On the identification of individual level pleiotropic, pure direct, and principal stratum direct effects without cross world assumptions

Jaffer M Zaidi; Tyler J VanderWeele

doi:10.1111/sjos.12464

. Author manuscript; available in PMC: 2024 Feb 5.

Published in final edited form as: Scand Stat Theory Appl. 2020 Apr 24;48(3):881–907. doi: 10.1111/sjos.12464

On the identification of individual level pleiotropic, pure direct, and principal stratum direct effects without cross world assumptions

Jaffer M Zaidi ¹, Tyler J VanderWeele ²

PMCID: PMC10839820 NIHMSID: NIHMS1945761 PMID: 38317823

Abstract

The analysis of natural direct and principal stratum direct effects has a controversial history in statistics and causal inference as these effects are commonly identified with either untestable cross world independence or graphical assumptions. This article demonstrates that the presence of individual level natural direct and principal stratum direct effects can be identified without cross world independence assumptions. We also define a new type of causal effect, called pleiotropy, that is of interest in genomics, and provide empirical conditions to detect such an effect as well. Our results are applicable for all types of distributions concerning the mediator and outcome.

Keywords: direct effects, mediation, missing data, natural direct effects, pleiotropy, principal stratum

1 ∣. INTRODUCTION

1.1 ∣. Natural and principal stratum direct effects

The analysis of natural direct and principal stratum direct effects has a long history in statistics and causal inference (Frangakis & Rubin, 2002; Pearl, 2009; Robins & Greenland, 1992; VanderWeele, 2015). All of these effects are defined in a counterfactual or potential outcome framework (Frangakis & Rubin, 2002; Pearl, 2009; Robins & Greenland, 1992; VanderWeele, 2015), but these effects are commonly identified using either untestable cross world independence assumptions or graphical assumptions (Frangakis & Rubin, 2002; Pearl, 2009; Robins & Greenland, 1992; VanderWeele, 2015). Consequently, the identification and use of these effects in research has elicited controversy from authorities across varied disciplines (Frangakis & Rubin, 2002; Pearl, 2009; Robins & Greenland, 1992; VanderWeele, 2015). The invited discussion section of a recent article elucidates some of the controversy surrounding natural direct and indirect effects (Imai, Tingley, & Yamamoto, 2013).

We detail a few of the issues surrounding natural direct effects. The first concern is that these effects are identified under unverifiable and untenable assumptions that usually offer no experimental design to detect and quantify individual level direct and indirect effects. The second grievance is that the cross world counterfactuals that are used to define natural direct and indirect effects might not even exist for all or a subset of the population, and therefore using cross world counterfactuals that might not even have a metaphysical interpretation or even exist in the simplest thought experiments present a deep challenge to scientists and statisticians who are concerned with interpreting the results of actual experiments. Statisticians have attempted to overcome these first two challenges through different approaches (Imai et al., 2013). In the online supplement, the first author details a few of his concerns regarding natural direct and indirect effects that are not shared with the second author.

The first contribution of this article demonstrates that individual level natural direct and principal stratum direct effects can sometimes be empirically detected and quantified without cross world independence assumptions. We first provide the simpler theory and methodology to quantify individual level direct effects in the setting of binary variables, and then follow up with the more complex theory and methods to quantify individual level direct effects in the setting of tuple or vector valued continuous mediator and/or outcome. The second contribution of this article is deriving lower bounds on the magnitude of the detected direct effect in comparison with the other causal effects. Such bounds enable statisticians to uncover mechanisms of the effect of a treatment on an outcome. Related results when one assumes that the treatment has a monotonic effect on the potential mediator are also derived. We also offer causally interpretable and user friendly sensitivity analysis for assessing the consequences of violations of the monotonicity assumption.

As a consequence of our work, the quantification of individual level direct effects can sometimes be carried out with no stronger identification assumptions than the assumptions needed to analyze total effects in randomized clinical trials (Imbens & Rubin, 2015; VanderWeele, 2015). Finally, we also define “individual level pleiotropic effects” to be present when a treatment is shown to cause two different outcomes in a single individual in the population. Such an effect is of interest in understanding the etiology of different phenotypes from single gene or allele.

2 ∣. QUANTIFICATION OF INDIVIDUAL LEVEL DIRECT EFFECTS

2.1 ∣. Notation and assumptions

Let $Y$ denote a binary outcome. In addition, suppose a binary treatment $X \in {0, 1}$ is randomized at baseline, and a binary variable $M \in {0, 1}$ is also observed at a time between measurement of $Y$ and the assignment of treatment $X$ . The individuals or units in our study, denoted by symbol $ω$ compose a finite population, denoted $Ω$ . Define counterfactuals $Y_{x} (ω)$ and $M_{x} (ω)$ to be the value of $Y$ and $M$ , respectively, had we set the value of the treatment $X$ to $x$ for individual $ω$ . The manipulated counterfactual $Y_{x, m} (ω)$ denotes the value of $Y (ω)$ had we set $X$ to $x$ and $M$ to $m$ . Finally, define the counterfactual $Y_{x, M_{x^{'}}} (ω)$ to be the value of $Y$ had we set the value of the treatment $X$ to $x$ and set the value of the variable $M$ to the value it would take had we set $X$ to some value $x^{'}$ , that could be different to $x$ for individual $ω$ . Denote $P_{c} (y_{1}, y_{0}, m_{1}, m_{0}) = P (Y_{1} = y_{1}, Y_{0} = y_{0}, M_{1} = m_{1}, M_{0} = m_{0})$ , which under the composition axiom is equal to $P (Y_{1, M_{1}} = y_{1}, Y_{0, M_{0}} = y_{0}, M_{1} = m_{1}, M_{0} = m_{0})$ .

The individual level total effect is defined as $Y_{1} (ω) - Y_{0} (ω)$ . The individual pure direct effect is defined as $Y_{1, M_{0}} (ω) - Y_{0, M_{0}} (ω)$ and the total indirect is defined as $Y_{1, M_{1}} (ω) - Y_{1, M_{0}} (ω)$ (Robins & Greenland, 1992). Similarly, the total direct is defined as $Y_{1, M_{1}} (ω) - Y_{0, M_{1}} (ω)$ , and the individual pure indirect effect is defined as $Y_{0, M_{1}} (ω) - Y_{0, M_{0}} (ω)$ (Robins & Greenland, 1992). These pure and total direct and indirect effects are sometimes also referred to as natural direct and indirect effects (Pearl, 2009). The individual controlled direct effect is defined as $Y_{1, m} (ω) - Y_{0, m} (ω)$ . We say there is an individual principal stratum direct effect if for some individual $ω$ for whom $M_{1} (ω) = M_{0} (ω) = m$ , we have $Y_{1} (ω) - Y_{0} (ω) \neq 0$ . Population total, total direct, pure direct, total indirect, pure indirect, and controlled direct effects are derived through taking expectations of the relevant individual level effects. The population level principal stratum direct effect of stratum $M_{1} (ω) = M_{0} (ω) = m$ is defined as $E (Y_{1} - Y_{0} ∣ M_{1} = M_{0} = m)$ for some $m \in {0, 1}$ (Frangakis & Rubin, 2002). For binary $M$ and $Y$ , the unstandardized principal stratum direct effects for some $m \in {0, 1}$ and $y \in {0, 1}$ is given as $P (Y_{1} = y, Y_{0} = 1 - y, M_{1} = m, M_{0} = m) - P (Y_{1} = 1 - y, Y_{0} = y, M_{1} = m, M_{0} = m)$ . The corresponding standardized principal stratum direct effects for some $y \in {0, 1}$ and $m \in {0, 1}$ is given as

\frac{P (Y_{1} = y, Y_{0} = 1 - y, M_{1} = m, M_{0} = m) - P (Y_{1} = 1 - y, Y_{0} = y, M_{1} = m, M_{0} = m)}{P (M_{1} = m, M_{0} = m)} .

The proportion of individuals for whom treatment causes (or prevents) the outcome among the principal stratum $M_{1} (ω) = M_{0} (ω) = m$ for $y = 1$ ( $y = 0$ , respectively) is denoted $P (Y_{1} = y, Y_{0} = 1 - y ∣ M_{1} = m, M_{0} = m)$ . Each of the principal stratum direct effects presented herein differs from the usual definition of $E (Y_{1} - Y_{0} ∣ M_{1} = M_{0} = m)$ provided by Frangakis and Rubin (2002). Only in the case of a binary outcome does the usual Frangakis and Rubin (2002) definition of a principal stratum direct effect $E (Y_{1} - Y_{0} ∣ M_{1} = M_{0} = m)$ and the standardized principal direct effect $P (Y_{1} = y, Y_{0} = 1 - y ∣ M_{1} = m, M_{0} = m) - P (Y_{1} = 1 - y, Y_{0} = y ∣ M_{1} = m, M_{0} = m)$ coincide with each other. We emphasize each type of causal estimand to highlight the relative dissimilarity and relations between each of these population level causal estimands for researchers to understand and deliberate on what needs to be estimated. In addition to the question of which causal estimand is of practical and scientific relevance, researchers will also need to ask how are these causal contrasts estimated solely from the observed data distribution?

With treatment randomized at baseline, one may make the “weak ignorability” assumption $(Y_{x}, M_{x}) ∐ X$ . This is the only assumption that is guaranteed by design of a randomized trial. Unlike typical approaches to identify natural direct and indirect effects, no cross world independence assumptions of the form $Y_{x, m} ∐ M_{x^{'}} ∣ X$ (Pearl, 2009) are used to derive our results. Throughout, we require the consistency assumption for both $Y_{x}$ and $M_{x}$ , which means that when $X (ω) = x$ then $Y_{x} (ω) = Y (ω)$ and $M_{x} (ω) = M (ω)$ . This assumption states that the value of $Y$ and $M$ that would have been observed if $X$ had been set to what in fact they were observed to be is equal respectively to the value of $Y$ and $M$ that was observed. In addition, the consistency assumption for $Y_{x, m}$ means that when $X (ω) = x$ and $M (ω) = m$ , then $Y_{x, m} (ω) = Y (ω)$ . Finally, we assume the composition axiom, which states that when $M_{x} (ω) = m$ for some $m \in {0, 1}$ , then $Y_{x, m} (ω) = Y_{x} (ω)$ . The randomization, consistency, and composition assumptions are the only assumptions necessary to derive our results. The assumptions needed to quantify such direct effects in observational studies is provided in the online supplement. Proofs of all results are given in the Appendix.

2.2 ∣. Identification of direct effects under randomization

Theorem 1. Suppose $X$ is randomized at baseline. If for some $y \in {0, 1}$ and $m \in {0, 1}$ , $P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) > 1$ , then there exists a nonempty subpopulation $Ω_{s} \subseteq Ω$ such that for every $ω \in Ω_{s}$ , $Y_{1} (ω) = y$ , $Y_{0} (ω) = 1 - y$ , and $M_{1} (ω) = M_{0} (ω) = m$ .

Corollary 1. Suppose $X$ is randomized at baseline. If the empirical condition $P (Y = 1, M = 0 ∣ X = 1) + P (Y = 0, M = 0 ∣ X = 0) > 1$ holds, then there exists a nonempty subset $Ω_{s} \subseteq Ω$ such that for every $ω \in Ω_{s}$ , $Y_{1, M_{1}} (ω) - Y_{0, M_{0}} (ω) = Y_{1, M_{1}} (ω) - Y_{0, M_{1}} (ω) = Y_{1, M_{0}} (ω) - Y_{0, M_{0}} (ω) = Y_{1} (ω) - Y_{0} (ω) = 1$ , with $M_{1} (ω) = M_{0} (ω) = 0$ , and $Y_{1, M_{1}} (ω) - Y_{1, M_{0}} (ω) = Y_{0, M_{1}} (ω) - Y_{0, M_{0}} (ω) = 0$ , that is, the total effect, total direct effect, pure direct effect, principal stratum direct effect with $M_{1} (ω) = M_{0} (ω) = 0$ are all equal to 1, and the total indirect and pure indirect effects for individual $ω$ are both zero.

