In this Section we present a novel analysis to bound the Probability of Causation for a case where a third variable, M, is involved in the causal pathway between the exposure X and the outcome Y [19, 41]. In particular, in this section, we will focus on the case of no direct effect, as intuitively described by Figure 5.1. Applications where this assumption might be plausible can be found in [38]. In this paper he presented various mechanisms of complete mediation such as: a tobacco prevention program reduces cigarette smoking by changing the social norms for tobacco use; exposure to negative life events affects blood pressure through the
5.4. COMPLETE MEDIATION 101
mediation of cognitive attributions to stress. Another situation in which such an assumption might be plausible is in the treatment of ovarian cancer, Silber et al. (2007) in [70], where X represents management either by a medical oncologist or by a gynaecological oncologist, M is the intensity of chemotherapy prescribed, and Y is death within 5 years.
We shall be interested in the case that M is observed in the experimental data but is not observed for Ann, and see how this additional experimental evidence can be used to refine the bounds on PCA.
X
M
Y
Figure 5.1: Directed Acyclic Graph representing a mediator M, responding to ex- posure X and affecting response Y . There is no “direct effect”, unmediated by M, of X on Y .
To formalize our assumption of “no direct effect”, we introduce M(x), the potential value of M for X ← x, and Y∗(m), the potential value of Y for M ← m, where the irrelevance of the value x of X to Y∗ encapsulates our assumption that X has no effect on Y over and above that transmitted through its influence on the mediator M. The potential value of Y for X ← x (in cases where there is no intervention on M, which we here assume) is then Y (x) := Y∗{M(x)}.
In the sequel we restrict to the case that all variables are binary, and define M := (M(0), M(1)), Y∗ := (Y∗(0), Y∗(1)), and Y := (Y (0), Y (1)). In particular, we have observable variables (X, M, Y ) = (X, M(X), Y (X)). We denote the bivariate distributions of the potential response pairs by
mab := P(M(0) = a, M(1) = b) yrs∗ := P(Y∗(0) = r, Y∗(1) = s) yrs := P(Y (0) = r, Y (1) = s). Then ma+ = P(M = a | X ← 0) m+b = P(M = b | X ← 1) yr+∗ = P(Y = r | M ← 0) y+s∗ = P(Y = s | M ← 1) yr+ = P(Y = r | X ← 0) y+s = P(Y = s | X ← 1),
where ma+ denotes P1b=0mab, etc.
In addition to the assumptions of § 5.1 we further suppose that none of the causal mechanisms depicted in Figure 5.1 are confounded—expressed mathematically by assuming mutual independence between X, M and Y∗ (both for experimental individuals, and for Ann). Then, ma+, m+b, y∗r+, y+s∗ , yr+, y+s are all estimable from experimental data in which X is randomized, and M and Y are observed. It is also easy to show the Markov property:
Y ⊥⊥ X | M. (5.12)
This observable property can serve as a test of the validity of our conditions. The assumed mutual independence implies
yrs = P(Y∗(M(0)) = r, Y∗(M(1)) = s) = 1 X a,b=0 P(Y∗(a) = r, Y∗(b) = s)P(M(0) = a, M(1) = b). This yields y00 = m00y0+∗ + (m01+ m10)y00∗ + m11y+0∗ y01 = m01y01∗ + m10y10∗ y10 = m01y10∗ + m10y01∗ (5.13) y11 = m00y1+∗ + (m01+ m10)y11∗ + m11y+1∗ , and yr+ = m0+yr+∗ + m1+y∗+r (5.14) y+s = m+0ys+∗ + m+1y+s∗ . (5.15) Suppose now that we observe XA= 1 and YA = 1, but do not observe MA. We have PCA= y01 y+1 = m01y ∗ 01+ m10y10∗ y+1 . (5.16)
The denominator of (5.16) is P(Y = 1 | X ← 1), which is estimable from the data.
