On the Monotonicity of a Nondifferentially Mismeasured Binary Confounder
-
Jose M. Peña
Abstract
Suppose that we are interested in the average causal effect of a binary treatment on an outcome when this relationship is confounded by a binary confounder. Suppose that the confounder is unobserved but a nondifferential proxy of it is observed. We show that, under certain monotonicity assumption that is empirically verifiable, adjusting for the proxy produces a measure of the effect that is between the unadjusted and the true measures.
1 Introduction
Suppose that we are interested in the average causal effect of a binary treatment A on an outcome Y when this relationship is confounded by a binary confounder C. Suppose also that C is nondifferentially mismeasured, meaning that (i) C is not observed and, instead, a binary proxy D of C is observed, and (ii) D is conditionally independent of A and Y given C. The causal graph to the left in Figure 1 represents the relationships between the random variables.
Causal graphs, where Y is a discrete or continuous random variable, and A, C and D are binary random variables. Moreover, C is unobserved.
[2] argues that adjusting for D produces a partially adjusted measure of the average causal effect of A on Y that is between the crude (i.e., unadjusted) and true (i.e., adjusted for C) measures. Ogburn and VanderWeele [4, Lemma 1] show that, although this result does not always hold, it does hold under some monotonicity condition in C. Specifically, E [Y|A, C] must be nondecreasing or nonincreasing in C. Since this condition can be interpreted as that the average causal effect of C on Y must be in the same direction among the treated (A = 1) and the untreated (A = 0), [4] argue that the condition is likely to hold in most applications in epidemiology. Unfortunately, the condition cannot be verified empirically because C is unobserved. Therefore, one has to rely on substantive knowledge to verify it. Moreover, the condition is sufficient but not necessary. [5] extend these results to the case where C takes more than two values. If there are at least two independent proxies of C, then [3] show that the causal effect of A on Y can be identified under certain rank condition.
In this paper, we prove that if the monotonicity condition holds in D, then it holds in C as well. Since D is observed, the monotonicity condition in D can be verified empirically. Therefore, if no substantive knowledge is available but data are, then combining our result with Lemma 1 by [4] may allow us to conclude that the partially adjusted effect is between the crude and the true ones and, thus, that the partially adjusted effect is a better approximation to the true effect than the crude one. We also report experiments showing that most random parameterizations of the causal graph to the left in Figure 1 result in a partially adjusted effect that lies between the crude and the true ones, although only half of them satisfy the monotonicity condition in D. This confirms that the condition is sufficient but not necessary. This result should be interpreted with caution because, in fields like epidemiology, one is not typically concerned with a random parameterization but, rather, with one carefully engineered by evolution. We provide a partial answer to this question by characterizing a nonmonotonic case (albeit empirically untestable) where the partially adjusted effect still lies between the crude and the true ones. Finally, we also prove that if the monotonicity condition holds in D, then it also holds in C when D is a driver of C rather than a proxy, i.e. D causes C. We illustrate the relevance of this result with an example on transportability of causal inference across populations.
The rest of the paper is organized as follows. Sections 2 and 3 present our results when D is a proxy and a driver of C, respectively. Section 4 closes with some discussion.
2 On a Proxy of the Confounder
Consider the causal graph to the left in Figure 1, where Y is a discrete or continuous random variable, and A, C and D are binary random variables. The graph entails the following factorization:
Let A take values a and ā, and similarly for C and D. Let A, D and Y be observed and let C be unobserved. Let
Since C is unobserved, RDtrue cannot be computed. It can be approximated by the unadjusted average causal effect or crude risk difference (RDcrude):
and by the partially adjusted average causal effect or observed risk difference (RDobs):
We say that E [Y|A, D] is nondecreasing in D if
Likewise, E [Y|A, D] is nonincreasing in D if
Moreover, E [Y|A, D] is monotone in D if it is nondecreasing or nonincreasing in D. Ogburn and VanderWeele [4, Lemma 1] show that if E [Y|A, C] is monotone in C, then E [Y|A, D] is monotone in D. The following theorem proves the converse result. The relevance of this result is as follows. Ogburn and VanderWeele [4, Result 1] show that if E [Y|A, C] is monotone in C, then RDobs lies between RDtrue and RDcrude. The antecedent of this rule cannot be verified empirically, because C is unobserved. Therefore, one must rely on substantive knowledge to apply the rule. The following theorem implies that, luckily, the rule also holds for D and, thus, that the antecedent can be verified empirically.