Theorem 1 and Corollary 1 allow for the empirical detection of individual natural and principal stratum direct effects. A somewhat related set of results to Theorem 1 are the empirical condition to detect individual level sufficient cause interaction (VanderWeele & Robins, 2008; VanderWeele & Richardson, 2012; Roland, 2013). Theorem 1 demonstrates that such logic can also be used to detect individual level natural and principal stratum direct effects. Each of the four empirical conditions provided in Theorem 1 will correctly identify when an individual principal stratum direct effect is present in the population. Readers familiar with literature on the falsification of the binary instrumental variable model (Balke & Pearl, 1997) will recognize that each of the empirical conditions in Theorem 1 as the complement of one of four “instrumental inequalities” (Balke & Pearl, 1997; Swanson, Hernán, Miller, Robins, & Richardson, 2018). The result closest to ours is the result of Cai, Kuroki, Pearl, and Tian (2008) who demonstrate that nonzero population average controlled direct effects can be identified under the “strong ignorability” assumption through rejecting the “instrumental inequality” (Cai et al., 2008), but their result does not make any statement about individual or population principal stratum or natural direct effects.

Richardson and Robins (2010) also discuss implications of falsifying the instrumental variable model, but their results focus on the controlled direct effect and make no mention of individual principal stratum direct effects. Sjölander (2009) provides bounds for the population natural direct effect. These bounds on the population natural direct effect are different to the Cai et al. (2008) bounds on the population controlled direct effect. It is simple to demonstrate that if the Sjölander (2009) lower bound (upper bound) of the natural direct effect is greater than zero (less than zero), respectively, then one of the inequalities associated with Theorem 1 holds. However, if we consider the converse, that is, we can have an inequality with Theorem 1 can hold, but the Sjölander (2009) bounds for the population natural direct effect could still include zero. The difference stems from Sjölander (2009) presenting bounds on the population average natural direct effects, while we present lower bounds on the proportion of individuals that display a particular form of a principal stratum direct effect and consequently a natural direct effect.

Other authors derive bounds on the principal stratum direct effects assuming some form of monotonicity assumption and additionally that the relevant principal stratum exists (Hudgens, Hoering, & Self, 2003; Zhang & Rubin, 2003), whereas our results demonstrates that when the inequality is satisfied the relevant principal stratum must in fact exist, as shown below, and provides information on the proportion of individuals that display such a principal stratum direct effect in comparison with other counterfactual responses. In comparison with other approaches, we only require randomization at baseline, and we bound the proportion of individuals that display a particular direct effect. To the best of our knowledge, we are the first to detect and quantify the magnitude of an individual level direct effect with only the assumptions that are guaranteed by a randomized experiment.

Natural direct and indirect effects are identified also if one assumes an underlying nonparametric structural equation models that imply the cross world independence assumption (Pearl, 2009). We do not make that assumption here. Moreover, under a nonparametric structural model framework, these natural direct effects fail to be identified if there exists a postbaseline confounder between $M$ and $Y$ that is also effected by treatment $X$ (Avin, Shpitser, & Pearl, 2005). However, Theorem 1 will still correctly identify nonzero direct effects even in the presence of a confounder between $M$ and $Y$ that is affected by treatment $X$ . Imai et al. (2013) use a more elaborate experimental design in which both treatment and mediator are randomized to identify the presence of indirect effects, but we are not assuming randomization of the mediator, only treatment. We now give an additional result on the distribution of counterfactual response types corresponding to direct effects.

Proposition 1. Suppose $X$ is randomized at baseline. For some $y \in {0, 1}$ and $m \in {0, 1}$ , the expression $P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) - 1$ is equal to

P_{c} (y, 1 - y, m, m) - {P_{c} (1 - y, y, m, m) + P_{c} (1 - y, 1 - y, m, 1 - m) + P_{c} (1 - y, y, m, 1 - m) + P_{c} (y, y, 1 - m, m) + P_{c} (1 - y, y, 1 - m, m) + P_{c} (y, 1 - y, 1 - m, 1 - m) + P_{c} (1 - y, 1 - y, 1 - m, 1 - m) + P_{c} (y, y, 1 - m, 1 - m) + P_{c} (1 - y, y, 1 - m, 1 - m)} .

The proposition above provides the full consequence of rejecting the instrumental inequality. The following corollary provides researchers with the ability to bound the magnitude of the direct effect of interest with only the assumption of randomization at baseline. We note here that such results are nowhere present in any text or article, despite the controversy surrounding direct and indirect effects.

Corollary 2. Suppose $X$ is randomized at baseline. For some $m \in {0, 1}$ and $y \in {0, 1}$ , the expression $P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) - 1$ is a lower bound of $P_{c} (y, 1 - y, m, m) - P_{c} (1 - y, y, m, m)$ .

If for some $m \in {0, 1}$ and $y \in {0, 1}$ , the inequality $P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) > 1$ holds, then the expression

\frac{P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) - 1}{P (M = m ∣ X = 1) - P (M = 1 - m ∣ X = 0)}

is a lower bound of

\frac{P_{c} (y, 1 - y, m, m) - P_{c} (1 - y, y, m, m)}{P (M_{1} = M_{0} = m)},

which relates to the standardized population principal stratum direct effect. This standardized population principal stratum direct effect is equivalent to

P (Y_{1} = y, Y_{0} = 1 - y ∣ M_{1} = m, M_{0} = m) - P (Y_{1} = 1 - y, Y_{0} = y ∣ M_{1} = m, M_{0} = m) .

Theorem 1 demonstrates that individual level principal stratum are identified solely assuming randomization and consistency of counterfactuals. Corollary 1 then demonstrates that this individual level principal stratum direct effect is also a natural direct effect using the composition axiom. Finally, Proposition 1 provides the full equality relationship between the observed world probability contrasts under investigation and counterfactual probabilities that inform mechanism of treatment. This relationship demonstrates that if for some $y \in {0, 1}$ and $m \in {0, 1}$ one rejects a null hypothesis of the form $P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) \leq 1$ , then the counterfactual distribution of ( $Y_{1, M_{1}}, Y_{0, M_{0}}, M_{1}, M_{0}$ ) is constrained. This constraint on the counterfactual distribution has implications for understanding mechanisms in the population. It informs us that the proportion of individuals for whom the treatment has a positive (or negative) total effect on the outcome and simultaneously the treatment does not change the mediator value from some fixed value $m \in {0, 1}$ is greater than the proportion of individuals for whom the treatment has a negative (or positive respectively) total effect and simultaneously the treatment does not change the mediator value from some fixed value $m \in {0, 1}$ .

Corollary 2 tells us the magnitude of the resulting direct effect. From Corollary 2, researchers can use $P (Y = 1, M = 0 ∣ X = 1) + P (Y = 0, M = 0 ∣ X = 0) - 1$ , which is a function only of the observed data distribution, as a lower bound on the risk difference of the proportion of individuals randomized at baseline for whom the treatment leaves the variable $M$ fixed at 0 and causes the outcome $Y$ and the proportion of individuals randomized at baseline for whom the treatment also leaves the variable $M$ fixed at 0 and prevents the outcome $Y$ . In counterfactual notation, this risk difference is given as $P_{c} (1, 0, 0, 0) - P_{c} (0, 1, 0, 0)$ . From Corollary 2, if $P (Y = 1, M = 0 ∣ X = 1) + P (Y = 0, M = 0 ∣ X = 0) > 1$ , then ${P (Y = 1, M = 0 ∣ X = 1) + P (Y = 0, M = 0 ∣ X = 0) - 1} ∕ {P (M = 0 ∣ X = 1) - P (M = 1 ∣ X = 1)}$ is a lower bound on the risk difference between the proportion of individuals for who treatment causes the outcome among the individuals in the principal strata $M_{1} (ω) = M_{0} (ω) = 0$ and the proportion of individuals for who treatment prevents the outcome among the principal strata $M_{1} (ω) = M_{0} (ω) = 0$ , which is denoted $P (Y_{1} = 1, Y_{0} = 0 ∣ M_{1} = M_{0} = 0) - P (Y_{1} = 0, Y_{0} = 1 ∣ M_{1} = M_{0} = 0)$ .

These theorems, propositions, and corollaries provided in this section cannot be improved upon without additional assumptions. In this section, we only assumed randomization at baseline and derived empirical conditions to detect and quantify the magnitude of individual level direct effects. Corollary 2 also provides the greatest lower bound on the proportion of individual level direct effects of interest when only assuming randomization at baseline in terms of the observed data distribution. To derive lower bounds that are greater or sharper than those presented in Corollary 2, monotonicity assumptions are required and are presented next.

Note the bound relating to the standardized population principal stratum direct effect presented in Corollary 2 requires the that $P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) > 1$ . The reason for this requirement is that we first need evidence for an individual level principal stratum direct effect. Otherwise, the statistician has no evidence with from the data alone that the population average principal stratum direct effect is nonzero because if for some $y \in {0, 1}$ and $m \in {0, 1}$ $P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) \leq 1$ then it is entirely possible that there is no individual such that $M_{1} (ω) = M_{0} (ω) = m$ and $Y_{1} (ω) = y, Y_{0} (ω) = 1 - y$ , in the observed data. If $P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) - 1$ is less than zero, then take zero as the lower bound of $P (Y_{1} = y, Y_{0} = 1 - y ∣ M_{1} = m, M_{0} = m)$ , since a probability is always nonnegative.

3 ∣. IDENTIFICATION OF DIRECT EFFECTS WITH MONOTONICITY ASSUMPTIONS

Now consider the setting of positive monotonicity defined as $M_{1} (ω) \geq M_{0} (ω)$ for all $ω \in Ω$ , that is, there is no individual for whom treatment prevents the mediator $M$ from occurring. While such monotonic effects are never verifiable, they are falsifiable and can sometimes be justified with subject matter knowledge. Under the assumption of monotonicity, there is a gain in the capacity to detect direct effects in comparison with the results from the previous section. In addition, if such assumptions are reasonable, the tighter bounds enable detection of two different principal stratum direct effects in a population. We offer causally interpretable and user friendly sensitivity analysis in the setting where the statistician believes that some individuals might violate the monotonicity assumptions to assess the consequence of how such assumptions can impact study conclusions.

3.1 ∣. Positive monotonicity of exposure on mediator

Theorem 2. Suppose $X$ is randomized at baseline. In addition, suppose that $X$ has a positive monotonic effect on $M$ . If for some $m \in {0, 1}$ and $y \in {0, 1}$ the inequality $P (Y = 1 - y, M = m ∣ X = 1 - m) > P (Y = 1 - y, M = m ∣ X = m)$ holds, then there exists a nonempty subpopulation $Ω_{s} \subseteq Ω$ such that for every $ω \in Ω_{s}$ , $Y_{1} (ω) = y m + (1 - m) (1 - y)$ , $Y_{0} (ω) = (1 - y) m + (1 - m) y$ , and $M_{1} (ω) = M_{0} (ω) = m$ .

Corollary 3. Suppose $X$ is randomized at baseline. In addition suppose that $X$ has a positive monotonic effect on $M$ . If the inequality $P (Y = 1, M = 0 ∣ X = 1) > P (Y = 1, M = 0 ∣ X = 0)$ holds, then there exists a nonempty subpopulation $Ω_{s} \subseteq Ω$ such that for every $ω \in Ω_{s}$ , the total effect, total direct effect, pure direct effect, principal stratum direct effect with $M_{1} (ω) = M_{0} (ω) = 0$ are all equal to 1, and the total indirect and pure indirect effects for $ω$ are both zero.

