary by treatment assignment, where only one value is actually observed (Si(Zi)). Second, principal stratification P with respect to post treatment variables S is a partition of the units whose sets are unions of sets in the basic principal stratification P 0 (Xxxxxxxxx & Xxxxx 2002a). Noteworthy properties of principal stratification include: (1) treatment as- signment does not affect the stratum SP , (2) an “exclusion” assumption can be made so that if treatment assignment does not effect Si it must not effect the final outcome, and (3) any principal effect (as defined above) is a causal effect. The first property, which states that the individual strata assignment is unaffected by treatment, is very similar to and can be considered an extension of SUTVA, which states that there is no interference between observed treat- ment Zobs and the potential outcomes. The second property allows for the principals of instrumental variables to be utilized when estimating the causal effect within a particular stratum of interest (Angrist et al. 1996, Xxxxx 2005). The principal effect would then be the comparison of outcome under var- ious treatment alternatives within a principal stratum SP = S, specifically i i the effect estimates between the sets ΣYi () : SP Σ and ΣYi () : SP Σ. However, this treatment effect may not be estimable in every strata. Principal effect of treatment on outcome cannot be measured if individuals in the strata, drops out of a study, or does not comply to treatment. Thus the principal or causal i effect of treatment is only estimated in the stratum SP where Si(1) = 1 and Si(0) = 1 (Xxxxxxxxx & Xxxxx 2002a).
Principal Stratification. If a patient is not alive under the treatment that is actually received, Si = 0, the outcome Yi cannot be measured. We may consider those outcomes that are not measured due to patient death as not defined on the set of real positive numbers, R+. Following the notation of Zhang & Xxxxx (2003), we can instead consider the non-observed outcomes to be *, extending our sample space to {R+, ∗}. In the presence of this censoring of the outcome by death, principal stratification using post treatment survival status allows for estimation of the treatment effect. Specifically, the Survivor Average Causal Effect (SACE) is defined as the mean difference in the outcomes of treated individuals compared to untreated individuals in the LL stratum, E(YLL,i(1)) − E(YLL,i(0)). The four potential principal strata are constructed by pairing indicators of survival by treatment scenarios at time of outcome measurement, as defined below. • LL = {i|Si(1) = 1, Si(0) = 1}, or those patients who would be alive at time to regardless of treatment. • LD = {i|Si(1) = 1, Si(0) = 0}, or those patients who at time to would be alive if they receive the PEG tube but would not be alive if they did not. • DL = {i|Si(1) = 0, Si(0) = 1}, or those patients who at time to would not be alive if they receive PEG treatment, but would be alive if they did not. • DD = {i|Si(1) = 0, Si(0) = 0}, or those patients who would not be alive at time to regardless of treatment.
Principal Stratification. Even if all outcomes are observed for living patients, the outcome Yi can- not be measured for those patients who are not alive at the time of outcome measurement, Si = 0. Those outcomes that are not measured due to patient death are considered not defined on the set of real positive numbers, R+. Fol- lowing the notation of Zhang & Xxxxx (2003), we can instead consider the non-observed outcomes to be *, extending our sample space to {R+, ∗}. In the presence of the censoring of the outcome by death, principal stratification using post treatment survival status allows for estimation of the treatment ef- fect. Specifically, the Survivor Average Causal Effect (SACE) is defined as the mean difference in the outcomes of treated individuals compared to untreated individuals in the LL stratum, E(YLL,i(1)) − E(YLL,i(0)). The four potential principal strata are constructed by pairing indicators of survival by treatment scenarios: LL, LD, DL, and DD. Individuals who are in the LL stratum are those who would be alive at time to regardless of treatment. Those individuals who are in the LD stratum would be alive if they received the PEG tube but would not be alive if they did not and those who are in the DL stratum would not be alive if they receive PEG treatment, but would be alive if they did not. Finally, individuals in the DD are those who would not be alive at time to regardless of treatment.
