Principal stratification has become an increasingly popular approach to thinking about certain classes of causal effects. The notion of principal stratification is most closely associated with a paper of

Frangakis and Rubin (2002). Although the idea of principal stratification had clear antecedents (

Robins, 1986;

Angrist et al., 1996),

Frangakis and Rubin (2002) proposed that this approach to thinking about causal effects be used to address a broad class of related problems concerning noncompliance, censoring-by-death, and surrogate outcomes. In his commentary,

Pearl (2011) has asked the causal inference community to reflect on and clarify the specific value of the “principal stratification framework.” Pearl offered a four-fold classification of what he sees as the uses and misuses of the principal stratification framework: (i) partitioning of response types, (ii) defining effects that approximate those of interest (he lists non-compliance as an example), (iii) defining effects that are of genuine interest (he lists censorship by death as an example), and (iv) imposing an intellectual restriction e.g. by not allowing for counterfactuals defined by interventions on an intermediate (he lists surrogate outcomes and mediation as examples). In what follows I will offer my own thoughts on this issue and briefly survey the range of applications which have employed principal stratification ideas. As will be seen below, I believe a more nuanced evaluation is merited. The utility of the framework varies considerably across applications. Moreover, I would likely put non-compliance in the third, rather than the second, of Pearl’s four categories. And perhaps more importantly, I think a sharp distinction should be drawn between surrogate outcomes and mediation; formally, the two applications look somewhat similar but the questions that are asked are in fact quite different and I believe that principal stratification holds more promise for the former than the latter. I will return to these points below.

We first review the notion of principal stratification itself. Stated briefly, if

*X* denotes some binary treatment and

*S* some post-treatment variable and if we let

*S*_{x} denote the potential outcome (

Rubin, 1974) for each individual that we would have observed had

*X*, possibly contrary to fact, been

*x*, then a principal stratum is simply a subgroup of individuals homogenous in their joint potential outcomes (

*S*_{0},

*S*_{1}). If

*S* is also binary then we have four principal strata: (

*S*_{0} = 0,

*S*_{1} = 0) sometimes called “never-takers”, (

*S*_{0} = 0,

*S*_{1} = 1) sometimes called “compliers”, (

*S*_{0} = 1,

*S*_{1} = 0) sometimes called “defiers”, and (

*S*_{0} = 1,

*S*_{1} = 1) sometimes called “always takers.” Suppose that, in addition, we have some other outcome

*Y* and let

*Y*_{x} denote the potential outcome for each individual that we would have observed had

*X*, possibly contrary to fact, been

*x*. The overall causal effect for the population is then given by

*E*[

*Y*_{1} –

*Y*_{0}]. However, we could also consider the causal effect of

*X* on

*Y* conditional on the principal stratum i.e.

*E*[

*Y*_{1} –

*Y*_{0}|

*S*_{0} =

*s*_{0}*, S*_{1} =

*s*_{1}]. This is what

Frangakis and Rubin (2002) call a “principal causal effect.” Conditioning on the principal stratum has certain advantages over conditioning on the observed post-treatment variable

*S*. In general, if we condition on a post-treatment variable

*S*, we will induce bias in analysis (see

Shpitser et al., 2010, for exceptions). However, the principal stratum, (

*S*_{0},

*S*_{1}), that an individual belongs to is essentially viewed as a pretreatment characteristic of an individual and thus we can, in principle, stratify on it as we could any other pretreatment variable. The difficulty is that we do not know who is in which principal stratum. As is discussed below, this creates problems for identifying quantities like

*E*[

*Y*_{1} –

*Y*_{0}|

*S*_{0} =

*s*_{0},

*S*_{1} =

*s*_{1}] from observed data and it also makes it difficult to know who the individuals are to which such estimates apply.

Applications of the notion of principal stratification to develop methodology to address a wide range of problems now abound. This approach has been used in the context of non-compliance, censoring by death, and the related problem of post-infection outcomes, and more recently to issues of surrogate outcomes and “mediation.” I will briefly discuss each of these with a special focus on the final topic, as this area of application has been somewhat more controversial.