Proposition 2. Suppose $X$ is randomized at baseline. In addition, suppose that $X$ has a positive monotonic effect on $M$ . For some $m \in {0, 1}$ and $y \in {0, 1}$ , the expression $P (Y = 1 - y, M = m ∣ X = 1 - m) - P (Y = 1 - y, M = m ∣ X = m)$ is equal to

P_{c} (y m + {1 - y} {1 - m}, {1 - y} m + {1 - m} y, m, m) - P_{c} (m {1 - y} + {1 - m} y, y m + {1 - y} {1 - m}, m, m) - P_{c} (m {1 - y} + {1 - m} y, y m + {1 - y} {1 - m}, 1, 0) - P_{c} (1 - y, 1 - y, 1, 0) .

Corollary 4. Suppose $X$ is randomized at baseline. In addition suppose that $X$ has a positive monotonic effect on $M$ . For $m \in {0, 1}$ and $y \in {0, 1}$ , the expression $P (Y = 1 - y, M = m ∣ X = 1 - m) - P (Y = 1 - y, M = m ∣ X = m)$ is a lower bound on the counterfactual risk difference

P_{c} (y m + {1 - y} {1 - m}, {1 - y} m + {1 - m} y, m, m) - P_{c} (m {1 - y} + {1 - m} y, y m + {1 - y} {1 - m}, m, m),

and when $P (Y = 1 - y, M = m ∣ X = 1 - m) > P (Y = 1 - y, M = m ∣ X = m)$ the expression

\frac{P (Y = 1 - y, M = m ∣ X = 1 - m) - P (Y = 1 - y, M = m ∣ X = m)}{P (M = m ∣ X = 0)}

is a lower bound on

\frac{(\begin{matrix} P_{c} (y m + {1 - y} {1 - m}, {1 - y} m + {1 - m} y, m, m) \\ - P_{c} (m {1 - y} + {1 - m} y, y m + {1 - y} {1 - m}, m, m) \end{matrix})}{P (M_{1} = M_{0} = m)},

which relates to the standardized principal stratum direct effect. This standardized population principal stratum direct effect is equivalent to

P (Y_{1} = y m + {1 - y} {1 - m}, Y_{0} = {1 - y} m + {1 - m} y ∣ M_{1} = m, M_{0} = m) - P (Y_{1} = m {1 - y} + y {1 - m}, Y_{0} = y m + {1 - y} {1 - m} ∣ M_{1} = m, M_{0} = m) .

Proposition 2 enables similar interpretations of distribution of the counterfactual contrasts that were provided for Proposition 2. If $P (Y = 1, M = 0 ∣ X = 1) > P (Y = 1, M = 0 ∣ X = 0)$ , then the proportion of individuals $ω$ randomized at baseline for whom treatment has a beneficial effect on $Y (ω)$ and $M_{1} (ω)$ and $M_{0} (ω)$ are still zero is greater than the proportion of individuals for whom treatment has a harmful effect on $Y (ω)$ and $M_{1} (ω)$ and $M_{0} (ω)$ are still zero. The “instrumental inequalities” assuming $X$ having a positive monotonic effect on $M$ are for $m \in {0, 1}$ and $y \in {0, 1}$ , $P (Y = 1 - y, M = m ∣ X = 1 - m) \leq P (Y = 1 - y, M = m ∣ X = m)$ (Balke & Pearl, 1997). These “instrumental inequalities” serve as null hypotheses of the corresponding principal stratum direct effect under investigation. At most two of the four inequalities associated with Theorem 2 can hold.

Example Consider a population denoted $Ω$ for which half of the individuals $ω_{1}$ follow counterfactual response type $Y_{1, M_{1}} (ω_{1}) = 1$ , $Y_{0, M_{0}} (ω_{1}) = 0$ , $M_{1} (ω_{1}) = 0$ and $M_{0} (ω_{1}) = 0$ and the other half of individuals $ω_{2}$ follow counterfactual response type $Y_{1, M_{1}} (ω_{2}) = 0$ , $Y_{0, M_{0}} (ω_{2}) = 1$ , $M_{1} (ω_{2}) = 1$ and $M_{0} (ω_{2}) = 1$ . We have $M_{1} (ω) \geq M_{0} (ω)$ for all $ω \in Ω$ . For such a population, the average pure direct effect, $E (Y_{1, M_{0}} - Y_{0, M_{0}})$ , and the total direct effect, $E (Y_{1, M_{1}} - Y_{0, M_{1}})$ , are both zero. However, the two null hypotheses associated with Theorem 2 are both falsified, indicating that the two principal stratum direct effects associated with individual response types $ω_{1}$ and $ω_{2}$ are both present in our population. As noted in Richardson and Robins (2010), at most one of the four inequalities associated with Theorem 1 can hold, which means that without the monotonicity assumption, one can at most detect one individual level principal stratum direct effect. However, two of the inequalities associated with Theorem 2 hold. We see that $P (Y = 1, M = 1 ∣ X = 0) > P (Y = 1, M = 1 ∣ X = 1)$ and $P (Y = 1, M = 0 ∣ X = 1) > P (Y = 1, M = 0 ∣ X = 0)$ in this example. This result means that both individual level principal stratum direct effects can be detected using our results from Theorem 2.

Earlier, for the setting where one does not assume $X$ has a positive monotonic effect on $Y$ , we saw Theorem 1 would quantify individual level principal stratum and natural direct effects even when the Sjölander (2009) bounds for the natural direct effect included zero. The key difference between our results and Sjölander (2009) results for the monotonicity setting is that we only assume that $X$ has a monotonic effect on $M$ , whereas Sjölander (2009) makes additional monotonicity assumptions. We are able to detect two different individual level principal stratum direct effects even under the scenario where the total effect or natural direct effect is zero. Detection of two different principal stratum direct effects is not previously described in literature (Balke & Pearl, 1997; Cai et al., 2008; Richardson & Robins, 2010; Sjölander, 2009; Swanson et al., 2018).

3.2 ∣. Sensitivity analysis for monotonicity assumptions

As demonstrated in Theorem 1, Corollary 1, and Proposition 1, we provide methods to detect individual level direct effects and characterize the magnitude of such individual level direct effects with only assumptions that are guaranteed by design. To the best of our knowledge, most methods assume that the nonparametric structural equation model, influence diagram, or directed acyclic graph that is used to point identify the direct effect is correctly specified and available to the analyst (Pearl, 2009). Therefore, most if not all approaches require assumptions that are not only unverifiable, but they also offer no experimental design where these assumptions are guaranteed by design.

Monotonicity assumptions are not guaranteed by design of experiment. However, without these monotonicity assumptions, the tests to detect individual level direct effects might be too stringent to detect even large individual level direct effects in the population. In such settings, where the investigator is unsure if the monotonicity assumptions hold for all individuals in the population, informative sensitivity analysis enables statisticians to interpret their results even when these monotonicity assumptions are violated and also precisely quantifies how violations of the monotonicity assumptions impact the magnitude of the proposed individual level direct effect. The next theorem provides all the required ingredients necessary to conduct informative and useful sensitivity analyses.

Theorem 3. Suppose $X$ is randomized at baseline. For $m \in {0, 1}$ and $y \in {0, 1}$ , the expression $P (Y = 1 - y, M = m ∣ X = 1 - m) - P (Y = 1 - y, M = m ∣ X = m) - r$ , where

r = P_{c} (y m + {1 - m} {1 - y}, {1 - y} m + {1 - m} y, 0, 1) + P_{c} (1 - y, 1 - y, 0, 1) - P_{c} (y {1 - m} + {1 - y} m, {1 - y} {1 - m} + y m, 1, 0) - P_{c} (1 - y, 1 - y, 1, 0)

is equal to the counterfactual risk difference

P_{c} (y m + {1 - m} {1 - y}, {1 - y} m + {1 - m} y, m, m) - P_{c} ({1 - y} m + y {1 - m}, {1 - y} {1 - m} + y m, m, m),

and the expression

\frac{P (Y = 1 - y, M = m ∣ X = 1 - m) - P (Y = 1 - y, M = m ∣ X = m) - r}{P (M = m ∣ X = 1) - q},

where $q = P (M_{1} = m, M_{0} \neq m)$ , is equal to

\frac{(\begin{matrix} P_{c} (y m + {1 - y} {1 - m}, {1 - y} m + {1 - m} y, m, m) \\ - P_{c} (m {1 - y} + {1 - m} y, y m + {1 - y} {1 - m}, m, m) \end{matrix})}{P (M_{1} = M_{0} = m)},

which relates to the standardized principal stratum direct effect. This standardized population principal stratum direct effect is equivalent to

P (Y_{1} = y m + {1 - y} {1 - m}, Y_{0} = {1 - y} m + {1 - m} y ∣ M_{1} = m, M_{0} = m) - P (Y_{1} = m {1 - y} + y {1 - m}, Y_{0} = y m + {1 - y} {1 - m} ∣ M_{1} = m, M_{0} = m) .

The quantities $r$ and $q$ are not point identified in randomized studies. Consequently, the statistician or scientist will need to consider a range of suitable values. A large $r$ will attenuate the proposed proportion of individual level direct effect as well as the standardized principal stratum direct effect, while in contrast a large $q$ will amplify the proposed standardized principal stratum direct effect. As $r$ and $q$ are functions only of potential outcomes, this enables scientists to directly consider values of sensitivity parameters that are intrinsically related to the physical experiment. Here, $r$ should never exceed $P (Y = 1 - y, M = m ∣ X = 1 - m) - P (Y = 1 - y, M = m ∣ X = m)$ and $q$ should never exceed $P (M = m ∣ X = 1)$ .

The following example from a hypothetical randomized experiment illustrates how sensitivity analysis parameter $r$ can be used to quantify how of individual level violations of the monotonicity assumption can impact the study findings. Suppose a statistician estimates $P (Y = 0, M = 1 ∣ X = 0) - P (Y = 0, M = 1 ∣ X = 1)$ as 0.3. Then, for her study, the statistician would need at least 30% of the individuals randomized in our study to follow counterfactual response types $Y_{1} (ω) = 1$ , $Y_{0} (ω) = 0$ , $M_{1} (ω) = 0$ , $M_{0} (ω) = 1$ or $Y_{1} (ω) = 0$ , $Y_{0} (ω) = 0$ , $M_{1} (ω) = 0$ , $M_{0} (ω) = 1$ for there to be no individual that follows the counterfactual response type $Y_{1} (ω) = 1$ , $Y_{0} (ω) = 0$ , $M_{1} (ω) = 1$ , $M_{0} (ω) = 1$ in her population. Similarly, if the scientist reasons that at most 10% of her study population follows counterfactual response types $Y_{1} (ω) = 1$ , $Y_{0} (ω) = 0$ , $M_{1} (ω) = 0$ , $M_{0} (ω) = 1$ or $Y_{1} (ω) = 0$ , $Y_{0} (ω) = 0$ , $M_{1} (ω) = 0$ , $M_{0} (ω) = 1$ , then she can conclude that at least 20% of her population follows counterfactual response type $Y_{1} (ω) = 1$ , $Y_{0} (ω) = 0$ , $M_{1} (ω) = 1$ , $M_{0} (ω) = 1$ . Prior studies or subject matter expertise can help statisticians understand how individual level violations of monotonicity assumptions can impact their findings.

If the statistician is completely averse to sensitivity analysis, then the results associated with Theorem 1, Corollary 1, Proposition 1, and Corollary 2 are all still available to her. However, such sensitivity analysis enables the statistician to detect and quantify individual level directs when methods that assume randomization alone are too stringent to detect and quantify individual level direct effects. In this example, if the statistician takes $r = P (Y = 1, M = 0 ∣ X = 1) + P (Y = 0, M = 0 ∣ X = 1)$ and $q = P (M = 0 ∣ X = 0)$ , she will essentially be conducting the same tests associated with Theorem 1 and Corollary 2 as expressions in Theorem 3 with the proposed $q$ and $r$ will take us back to exactly the same expressions as those provided in Corollary 2.