As for the numerator, this can be expressed as
5.4. COMPLETE MEDIATION 103
with µ = m01, η = y∗01, A = y∗+0− y0+∗ , and B = m+0− m0+. Note that A and B are identified from the data, whereas for µ and η we can only obtain inequalities:
max{0, −B} ≤ µ ≤ min{m0+, m+1} max{0, −A} ≤ η ≤ min{y∗
0+, y+1∗ }, so that
|B/2| ≤ µ + B/2 ≤ min{12(m0++ m+0),12(m1++ m+1)} |A/2| ≤ η + A/2 ≤ min{1
2(y ∗ 0++ y+0∗ ),12(y ∗ 1++ y+1∗ )}. (5.18) The lower (respectively, upper) limit for (5.17) will be when µ + B/2 and η + A/2 are both at their lower (respectively, upper) limits. In particular, the lower limit for (5.17) is max{0, AB}. Using (5.14) and (5.15), we compute AB = y+1− y1+, which leads to the lower bound
PCA≥ 1 −
P(Y = 1 | X ← 0)
P(Y = 1 | X ← 1) = 1 − 1 RR,
exactly as for the case that M was not observed. Thus the possibility to observe a mediating variable in the experimental data has not improved our ability to lower bound PCA.
We do however obtain an improved upper bound. Taking into account the various possible choices for the upper bounds in (5.18), the upper bound for the numerator of (5.16), in terms of experimentally estimable quantities, is given in Table 5.3. m1++ m+1 ≥ 1 m1++ m+1 < 1 y∗ 1++ y+1∗ ≥ 1 m0+y0+∗ + m+0y+0∗ m1+y+0∗ + m+1y0+∗ y∗ 1++ y+1∗ < 1 m0+y+1∗ + m+0y1+∗ m1+y1+∗ + m+1y+1∗
Table 5.3: Upper bound for the numerator of the PCA in complete mediation In § 5.6 will shown that this upper bound is never greater than that in (5.4), which ignores the mediator M, and is strictly smaller unless y∗
1+ + y∗+1 ≥ 1 and m1++ m+1 = 1.
5.4.1
Identifiability under monotonicity
Without any assumption about the data generating process, in § 5.4 we calculated a lower and an upper bound for the probability of causation when a mediator com- pletely justifies the causal relation between exposure and outcome. In this section we will show how PCA can be identify under the monotonicity assumption.
Definition 5.4.1 (Monotonicity) A variable Y is monotonic relative to X in a causal model, if and only if the bivariate distribution of Y (x) is monotonic in x:
P [Y (0) = 1, Y (1) = 0] = 0
For the particular case of no direct effect of X on Y , as described by the DAG in Figure 5.1, we obtained the following result:
Theorem 5.4.1 (Monotonicity in mediation) If M is monotonic relative to X and Y is monotonic relative to M then Y is monotonic relative to X.
Proof 5.4.1 If M is monotonic relative to X then m10 = 0. If Y is monotonic relative to M then y∗
10 = 0. Then, from (5.13), y10 will be zero. The reverse implication is not always true.
Then if M is monotonic relative to X and Y is monotonic relative to M the Prob- ability of Causation will be
PCA = 1 − 1 RR.
Under the monotonicity assumption we obtained a result consistent with what Tian and Pearl (2000) found in [76]. This can perhaps be considered as a proof of the validity of this method.
5.4.2
Example
Let us suppose that Ann’s children filled a criminal lawsuit against a pharmaceutical manufacturer claiming that Ann died after taking one of their drugs. On the other hand, the manufacturer claims that is a rare (binary) side effect of the drug that cause the death rather than the drug itself. Suppose we obtain the following values from the data:
P(M = 1 | X ← 1) = 0.25 P(M = 1 | X ← 0) = 0.025
P(Y = 1 | M ← 1) = 0.9 P(Y = 1 | M ← 0) = 0.1.
Again, these imply the values given in Table 5.1. According to the four possible combinations in Table 5.3, we find 0.60 ≤ PCA ≤ 0.76; whereas without taking account of the mediator we would have no non-trivial upper bound. In § 5.6 we will compare these results with the simple analysis of X on Y and with the case of partial mediation.