Thus, the sensitivity analysis provided here can be viewed as a compromise between the stringent setting where only baseline randomization is assumed and the more flexible setting where in addition to baseline randomization the statistician assumes positive monotonicity. Some fixed $r$ in Theorem 3 will provide the true value of $P_{c} (y, 1 - y, m, m) - P_{c} (1 - y, y, m, m)$ and for a fixed $r$ and $q$ in Theorem 3 will provide the standardized principal stratum direct effect, but unfortunately there is no mechanism through which the statistician can point identify this $r$ and $q$ . Consequently, a range of suitable values for $r$ and $q$ will need to be considered, and scientists can deliberate whether the proposed values are sensible.

4 ∣. PLEIOTROPIC EFFECTS

In addition to variables and assumptions provided in Section 2, let $Z$ denote an additional binary outcome. As before, assume treatment $X$ is randomized, so that $X ∐ (Z_{x}, Y_{x})$ . We also require the consistency assumption for $Y_{x}$ and $Z_{x}$ . This section will define pleiotropic effects, which are of importance in the context of making inferences about the effect of a single gene or treatment on two distinct outcomes or phenotypes (Solovieff, Cotsapas, Lee, Purcell, & Smoller, 2013). The results provided herein enable scientists to discover such effects. Pleiotropic effects are of interest in the detection of genetic variants that may operate through multiple pathways or mechanisms.

Pleiotropic effects are also of interest in the context of Mendelian randomization studies that use a genetic variant as an instrumental variable because pleiotropic effects will often indicate that the exclusion restriction required for the instrumental variable analysis is violated. We give a formal counterfactual definition for the intuitive notion of pleiotropic effects as follows.

This section does not use notation $Y_{x, m}$ or $Y_{x, M_{x^{'}}}$ , from the previous three sections.

Definition 1 (Pleiotropic Effects). A treatment or exposure $X$ is defined to have an individual level pleiotropic effect on outcomes $Y$ and $Z$ if there exists an individual $ω \in Ω$ for whom one of the following four counterfactual response types holds:

Y_{1} (ω) = 1, Y_{0} (ω) = 0, Z_{1} (ω) = 1, Z_{0} (ω) = 0; Y_{1} (ω) = 1, Y_{0} (ω) = 0, Z_{1} (ω) = 0, Z_{0} (ω) = 1; Y_{1} (ω) = 0, Y_{0} (ω) = 1, Z_{1} (ω) = 1, Z_{0} (ω) = 0; Y_{1} (ω) = 0, Y_{0} (ω) = 1, Z_{1} (ω) = 0, Z_{0} (ω) = 1 .

For the first condition, the treatment $X$ is causative for both $Y$ and $Z$ . For the second condition, the treatment $X$ is causative for $Y$ and preventative for $Z$ . For the third condition, the treatment is preventative for $Y$ and causative for $Z$ . Finally, for the last condition, $X$ is preventative for both $Z$ and $Y$ .

Theorem 4. Suppose $X$ is randomized at baseline. If $P (Y = 1, Z = 1 ∣ X = 1) + P (Y = 0, Z = 0 ∣ X = 0) > 1$ , then there exists a nonempty subpopulation $Ω_{s} \subseteq Ω$ such that for all $ω \in Ω_{s}$ , $Y_{1} (ω) = 1$ , $Y_{0} (ω) = 0$ and $Z_{1} (ω) = 1$ , $Z_{0} (ω) = 0$ .

The expression $P (Y = 1, Z = 1 ∣ X = 1) + P (Y = 0, Z = 0 ∣ X = 0) - 1$ is equal to

P (Y_{1} = 1, Y_{0} = 0, Z_{1} = 1, Z_{0} = 0) - \sum_{(y_{1}, z_{1}) \in {0, 1}^{2} ∖ {1, 1}} P (Y_{1} = y_{1}, Z_{1} = z_{1}, (Y_{0}, Z_{0}) \neq (0, 0)),

and is therefore a lower bound for $P (Y_{1} = 1, Y_{0} = 0, Z_{1} = 1, Z_{0} = 0)$ .

The other three counterfactual response types can be likewise detected using analogous results collected in the online supplement. Theorem 4 enables the detection of individual level pleiotropic effects, but does not distinguish between two different forms of individual level pleiotropy. The first situation is where $X$ causes $Y$ which in turn causes $Z$ (or $X$ causes $Z$ which in turn causes $Y$ ). The second situation is where $X$ causes $Y$ through a pathway not through $Z$ and $X$ causes $Z$ through a pathway not through $Y$ . The first situation is called “mediated pleiotropy” and the second situation is known as “biologic pleiotropy” (Solovieff et al., 2013). Theorem 4 allows an investigator to detect if at least one of these two types of pleiotropy is present.

5 ∣. DATA ANALYSIS

Yerushalmy (1964) noted that maternal smoking $X \in {0, 1}$ is associated with lower infant mortality $Y \in {0, 1}$ among low birth weight infants $M \in {0, 1}$ . Other authors after Yerushalmy (Hernández-Díaz, Schisterman, & Hernán, 2006; VanderWeele, Mumford, & Schisterman, 2012) have replicated this finding and referred to this finding as the birth weight paradox, as the direction of the association is opposite of what is expected. Here, $X = 1$ denotes that the mother smoked cigarettes, and $X = 0$ denotes that the mother did not smoke cigarettes; $M = 1$ denotes that the mother had a low birth weight baby (less than 2,500 g), and $M = 0$ denotes that the mother did not have a low birth weight baby; $Y = 1$ denotes that the baby died 1 year after birth, and $Y = 0$ denotes that the baby lived at least 1 year after birth. Specifically, Yerushalmy (1964) reported finding that, among low birth weight infants of less than 2,500 g, the observed risk of infant mortality was greater for nonsmoking mothers than smoking mothers. The “paradox” stems from the realization that smoking is known to be detrimental to human health, but in the analysis of Yerushalmy (1964) it seemingly appeared as though for babies that are born with low birth weight maternal smoking was protective for infant mortality. The two sets of 2 × 2 contingency tables are provided to ensure that our results are reproducible.

Table 3 in Yerushalmy (2014) provides us with the raw numbers for the first row of the multiway contingency table provided in Table 1. The second row of Table 1 comes from using table 2 of Yerushalmy (2014) to calculate the raw numbers of neonatal deaths for smokers and nonsmokers from the mortality rates. The column totals are provided by Yerushalmy (2014) as well as the cell entries for the very first row. With the mortality rates for smokers and nonsmokers, we are able to recreate the raw numbers of infant mortality for normal birth weight babies.

TABLE 1.

Yerushalmy's infant mortality data

	$X = 1$				$X = 0$
	$Y = 1$	$Y = 0$	Total		$Y = 1$	$Y = 0$	Total
$M = 1$	27	210	237	$M = 1$	43	154	197
$M = 0$	15	3,474	3,489	$M = 0$	24	5,846	5,870
Total	42	3,684	3,726	Total	67	6,000	6,067

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In Yerushalmy's data of white mothers, the risk difference of infant mortality comparing low birth weight infants whose mothers smoke versus those who do not is $P (Y = 1 ∣ X = 1, M = 1) - P (Y = 1 ∣ X = 0, M = 1) = 27 ∕ 237 - 43 ∕ 197 = 0.114 - 0.218 = - 0.104$ with a two-sided confidence interval of (−0.18, −0.03) (Yerushalmy, 2014). This result might erroneously, foolishly, and catastrophically interpreted as evidence that maternal smoking reduces the risk of infant mortality for low birth weight babies (Yerushalmy, 1964). However, if one posits the existence of a nonparametric structural equation model that generates the observed data (Pearl, 2009), such analysis ignores the possibility of unmeasured factors that are common causes of both low birth weight $M$ and infant mortality $Y$ . Previous research speculated that malnutrition or birth defects could act as common causes of low birth weight and infant mortality (Hernández-Díaz et al., 2006; VanderWeele et al., 2012).

Even if the relationship between maternal smoking and birth weight is not confounded, and also that between maternal smoking and infant mortality, an unmeasured common cause of low birth weight and infant mortality could introduce bias of the analysis. The results provided in our article do not require having data on, or having controlled for, potential common causes of low birth weight and infant mortality. Furthermore, our analysis is agnostic to whether the data are generated by nonparametric structural equation models, influence diagrams, or directed acyclic graphs (Pearl, 2009) with untestable and unverifiable assumptions, which is required for prior research concerning the birth weight paradox.

We will assume that the population of white mothers who smoke are exchangeable with mothers who do not smoke, which is denoted $(Y_{x}, M_{x}) ∐ X$ . We will also assume that there is no baby that would be born with normal birth weight had the mother smoked, but this same baby would be born with low birth weight had the mother not smoked, which is the positive monotonicity assumption $M_{1} (ω) \geq M_{0} (ω)$ for all $ω \in Ω$ . Using Yerushalmy's data, we find that the risk of giving birth to a low birth weight baby is nearly double for smokers compared with nonsmoking mothers. Vast literature on reproductive, perinatal, and pediatric epidemiology provides strong evidence that smoking is likely to cause low birth weight babies (Hebel, Fox, & Sexton, 1988; Hebel, Nowicki, & Sexton, 1985; Sexton & Hebel, 1984).

We can now apply Theorem 2, Proposition 2, and Corollary 4. To detect whether there exists at least one baby that would be born with low birth weight regardless of the mother's smoking status and maternal smoking prevents infant mortality for the same baby, we would attempt to falsify the null hypothesis $P (Y = 1, M = 1 ∣ X = 0) \leq P (Y = 1, M = 1 ∣ X = 1)$ . If we falsify this null hypothesis, then from Corollary 4, we can conclude that the proportion of babies that are born with low birth weight regardless of maternal smoking status and smoking protects the baby from infant mortality is greater than the proportion of babies that are born with low birth weight regardless of maternal smoking status and smoking causes the baby's death within the first year of birth. From Proposition 2, the risk difference $P (Y = 1, M = 1 ∣ X = 0) - P (Y = 1, M = 1 ∣ X = 1)$ is equal to $P_{c} (0, 1, 1, 1) - {P_{c} (1, 0, 1, 1) + P_{c} (1, 0, 1, 0) + P_{c} (1, 1, 1, 0)}$ . Therefore, the same risk difference is a lower bound on $P_{c} (0, 1, 1, 1) - P_{c} (1, 0, 1, 1)$ from Corollary 4.

From Corollary 4, the lower bound on the proportion of babies that would be born with low birth weight regardless of maternal smoking and these babies would be prevented from dying in its first year from maternal smoking is given as $P (Y = 1, M = 1 ∣ X = 0) - P (Y = 1, M = 1 ∣ X = 1)$ . Using Yerushalmy's data this risk difference is 43/6067 − 27/3726 = 0.0071 − 0.0072 = −0.0001 with a one-sided 95% confidence interval of (−0.003, 1). We therefore do not find evidence that smoking prevents infant mortality for babies that would be born low birth weight regardless of maternal smoking status. Had we found evidence for such babies in our population using Corollary 4, we could have applied Theorem 3 to assess the consequences of the monotonicity assumptions on exaggerating the proportion of babies that regardless of maternal smoking status are born with low birth weight and maternal smoking prevents infant mortality.

To test the more reasonable hypothesis, given the preponderance of scientific evidence of the harmful effect of smoking on human health, that smoking actually causes infant mortality for babies that would be born low birth weight regardless of maternal smoking, we again can apply Theorem 2, Proposition 2, and Corollary 4. From Theorem 2, to detect whether there exists at least one baby that would be born with low birth weight regardless of maternal smoking and maternal smoking causes infant mortality for the same baby, we would attempt to falsify the null hypothesis $P (Y = 0, M = 1 ∣ X = 0) \leq P (Y = 0, M = 1 ∣ X = 1)$ . If we falsify this null hypothesis, then from Corollary 4, we can conclude that the proportion of babies that are born with low birth weight regardless of maternal smoking status and smoking causes the baby's infant mortality is greater than the proportion of babies that are born with low birth weight regardless of maternal smoking status and smoking prevents the baby's infant mortality within the first year of birth.

From Proposition 2, the risk difference $P (Y = 0, M = 1 ∣ X = 0) - P (Y = 0, M = 1 ∣ X = 1)$ is equal to $P_{c} (1, 0, 1, 1) - {P_{c} (0, 1, 1, 1) + P_{c} (0, 1, 1, 0) + P_{c} (0, 0, 1, 0)}$ . From Corollary 4, the same risk difference is also a lower bound on $P_{c} (1, 0, 1, 1) - P_{c} (0, 1, 1, 1)$ . From Theorem 2 and Corollary 4, the lower bound on the proportion of babies that would be born low birth weight regardless of maternal smoking and these same babies would die in its first year from maternal smoking is given as $P (Y = 0, M = 1 ∣ X = 0) - P (Y = 0, M = 1 ∣ X = 1)$ . Using Yerushalmy's data, this risk difference is 154/6067 − 210/3726 = 0.025 − 0.056 = −0.031 with a one-sided 95% confidence interval of (−0.038, 1). Therefore, using Corollary 4, we do not find evidence that smoking causes infant mortality for babies that would be born low birth weight regardless of maternal smoking status. Had we found evidence for such babies in our population using Corollary 4, we could have applied Theorem 3 to assess the consequences of the monotonicity assumptions on exaggerating the proportion of babies that regardless of maternal smoking status are born with low birth weight and maternal smoking causes infant mortality.

To understand why we are unable to find statistical evidence for this second more reasonable hypothesis, we can conduct a sensitivity analysis similar to what is presented in Theorem 3. As stated earlier, under the reasonable monotonicity assumption that smoking never prevents low birth weight, the risk difference $P (Y = 0, M = 1 ∣ X = 0) - P (Y = 0, M = 1 ∣ X = 1)$ is equal to $P_{c} (1, 0, 1, 1) - {P_{c} (0, 1, 1, 1) + P_{c} (0, 1, 1, 0) + P_{c} (0, 0, 1, 0)}$ . This counterfactual risk difference includes the term $P_{c} (0, 0, 1, 0)$ , which is the proportion of babies that as a result of smoking would not die in their first year after birth regardless of maternal smoking, but would be born of low birth weight only if the mother smoked. From yet another application of Corollary 4 (interchanging the roles of $M$ and $Y$ ), under the reasonable assumption that smoking never prevents infant mortality, a lower bound on this proportion $P_{c} (0, 0, 1, 0)$ is given by $P (M = 1, Y = 0 ∣ X = 1) - P (M = 1, Y = 0 ∣ X = 0) = 210 ∕ 3726 - 154 ∕ 6067 = 0.031$ , which has a one-sided 95% confidence interval of (0.024, 1), suggesting that $P_{c} (0, 0, 1, 0)$ is at least 0.24.

Under the monotonicity assumption that smoking never prevents a low birth weight baby, the risk difference $P (Y = 0, M = 1 ∣ X = 0) + P_{c} (0, 0, 1, 0) - P (Y = 0, M = 1 ∣ X = 1)$ is equal to the counterfactual expression $P_{c} (1, 0, 1, 1) - {P_{c} (0, 1, 1, 1) + P_{c} (0, 1, 1, 0)}$ . While we cannot point identify the risk difference $P (Y = 0, M = 1 ∣ X = 0) + P_{c} (0, 0, 1, 0) - P (Y = 0, M = 1 ∣ X = 1)$ , it appears as though it is nonnegative in Yerushalmy's data. Therefore, Yerushalmy's data contains no evidence that smoking prevents infant mortality for babies that regardless of maternal smoking would be born with low birth weight from the very first hypothesis test that we conducted. In fact, under realistic monotonicity assumptions and reasonable sensitivity analyses, the evidence is suggestive that maternal smoking actually causes infant mortality for babies that regardless of maternal smoking would be born with low birth weight. Hypothesis tests to detect whether regardless of maternal smoking a baby would be born with normal birth weight and smoking causes infant mortality could likewise be conducted using our methods.

6 ∣. GENERALIZATIONS AND EXTENSIONS

6.1 ∣. Multivariate direct effects with continuous and discrete outcomes and mediators

Our previous results are derived for a binary outcome $Y$ and binary mediator $M$ . We now provide results for tuple valued $Y$ and $M$ , each of which has support on $R^{n_{y}}$ and $R^{n_{m}}$ , where $n_{y}$ and $n_{m}$ denote the number of elements in the corresponding tuple. Here, $Y$ and $M$ could have continuous and/or discrete components. The special case of only continuous components is provided in the online supplement. All of our previous results can be interpreted as specializations of the results presented in this section. Treatment $X$ is still binary, but categorical or ordinal treatments can be suitably dichotomized to apply the results in this section at any level of the dichotomization. In this section, we allow $Y_{1} (ω)$ , $Y_{0} (ω)$ , $M_{1} (ω)$ , and $M_{0} (ω)$ to be stochastic counterfactuals that have distribution function $F_{Y_{1}, Y_{0}, M_{1}, M_{0}} (y_{a}, y_{b}, m_{a}, m_{b}) = P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b})$ , where $y_{a}$ and $y_{b}$ are subsets of $R^{n_{y}}$ and $m_{a}$ and $m_{b}$ are subsets of $R^{n_{m}}$ . We still require the randomization assumption that ( $Y_{x}$ , $M_{x}$ ) is independent of $X$ and the consistency assumption that when $X (ω) = x, Y_{x} (ω) = Y (ω)$ and $M_{x} (ω) = M (ω)$ .

Proposition 3. Suppose $X$ is randomized at baseline. For some $y_{a} \subset R^{n_{y}}$ and $m_{a} \subset R^{n_{m}}$ , the contrast

P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1

is equal to

P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - {P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a})},

and therefore is a sharp lower bound of $P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})$ .

When $P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) > 1$ , the expression

\frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M \in m_{a} ∣ X = 1) - P (M \notin m_{a} ∣ X = 0)}

is a sharp lower bound on the standardized principal stratum direct effect

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} = m_{a})},

which is equivalent to

P (Y_{1} \in y_{a}, Y_{0} \notin y_{a} ∣ M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a} ∣ M_{1} \in m_{a}, M_{0} \in m_{a}) .

This demonstrates that our results apply are also applicable to continuous random variables. This work demonstrates that quantification of direct effects can be conducted with only design based assumptions. Such design based assumptions make the study of direct effects possible with randomized trials. Consider now the analogous results under monotonicity assumptions.

Proposition 4. Suppose $X$ is randomized at baseline. In addition, suppose there exists no individual $ω \in Ω$ of response type $M_{1} (ω) \notin m_{a}$ and $M_{0} (ω) \in m_{a}$ . For some $y_{a} \subset R^{n_{y}}$ and $m_{a} \subset R^{n_{m}}$ , the contrast

P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1)

is a sharp lower bound on

P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) .

When there exists no individual $ω \in Ω$ of response type $M_{1} (ω) \notin m_{a}$ and $M_{0} (ω) \in m_{a}$ and

P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) > 0

then the empirical condition

\frac{P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1)}{P (M \in m_{a} ∣ X = 0)}

serves as a sharp lower bound on the standardized principal stratum direct effect

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})},

which is equivalent to

P (Y_{1} \in y_{a}, Y_{0} \notin y_{a} ∣ M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a} ∣ M_{1} \in m_{a}, M_{0} \in m_{a}) .

For simple sensitivity analysis for the monotonicity assumptions, we present the following results. These results enable scientists to directly consider how violations of monotonicity assumptions impact the findings of their results. Monotonicity assumptions enable scientists to uncover individual level direct effects in cases where without such assumptions the resulting tests are too stringent. Consequently, informative and simple sensitivity analyses that enable scientists to understand how such assumptions impact study results is important.

Proposition 5. Suppose $X$ is randomized at baseline. The expression $P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) - r$ , where

r = {P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a})},

is equal to the unstandardized principal stratum direct effect, $P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})$ . In addition, the expression

\frac{P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) - r}{P (M \in m_{a} ∣ X = 0) - q},

where $q = P (M_{1} \notin m_{a}, M_{0} \in m_{a})$ , is equal to the standardized principal stratum direct effect

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})},

which is equivalent to

P (Y_{1} \in y_{a}, Y_{0} \notin y_{a} ∣ M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a} ∣ M_{1} \in m_{a}, M_{0} \in m_{a}) .

Note our results in previous sections with binary $Y$ and $M$ are specializations of the results provided here with special sets for $y_{a}$ and $m_{a}$ . The full generality of our results provided here are appealing, but the specificity of our previous results are easier to read and comprehend. Under randomization, the results provided in this section demonstrate that coarsening a continuous or ordinal outcome or mediator and still provide useful results.

6.2 ∣. Direct effects through time with and without missing data

When there are no missing data, instead of examining $Y$ and $M$ at one specific time after randomization, the scientist can consider stochastic processes ( $Y (ω, t)$ , $M (ω, t_{1})$ ) for any set of times $t$ and $t_{1} \leq t$ after randomization and apply any or all of our results for each specific $t$ and $t_{1}$ . All the statistician has to do is replace $Y (ω)$ and $M (ω)$ with $Y (ω, t)$ and $M (ω, t_{1})$ , respectively, and $Y_{x} (ω)$ and $M_{x} (ω)$ with $Y_{x} (ω, t)$ and $M_{x} (ω, t_{1})$ in all our theorems, propositions, and corollaries. We provide the full set of results in the online supplement for completeness.

If the statistician has missing data or censoring in the dataset, then this can be easily handled. The tuple-valued $M (ω, t_{1})$ for each specific $t$ and $t_{1}$ can include a variable $R (ω, t_{1})$ , called response indicators, that takes value 0 if either $Y$ or $M$ is missing for individual $ω$ at time $t$ for $y$ or $t_{1}$ for $M$ , and 1 otherwise for each individual $ω$ at time pair ( $t$ , $t_{1}$ ), and then apply Proposition 3, 4, or 14 as required. If the the statistician is concerned only for total effects for $Y$ and has missing data for this outcome $Y$ , then take $M = R$ and use any of our results above. This approach examines the individual level principal stratum direct effect on $Y$ for those who would have no missing data regardless of treatment taken, and related sensitivity analysis would point identify the average treatment effect for $Y$ . For more refined handling of the missing data as well as sensitivity analysis, we refer the reader to the online supplement of this article which provides methods to handle missing data for the mediator and/or outcome with and without monotonicity assumptions.

The time index can be tuple valued as well, or $t = (t_{1}, \dots, t_{k})$ . In this setting, $Y (t)$ would denote the trajectories $Y (t) = (Y (t_{1}), \dots, Y (t_{k}))$ and $M (t) = (M (t_{1}), \dots, M (t_{k}))$ . The counterfactuals are similarly $Y_{x} (t) = (Y_{x} (t_{1}), \dots, Y_{x} (t_{k}))$ , and $M_{x} (t) = (M (t_{1}), \dots, M_{x} (t_{k}))$ . Substituting these factuals and counterfactuals in our results above, scientists are able to examine individual level direct effects through time. If $M (t)$ and $M_{x} (t)$ are taken to be the empty set, then we return to the setting of total effects for $Y (t)$ through time.

7 ∣. CONCLUSION

This work demonstrates that quantifying individual level natural direct effects, principal stratum direct effects, and pleiotropic effects can be implemented with no stronger identification assumptions than testing for a nonzero total effect. The identifiability assumptions that are needed to test the efficacy of a drug in a randomized clinical trial are exactly the identifiability assumptions needed to test for the relevant individual level direct effects. This embeds testing for these direct effects firmly within the Neyman-Pearson paradigm. Previous literature studied the “instrumental inequalities” and made contributions to understanding some of their consequences, including that falsification of the “instrumental inequality” provides evidence for a nonzero population average controlled direct effect (Cai et al., 2008). Our article is the first to establish tests for an individual level principal stratum direct and natural direct effects without unverifiable and at times uninterpretable cross world or graphical assumptions. We also quantify the magnitude of such direct effects. Under monotonicity, our conditions can be used to detect individual level natural direct effects even when the bounds for the population average natural direct effects include zero. We also generate results that provide information on the magnitude of the corresponding principal stratum direct effects. We enable statisticians to conduct sensitivity analysis of monotonicity assumptions to yield interpretable results for scientific investigation. Our methodology can handle all types of outcomes and mediators: binary, categorical, continuous, mixed, multivariate, and multivariate measurements with missing data that are stochastic processes through time. In addition, we define a new causal effect, individual level pleiotropy, in the counterfactual or potential outcome framework, and derive the associated empirical conditions which could be used to detect such an effect in a population.

Supplementary Material

Supplement May 20

NIHMS1945761-supplement-Supplement_May_20.pdf^{(515.1KB, pdf)}

ACKNOWLEDGEMENTS

The authors thank James M. Robins, Eric J. Tchetgen Tchetgen, Michael Hudgens, and Linbo Wang for helpful discussions. The authors also acknowledge useful comments from two anonymous reviewers and the associate editor. This work was supported by the National Institutes of Health, Grant/Award Number: 5T32AI007358-30; National Science Foundation, Grant/Award Number: DMS-1127914.

Funding information

National Institute of Allergy and Infectious Diseases, Grant/Award Number: 5 T32 AI 7358-30; National Science Foundation, Grant/Award Number: DMS-1127914

APPENDIX

A.1. Proofs of theorems

Proof of Theorem 1. We prove the contrapositive. Assume that for some $y \in {0, 1}$ and $m \in {0, 1}$ no individual $ω$ of response type $Y_{1} (ω) = y$ , $Y_{0} (ω) = 1 - y$ , $M_{1} (ω) = m$ and $M_{0} (ω) = m$ exists in our population. Then for all individuals $ω$ in our population $Ω$ , $I {(Y_{1} (ω) = y, M_{1} (ω) = m} + I {Y_{0} (ω) = 1 - y, M_{0} (ω) = m} \leq 1$ . Here, I(·) denotes the usual indicator function.

Taking expectations,

E [I {Y_{1} (ω) = y, M_{1} (ω) = m} + I {Y_{0} (ω) = 1 - y, M_{0} (ω) = m}] \leq 1 \Leftrightarrow P {Y_{1} (ω) = y, M_{1} (ω) = m} + P {Y_{0} (ω) = 1 - y, M_{0} (ω) = m} \leq 1 \Leftrightarrow P {Y_{1} (ω) = y, M_{1} (ω) = m ∣ X = 1} + P {Y_{0} (ω) = 1 - y, M_{0} (ω) = m ∣ X = 0} \leq 1 \Leftrightarrow P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) \leq 1 .

The second to third line uses $(M_{x}, Y_{x}) ∐ X$ and the third to last line uses consistency of counterfactuals. ■

Proof of Corollary 1. From Theorem 1, we have that there exists a nonempty subpopulation $Ω_{s} \subseteq Ω$ such that for every $ω \in Ω_{s}$ , $Y_{1} (ω) = 1$ , $Y_{0} (ω) = 0$ , $M_{1} (ω) = 0$ , and $M_{0} (ω) = 0$ . From the composition axiom and $M_{1} (ω) = M_{0} (ω) = 0$ , we have that $Y_{1} (ω) = Y_{1, M_{1}} (ω) = Y_{1, M_{0}} (ω)$ and $Y_{0} (ω) = Y_{0, M_{0}} (ω) = Y_{0, M_{1}} (ω)$ , which provides the following results for all $ω \in Ω_{s}$ : (1) $Y_{1} (ω) - Y_{0} (ω) = 1$ ; (2) $Y_{1, M_{0}} (ω) - Y_{0, M_{0}} (ω) = Y_{1, M_{1}} (ω) - Y_{0, M_{0}} (ω) = 1$ ; (3) $Y_{1, M_{1}} (ω) - Y_{1, M_{0}} (ω) = Y_{1, M_{1}} (ω) - Y_{1, M_{1}} (ω) = 0$ ; (4) $Y_{1, M_{1}} (ω) - Y_{0, M_{1}} (ω) = Y_{1, M_{1}} (ω) - Y_{0, M_{0}} (ω) = 1$ ; (5) $Y_{0, M_{1}} (ω) - Y_{0, M_{0}} (ω) = Y_{0, M_{0}} (ω) - Y_{0, M_{0}} (ω) = 0$ ; (6) $Y_{1, 0} (ω) - Y_{0, 0} (ω) = Y_{1, M_{1}} (ω) - Y_{0, M_{0}} (ω) = 1$ ; and (7) $Y_{1} (ω) - Y_{0} (ω) = 1$ and $M_{1} (ω) - M_{0} (ω) = 0$ . ■

Proof of Proposition 2 and Corollary 2. Taking expectation of $Y_{1} M_{1} + (1 - Y_{0}) M_{0} - 1$ , $(1 - Y_{1}) M_{1} + Y_{0} M_{0} - 1$ , $Y_{1} (1 - M_{1}) + (1 - Y_{0}) (1 - M_{0}) - 1$ , and $(1 - Y_{1}) (1 - M_{1}) + Y_{0} (1 - M_{0}) - 1$ gives the result of Proposition 1. A table with the relevant frequencies of individuals $ω \in Ω$ is provided in the online supplement. A proof of these results using stochastic counterfactuals is also provided in the online supplement.

The proof that

\frac{P (Y = y, M = m ∣ X = 1) + P (Y = 1 - y, M = m ∣ X = 0) - 1}{P (M = m ∣ X = 1) - P (M = m - 1 ∣ X = 0)}

is a valid lower bound of $(P_{c} (y, 1 - y, m, m) - P_{c} (1 - y, y, m, m)) ∕ P (M_{1} = M_{0} = m)$ is too lengthy for the main text. We provide it in the online supplement.

We provide a sketch proof regarding the lower bound of $(P_{c} (y, 1 - y, m, m) - P_{c} (1 - y, y, m, m)) ∕ P (M_{1} = M_{0} = m)$ . Without loss of generality, consider the case where $y = 1$ and $m = 1$ . Note, when $P (Y = 1, M = 1 ∣ X = 1) + P (Y = 0, M = 1 ∣ X = 0) < 1$ , then the expression $(P_{c} (1, 0, 1, 1) - P_{c} (0, 1, 1, 1)) ∕ P (M_{1} = M_{0} = 1)$ can always be reformulated, using arguments of consistency and randomization, to equal

A \cdot \frac{P (Y = 1, M = 1 ∣ X = 1) + P (Y = 0, M = 1 ∣ X = 0) - 1}{P (M = 1 ∣ X = 1) - P (M = 0 ∣ X = 0)},

where $A$ is a positive constant always greater than equal to 1. Since constant $A$ is always greater than equal to 1, $(P_{c} (1, 0, 1, 1) - P_{c} (0, 1, 1, 1)) ∕ P (M_{1} = M_{0} = 1)$ is bounded below from

\frac{P (Y = 1, M = 1 ∣ X = 1) + P (Y = 0, M = 1 ∣ X = 0) - 1}{P (M = 1 ∣ X = 1) - P (M = 0 ∣ X = 0)} .

Now, for the last portion of Corollary 2, note $(P_{c} (1, 0, 1, 1) - P_{c} (0, 1, 1, 1)) ∕ P (M_{1} = M_{0} = 1)$ is equal to $P (Y_{1} = 1, Y_{0} = 0 ∣ M_{1} = 1, M_{0} = 1) - P (Y_{1} = 0, Y_{0} = 1 ∣ M_{1} = 1, M_{0} = 1)$ . This completes our proof.

Proof of Theorem 2. We prove the contrapositive. Assume that no individual $ω$ of response type $Y_{1} (ω) = 1$ , $Y_{0} (ω) = 0$ , $M_{1} (ω) = 0$ and $M_{0} (ω) = 0$ exists in our population. Then for all individuals $ω$ in our population $Ω$ ,

$I {Y_{1} (ω) = 1$ , $M_{1} (ω) = 0} - I {Y_{0} (ω) = 1$ , $M_{0} (ω) = 0} \leq 0$ . This last assertion is true after examining a counterfactual table that is provided in the online supplement. Taking expectations, we have

P {Y_{1} (ω) = 1, M_{1} (ω) = 0} - P {Y_{0} (ω) = 1, M_{0} (ω) = 0} \leq 0 \Leftrightarrow P {Y_{1} (ω) = 1, M_{1} (ω) = 0 ∣ X = 1)} - P {Y_{0} (ω) = 1, M_{0} (ω) = 0 ∣ X = 0} \leq 0 \Leftrightarrow P (Y = 1, M = 0 ∣ X = 1) - P (Y = 1, M = 0 ∣ X = 0) \leq 0 .

The first to second line uses $(M_{x}, Y_{x}) ∐ X$ and the second to third line uses consistency of counterfactuals. ■

Proof of Corollary 3. Similar to Corollary 1. ■

Proof of Proposition 2 and Corollary 4. Taking expectation of $Y_{1} (1 - M_{1}) - Y_{0} (1 - M_{0})$ , $Y_{0} M_{0} - Y_{1} M_{1}$ , $(1 - Y_{1}) M_{1} - (1 - Y_{0}) M_{0}$ , and $(1 - Y_{1}) (1 - M_{1}) - (1 - Y_{0}) (1 - M_{0})$ gives the result. A table with the relevant frequencies of individuals $ω \in Ω$ is provided in the online supplement.

The proof of Corollary 4 is provided next. Under monotonicity, $P (Y = y, M = m ∣ X = 1 - m) - P (Y = y, M = m ∣ X = m)$ is lower bound of $P_{c} (m {1 - y} + y {1 - m}, y m + {1 - m} {1 - y}, m, m) - P_{c} (y m + {1 - m} {1 - y}, m {1 - y} + y {1 - m}, m, m)$ , and therefore $P (M_{1} = M_{0} = m) > 0$ . Now, $P (M = m ∣ X = 1) = P (M_{1} = m)$ and $P (M = m ∣ X = 0) = P (M_{0} = m)$ are both upper bounds to $P (M_{1} = m, M_{0} = m)$ . We have $P (M = m ∣ X = 0) \leq P (M = m ∣ X = 1)$ by the positive monotonicity assumption. Now, dividing a nonnegative lower bound of a numerator by a nonegative upper bound of the denominator will give a lower bound in of the ratio. Choose $P (M = m ∣ X = 0)$ as the denominator instead of $P (M = m ∣ X = 1)$ to get greater lower bound. This comples the proof. ■

Proof of Theorem 3. For $y \in {0, 1}$ and $m \in {0, 1}$ , taking expectation of $I (Y_{m} = y) I (M_{m} = m) - I (Y_{1 - m} = y) I (M_{1 - m} = m)$ and then subtracting $r$ as defined in Theorem 3 gives the result. For the population average principal stratum direct effect, note that $P (M = m ∣ X = 1) = P (M_{1} = m) = \sum_{j \in {0, 1}} P (M_{1} = m, M_{0} = j)$ . Subtracting $q$ in the denominator leaves us with $P (M_{1} = m, M_{0} = m)$ , and hence we have standardized the ratio appropriately. ■

Proof of Theorem 4. We prove the contrapositive. Assume that no individual $ω$ of response type $Y_{1} (ω) = 1$ , $Y_{0} (ω) = 0$ , $Z_{1} (ω) = 1$ and $Z_{0} (ω) = 0$ exists in our population. Then for all individuals $ω$ in our population $Ω$ ,

$I {Y_{1} (ω) = y, Z_{1} (ω) = 1} + I {Y_{0} (ω) = 0, Z_{0} (ω) = 0} \leq 1$ . Taking expectations,

P {Y_{1} (ω) = y, Z_{1} (ω) = 1} + P {Y_{0} (ω) = 0, Z_{0} (ω) = 0} \leq 1 \Leftrightarrow P {Y_{1} (ω) = 1, Z_{1} (ω) = 1 ∣ X = 1} + P {Y_{0} (ω) = 0, Z_{0} (ω) = 0 ∣ X = 0} \leq 1 \Leftrightarrow P (Y = 1, Z = 1 ∣ X = 1) + P (Y = 0, Z = 0 ∣ X = 0) \leq 1 .

The first to second line uses $(Z_{x}, y_{x}) ∐ X$ and the second to third line uses consistency of counterfactuals.

To prove the bounding of $P (Y_{1} = 1, Y_{0} = 0, Z_{1} = 1, Z_{0} = 0)$ , use the randomization and consistency assumptions. Note,

P (Y = 1, Z = 1 ∣ X = 1) + P (Y = 0, Z = 0 ∣ X = 0) - 1 = P (Y_{1} = 1, Z_{1} = 1) + P (Y_{0} = 0, Z_{0} = 0) - 1 = \sum_{(y_{0}, z_{0}) \in {0, 1}^{2}} P (Y_{1} = 1, Z_{1} = 1, Y_{0} = y_{0}, Z_{0} = z_{0}) - P ((Y_{0}, Z_{0}) \neq (0, 0)) = \sum_{(y_{0}, z_{0}) \in {0, 1}^{2}} P (Y_{1} = 1, Z_{1} = 1, Y_{0} = y_{0}, Z_{0} = z_{0}) - \sum_{(y_{1}, z_{1}) \in {0, 1}^{2}} P (Y_{1} = y_{1}, Z_{1} = z_{1}, (Y_{0}, Z_{0}) \neq (0, 0)) = P (Y_{1} = 1, Y_{0} = 0, Z_{1} = 1, Z_{0} = 0) - \sum_{(y_{1}, z_{1}) \in {0, 1}^{2} ∖ {1, 1}} P (Y_{1} = y_{1}, Z_{1} = z_{1}, (Y_{0}, Z_{0}) \neq (0, 0)) .

Consequently, $P (Y = 1, Z = 1 ∣ X = 1) + P (Y = 0, Z = 0 ∣ X = 0) - 1$ is a lower bound of $P (Y_{1} = 1, Y_{0} = 0, Z_{1} = 1, Z_{0} = 0)$ . ■

Proof of Proposition 3. For some $m_{a}$ , $m_{b}$ that are subsets of $R^{n_{m}}$ and $y_{a}$ , $y_{b}$ that are subsets of $R^{n_{y}}$ , begin with the empirical condition

P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{b}, M \in m_{b} ∣ X = 0) - 1 = P (Y \in y_{a}, M \in m_{a} ∣ X = 1) - P (Y \in y_{b} or M \notin m_{b} ∣ X = 0) = P (Y \in y_{a}, M \in m_{a} ∣ X = 1) - P (Y \in y_{b}, M \in m_{b} ∣ X = 0) - P (Y \notin y_{b}, M \notin m_{b} ∣ X = 0) - P (Y \in y_{b}, M \notin m_{b} ∣ X = 0) .

(A1)

Under randomization and consistency of counterfactuals, the empirical condition (A1) is equal to

P (Y_{1} \in y_{a}, M_{1} \in m_{a}) - P (Y_{0} \in y_{b}, M_{0} \in m_{b}) - P (Y_{0} \notin y_{b}, M_{0} \notin m_{b}) - P (Y_{0} \in y_{b}, M_{0} \notin m_{b}) .

Use the law of total probability. The empirical condition (A1) is equal to

P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{b}, M_{1} \in m_{a}, M_{0} \notin m_{b}) - {P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \notin m_{a}, M_{0} \in m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \notin m_{a}, M_{0} \in m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{b}, M_{1} \in m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{b}, M_{1} \in m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{b}, M_{1} \notin m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{b}, M_{1} \notin m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \notin m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \notin m_{a}, M_{0} \notin m_{b})} .

Simplifying this last expression, we have that the empirical condition (A1) is equal to

P (Y_{1} \in y_{a}, Y_{0} \notin y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b}) - P {(Y}_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b}) - {P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \notin m_{a}, M_{0} \in m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \notin m_{a}, M_{0} \in m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{b}, M_{1} \in m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{b}, M_{1} \notin m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{b}, M_{1} \notin m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{b}, M_{1} \notin m_{a}, M_{0} \notin m_{b}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \notin m_{a}, M_{0} \notin m_{b}) .}

(A2)

Given that probability is greater than equal to zero and less than equal to 1, the empirical condition (A1) is a lower bound on $P (Y_{1} \in y_{a}, Y_{0} \notin y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{b}, M_{1} \in m_{a}, M_{0} \in m_{b})$ .

If we take $y_{b} = y_{a}$ and $m_{b} = m_{a}$ , then $P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1$ serves as a lower bound on $P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})$ .

In addition,

P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1 = P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - {P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) .}

This is true if and only if

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} = \frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} .

Using the identity,

P (M_{1} \notin m_{a}, M_{0} \notin m_{a}) = P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \notin m_{a}),

(A3)

this equation for the standardized principal stratum direct effect simplifies to

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} = \frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{P (M_{1} \notin m_{a}, M_{0} \notin m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} .

Denote

x = P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a})

out of space considerations. Note, $x \in [0, 1)$ . With some algebra, the standardized principal stratum direct effect is equal to

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} = \frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{x + P (M_{1} \notin m_{a}, M_{0} \notin m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} . = \frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a})} \times (\frac{P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} + \frac{(x + P (M_{1} \notin m_{a}, M_{0} \notin m_{a}))}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} \frac{(P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}))}{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}) .

We want to show that the next expression

(\frac{P (M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a})} + \frac{(x + P (M_{1} \notin m_{a}, M_{0} \notin m_{a}))}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} \frac{(P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}))}{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1})

(A4)

is greater than or equal to one to show that $\frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \in m_{a})}$ is a lower bound of

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} .

Through inspection, expression (A4) is greater than or equal to one if and only if

\frac{(P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}))}{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1} \geq 1 .

Identity (A3) provides the equality

P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1 = P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - {P (Y_{1} \in y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) + P (M_{1} \notin m_{a}, M_{0} \notin m_{a})} .

The fraction

\frac{(P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}))}{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}

is greater than equal to one, if and only if the numerator is greater than the denominator. Here, we have that the denominator is positive as a condition of our theorem. The only positive term in the denominator is $P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})$ , which is less than equal to $P (M_{1} \in m_{a}, M_{0} \in m_{a})$ . Also, the numerator is greater than zero, as

P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}) \geq P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}) .

Therefore,

\frac{(P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}))}{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1} \geq 1 .

Consequently, our expression (A4) is greater than or equal to one, which in turn implies that

\frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a})} \leq \frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})} .

Therefore,

\frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a})}

is a lower bound on

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})},

when $P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1 < 0$ .

Now, note

P (M_{1} \in m_{a}, M_{0} \in m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}) = P (M_{1} \in m_{a}, M_{0} \in m_{a}) + P (M_{1} \in m_{a}, M_{0} \notin m_{a}) - P (M_{1} \in m_{a}, M_{0} \notin m_{a}) - P (M_{1} \notin m_{a}, M_{0} \notin m_{a}) = P (M_{1} \in m_{a}) - P (M_{0} \notin m_{a}) = P (M \in m_{a} ∣ X = 1) - P (M \notin m_{a} ∣ X = 0) .

Consequently,

\frac{P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1}{P (M \in m_{a} ∣ X = 1) - P (M \notin m_{a} ∣ X = 0)}

is a lower bound of

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})}

in the case when $P (Y \in y_{a}, M \in m_{a} ∣ X = 1) + P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - 1 > 0$ . ■

Proof of Proposition 4. From the randomization assumption, consistency of counterfactuals, and an application of the law of total probability,

P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) = P (Y_{0} \notin y_{a}, M_{0} \in m_{a}) - P (Y_{1} \in y_{a}, M_{1} \in m_{a}) = P (Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) + P (Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) .

Another application of the law of total probability, we have

P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) = P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) .

Simplifying this equality,

P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) = P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) + {P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a})} .

The monotonicity assumption that there is no individual of response type $M_{1} (ω) \notin m_{a}$ and $M_{0} (ω) \in m_{a}$ and $P (Y \in y_{a}, M \in m_{a} ∣ X = 0) > P (Y \in y_{a}, M \in m_{a} ∣ X = 1)$ , yields

{P (Y_{1} \notin y_{a}, Y_{0} \notin Y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a})} \leq 0,

and $P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1)$ is less than $P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})$ . We also have that $P (M \in m_{a}) ∣ X = 0)$ is an upper bound on $P (M_{1} \in m_{a}, M_{0} \in m_{a})$ .

Provided the numerator is positive,

\frac{(P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1))}{P (M \in m_{a} ∣ X = 0)}

is a lower bound on

\frac{(\begin{matrix} P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) \\ - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) \end{matrix})}{P (M_{1} \in m_{a}, M_{0} \in m_{a})},

and equivalently $P (Y_{1} \in y_{a}, Y_{0} \notin y_{a} ∣ M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a} ∣ M_{1} \in m_{a}, M_{0} \in m_{a})$ . This completes the proof.

Proof of Proposition 5. From the proof of Proposition 4 above, we had

P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) = P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) + {P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a})} .

Thus, $P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) - r$ , where

r = {P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) + P (Y_{1} \in y_{a}, Y_{a} \notin y_{a}, M_{1} \notin m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \notin m_{a})}

is equal to $P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})$ . Now, under randomization and consistency of counterfactuals,

P (M \in m_{a} ∣ X = 0) = P (M_{0} \in m_{a}) = P (M_{1} \in m_{a}, M_{0} \in m_{a}) + P (M_{1} \notin m_{a}, M_{0} \in m_{a}) .

Therefore,

P (M_{1} \in m_{a}, M_{0} \in m_{a}) = P (M \in m_{a} ∣ X = 0) - P (M_{1} \notin m_{a}, M_{0} \in m_{a}) .

This means that the expression

\frac{P (Y_{1} \in y_{a}, Y_{0} \notin y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a}) - P (Y_{1} \notin y_{a}, Y_{0} \in y_{a}, M_{1} \in m_{a}, M_{0} \in m_{a})}{P (M \in m_{a} ∣ X = 0) - P (M_{1} \notin m_{a}, M_{0} \in m_{a})}

is equal to

\frac{P (Y \notin y_{a}, M \in m_{a} ∣ X = 0) - P (Y \notin y_{a}, M \in m_{a} ∣ X = 1) - r}{P (M \in m_{a} ∣ X = 0) - q},

where $q = P (M_{1} \in m_{a}, M_{0} \in m_{a})$ . This completes the proof. ■

Footnotes

SUPPORTING INFORMATION

Additional supporting information may be found online in the Supporting Information section at the end of this article.

REFERENCES

Avin C, Shpitser I, & Pearl J Identifiability of path-specific effects. Paper presented at: Proceedings of the 19th International Joint Conference on Artificial Intelligence (pp. 357–363). Morgan Kaufmann Publishers Inc; 2005. [Google Scholar]
Balke A, & Pearl J (1997). Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association, 92(439), 1171–1176. [Google Scholar]
Cai Z, Kuroki M, Pearl J, & Tian J (2008). Bounds on direct effects in the presence of confounded intermediate variables. Biometrics, 64(3), 695–701. [DOI] [PubMed] [Google Scholar]
Frangakis CE, & Rubin DB (2002). Principal stratification in causal inference. Biometrics, 58(1), 21–29. [DOI] [PMC free article] [PubMed] [Google Scholar]
Hebel JR, Fox NL, & Sexton M (1988). Dose-response of birth weight to various measures of maternal smoking during pregnancy. Journal of Clinical Epidemiology, 41, 483–489. [DOI] [PubMed] [Google Scholar]
Hebel JR, Nowicki P, & Sexton M (1985). The effect of antismoking intervention during pregnancy: An assessment of interactions with maternal characteristics. American Journal of Epidemiology, 122, 135–148. [DOI] [PubMed] [Google Scholar]
Hernández-Díaz S, Schisterman EF, & Hernán MA (2006). The birth weight "paradox uncovered?". American Journal of Epidemiology, 164(11), 1115–1120. [DOI] [PubMed] [Google Scholar]
Hudgens MG, Hoering A, & Self SG (2003). On the analysis of viral load endpoints in HIV vaccine trials. Statistics in Medicine, 22, 2281–2298. [DOI] [PubMed] [Google Scholar]
Imai K, Tingley D, & Yamamoto T (2013). Experimental designs for identifying causal mechanisms. Journal of the Royal Statistical Society: Series A (Statistics in Society), 176(1), 5–51. [Google Scholar]
Imbens GW, & Rubin DB (2015). Causal inference in statistics, social, and biomedical sciences. Cambridge, MA: Cambridge University Press. [Google Scholar]
Pearl J (2009). Causality. Cambridge, MA: Cambridge University Press. [Google Scholar]
Richardson TS, & Robins JM (2010). Analysis of the binary instrumental variable model. Heuristics, Probability and Causality: A Tribute to Judea Pearl, 415–444. [Google Scholar]
Robins JM, & Greenland S (1992). Identifiability and exchangeability for direct and indirect effects. Epidemiology, 3(2) 143–155. [DOI] [PubMed] [Google Scholar]
Roland RR (2013). Probabilistic causality and detecting collections of interdependence patterns. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(4), 705–723. [Google Scholar]
Sexton M, & Hebel JR (1984). A clinical trial of change in maternal smoking and its effect on birth weight. JAMA, 251(February), 911–915. [PubMed] [Google Scholar]
Sjölander A. (2009). Bounds on natural direct effects in the presence of confounded intermediate variables. Statistics in Medicine, 28(4), 558–571. [DOI] [PubMed] [Google Scholar]
Solovieff N, Cotsapas C, Lee PH, Purcell SM, & Smoller JW (2013). Pleiotropy in complex traits: Challenges and strategies. Nature Reviews Genetics, 14(7), 483–495. [DOI] [PMC free article] [PubMed] [Google Scholar]
Swanson SA, Hernán MA, Miller M, Robins JM, & Richardson TS (2018). Partial identification of the average treatment effect using instrumental variables: review of methods for binary instruments, treatments, and outcomes. Journal of the American Statistical Association, 113(522), 933–947. 10.1080/01621459.2018.1434530 [DOI] [PMC free article] [PubMed] [Google Scholar]
VanderWeele TJ (2015). Explanation in causal inference: Methods for mediation and interaction. Oxford, UK: Oxford University Press. [Google Scholar]
VanderWeele TJ, Mumford SL, & Schisterman EF (2012). Conditioning on intermediates in perinatal epidemiology. In Epidemiology (Vol. 23, p. 1). Cambridge, MA: Cambridge University Press. [DOI] [PMC free article] [PubMed] [Google Scholar]
VanderWeele TJ, & Richardson TS (2012). General theory for interactions in sufficient cause models with dichotomous exposures. The Annals of Statistics, 40(4), 2128–2161. [DOI] [PMC free article] [PubMed] [Google Scholar]
VanderWeele TJ, & Robins JM (2008). Empirical and counterfactual conditions for sufficient cause interactions. Biometrika, 95(1), 49–61. [Google Scholar]
Yerushalmy J. (1964). Mother's cigarette smoking and survival of infant. American Journal of Obstetrics & Gynecology, 88(4), 505–518. [DOI] [PubMed] [Google Scholar]
Yerushalmy J. (2014). The relationship of parents' cigarette smoking to outcome of pregnancy–implications as to the problem of inferring causation from observed associations. International Journal of Epidemiology, 43(5), 1355–1366. [DOI] [PubMed] [Google Scholar]
Zhang JL, & Rubin DB (2003). Estimation of causal effects via principal stratification when some outcomes are truncated by “death”. Journal of Educational and Behavioral Statistics, 28(4), 353–368. [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Supplement May 20

NIHMS1945761-supplement-Supplement_May_20.pdf^{(515.1KB, pdf)}

[R1] Avin C, Shpitser I, & Pearl J Identifiability of path-specific effects. Paper presented at: Proceedings of the 19th International Joint Conference on Artificial Intelligence (pp. 357–363). Morgan Kaufmann Publishers Inc; 2005. [Google Scholar]

[R2] Balke A, & Pearl J (1997). Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association, 92(439), 1171–1176. [Google Scholar]

[R3] Cai Z, Kuroki M, Pearl J, & Tian J (2008). Bounds on direct effects in the presence of confounded intermediate variables. Biometrics, 64(3), 695–701. [DOI] [PubMed] [Google Scholar]

[R4] Frangakis CE, & Rubin DB (2002). Principal stratification in causal inference. Biometrics, 58(1), 21–29. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R5] Hebel JR, Fox NL, & Sexton M (1988). Dose-response of birth weight to various measures of maternal smoking during pregnancy. Journal of Clinical Epidemiology, 41, 483–489. [DOI] [PubMed] [Google Scholar]

[R6] Hebel JR, Nowicki P, & Sexton M (1985). The effect of antismoking intervention during pregnancy: An assessment of interactions with maternal characteristics. American Journal of Epidemiology, 122, 135–148. [DOI] [PubMed] [Google Scholar]

[R7] Hernández-Díaz S, Schisterman EF, & Hernán MA (2006). The birth weight "paradox uncovered?". American Journal of Epidemiology, 164(11), 1115–1120. [DOI] [PubMed] [Google Scholar]

[R8] Hudgens MG, Hoering A, & Self SG (2003). On the analysis of viral load endpoints in HIV vaccine trials. Statistics in Medicine, 22, 2281–2298. [DOI] [PubMed] [Google Scholar]

[R9] Imai K, Tingley D, & Yamamoto T (2013). Experimental designs for identifying causal mechanisms. Journal of the Royal Statistical Society: Series A (Statistics in Society), 176(1), 5–51. [Google Scholar]

[R10] Imbens GW, & Rubin DB (2015). Causal inference in statistics, social, and biomedical sciences. Cambridge, MA: Cambridge University Press. [Google Scholar]

[R11] Pearl J (2009). Causality. Cambridge, MA: Cambridge University Press. [Google Scholar]

[R12] Richardson TS, & Robins JM (2010). Analysis of the binary instrumental variable model. Heuristics, Probability and Causality: A Tribute to Judea Pearl, 415–444. [Google Scholar]

[R13] Robins JM, & Greenland S (1992). Identifiability and exchangeability for direct and indirect effects. Epidemiology, 3(2) 143–155. [DOI] [PubMed] [Google Scholar]

[R14] Roland RR (2013). Probabilistic causality and detecting collections of interdependence patterns. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(4), 705–723. [Google Scholar]

[R15] Sexton M, & Hebel JR (1984). A clinical trial of change in maternal smoking and its effect on birth weight. JAMA, 251(February), 911–915. [PubMed] [Google Scholar]

[R16] Sjölander A. (2009). Bounds on natural direct effects in the presence of confounded intermediate variables. Statistics in Medicine, 28(4), 558–571. [DOI] [PubMed] [Google Scholar]

[R17] Solovieff N, Cotsapas C, Lee PH, Purcell SM, & Smoller JW (2013). Pleiotropy in complex traits: Challenges and strategies. Nature Reviews Genetics, 14(7), 483–495. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R18] Swanson SA, Hernán MA, Miller M, Robins JM, & Richardson TS (2018). Partial identification of the average treatment effect using instrumental variables: review of methods for binary instruments, treatments, and outcomes. Journal of the American Statistical Association, 113(522), 933–947. 10.1080/01621459.2018.1434530 [DOI] [PMC free article] [PubMed] [Google Scholar]

[R19] VanderWeele TJ (2015). Explanation in causal inference: Methods for mediation and interaction. Oxford, UK: Oxford University Press. [Google Scholar]

[R20] VanderWeele TJ, Mumford SL, & Schisterman EF (2012). Conditioning on intermediates in perinatal epidemiology. In Epidemiology (Vol. 23, p. 1). Cambridge, MA: Cambridge University Press. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R21] VanderWeele TJ, & Richardson TS (2012). General theory for interactions in sufficient cause models with dichotomous exposures. The Annals of Statistics, 40(4), 2128–2161. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R22] VanderWeele TJ, & Robins JM (2008). Empirical and counterfactual conditions for sufficient cause interactions. Biometrika, 95(1), 49–61. [Google Scholar]

[R23] Yerushalmy J. (1964). Mother's cigarette smoking and survival of infant. American Journal of Obstetrics & Gynecology, 88(4), 505–518. [DOI] [PubMed] [Google Scholar]

[R24] Yerushalmy J. (2014). The relationship of parents' cigarette smoking to outcome of pregnancy–implications as to the problem of inferring causation from observed associations. International Journal of Epidemiology, 43(5), 1355–1366. [DOI] [PubMed] [Google Scholar]

[R25] Zhang JL, & Rubin DB (2003). Estimation of causal effects via principal stratification when some outcomes are truncated by “death”. Journal of Educational and Behavioral Statistics, 28(4), 353–368. [Google Scholar]

PERMALINK

On the identification of individual level pleiotropic, pure direct, and principal stratum direct effects without cross world assumptions

Jaffer M Zaidi

Tyler J VanderWeele

Abstract

1 ∣. INTRODUCTION

1.1 ∣. Natural and principal stratum direct effects

2 ∣. QUANTIFICATION OF INDIVIDUAL LEVEL DIRECT EFFECTS

2.1 ∣. Notation and assumptions

2.2 ∣. Identification of direct effects under randomization

3 ∣. IDENTIFICATION OF DIRECT EFFECTS WITH MONOTONICITY ASSUMPTIONS

3.1 ∣. Positive monotonicity of exposure on mediator

3.2 ∣. Sensitivity analysis for monotonicity assumptions

4 ∣. PLEIOTROPIC EFFECTS

5 ∣. DATA ANALYSIS

TABLE 1.

6 ∣. GENERALIZATIONS AND EXTENSIONS

6.1 ∣. Multivariate direct effects with continuous and discrete outcomes and mediators

6.2 ∣. Direct effects through time with and without missing data

7 ∣. CONCLUSION

Supplementary Material

ACKNOWLEDGEMENTS

Funding information

APPENDIX

A.1. Proofs of theorems

Footnotes

REFERENCES

Associated Data

Supplementary Materials

ACTIONS

PERMALINK

RESOURCES

Cite

Add to Collections

PERMALINK

On the identification of individual level pleiotropic, pure direct, and principal stratum direct effects without cross world assumptions

Jaffer M Zaidi

Tyler J VanderWeele

Abstract

1 ∣. INTRODUCTION

1.1 ∣. Natural and principal stratum direct effects

2 ∣. QUANTIFICATION OF INDIVIDUAL LEVEL DIRECT EFFECTS

2.1 ∣. Notation and assumptions

2.2 ∣. Identification of direct effects under randomization

3 ∣. IDENTIFICATION OF DIRECT EFFECTS WITH MONOTONICITY ASSUMPTIONS

3.1 ∣. Positive monotonicity of exposure on mediator

3.2 ∣. Sensitivity analysis for monotonicity assumptions

4 ∣. PLEIOTROPIC EFFECTS

5 ∣. DATA ANALYSIS

TABLE 1.

6 ∣. GENERALIZATIONS AND EXTENSIONS

6.1 ∣. Multivariate direct effects with continuous and discrete outcomes and mediators

6.2 ∣. Direct effects through time with and without missing data

7 ∣. CONCLUSION

Supplementary Material

ACKNOWLEDGEMENTS

Funding information

APPENDIX

A.1. Proofs of theorems

Footnotes

REFERENCES

Associated Data

Supplementary Materials

ACTIONS

PERMALINK

RESOURCES

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