Theorem 1
Consider the causal graph to the left in Figure 1. If E[Y|A, D] is monotone in D, then E[Y|A, C] is monotone in C.
Proof. Assume to the contrary that E[Y|A, C] is not monotone in C, i.e.
or
This gives four cases to consider: Whether Equation 2 or 3 holds, and whether Equation 4 or 5 holds. Hereinafter, we focus on the first case. The other cases are similar.
Assume that Equations 2 and 4 hold. We show next that the first inequalities in Equations 2 and 4 imply that
because Y is conditionally independent of D given A and C due to the causal graph under consideration and, thus,
because
Furthermore,
where
is known as the log odds, and σ() is known as the logistic sigmoid function [1, Section 4.2]. Note that σ() is an increasing function. Then,
because A is conditionally independent of D given C due to the causal graph under consideration and, thus,
Likewise, the second inequalities in Equations 2 and 4 imply that
which contradicts Equation 6 unless equality holds. However, equality only occurs if
Corollary 2
Consider the causal graph to the left in Figure 1. If E[Y|A, D] is monotone in D, then RDobs lies between RDtrue and RDcrude.
Proof. The result follows directly from Theorem 1 and Ogburn and VanderWeele [4, Result 1].
2.1 Experiments
In this section, we report some experiments that shed additional light on the relationships between the various risk differences. Specifically, we randomly parameterized 10000 times the causal graph to the left in Figure 1 by parameterizing the terms in the right-hand side of Equation 1 with parameter values drawn from a uniform distribution. [1]For each parameterization, we then computed RDtrue, RDobs and RDcrude. The results are reported in Table 1. Of the 10000 runs, 4891 were monotone in C and also in D, as expected from Ogburn and VanderWeele [4, Lemma 1]. There were no other runs that were monotone in D, as expected from Theorem 1. In all these 4891 runs, RDobs was between RDtrue and RDcrude, as expected from Corollary 2 and Ogburn and VanderWeele [4, Result 1]. It is also worth noticing from the table that the 10000 runs are rather evenly distributed among the different entries. Finally, 4460 of the 5109 runs where the monotonicity assumption did not hold still resulted in that RDobs was between RDtrue and RDcrude. In other words, although half of the runs violated the monotonicity assumption, few of them resulted in RDobs being outside the range of RDtrue and RDcrude. In total, RDobs was between RDtrue and RDcrude in 94 % of the runs. Therefore, RDobs was a better approximation to RDtrue than RDcrude in most of the runs. We investigate further this question in the next section, where we characterize a nonmonotonic case where RDobs still lies between RDtrue and RDcrude.
Results of 10000 random parameterizations of the causal graph to the left in Figure 1.
| In-between | Nondec. in D | Noninc. in D | Neither | |
|---|---|---|---|---|
| 2430 | Nondec. in C | 1175 | 1255 | 0 |
| 2461 | Noninc. in C | 1225 | 1236 | 0 |
| 4460 | Neither | 0 | 0 | 5109 |
The plots in Figure 2 show some additional descriptive statistics for the runs where RDobs belonged to the interval between RDtrue and RDcrude. The top left plot shows that most intervals were quite small and, thus, that RDobs was a good approximation to RDtrue in most cases. However, the top right plot shows that RDobs was typically closer to RDcrude than to RDtrue. The bottom left plot is a zoom of the previous plot at the smallest intervals. Finally, the bottom right plot shows that the lower the correlation between C and D when measured by the Youden index, the closer RDobs was to RDcrude. In summary, RDobs seems to be a good approximation to RDtrue, but it seems to be biased towards RDcrude. This is a problem when the interval between RDcrude and RDtrue is large. However, the length of the interval is unknown in practice, and we doubt substantive knowledge may provide hints on it. The bias seems to decrease with increasing correlation between C and D. Although this correlation is unknown in practice, substantive knowledge may give hints on it.
(tl) Histogram of interval length. (tr) Distance between RDobs and RDtrue relative to interval length. (bl) Zoom of previous plot. (br) Distance between RDobs and RDtrue relative to interval length, as a function of correlation between C and D when measured by Youden index.
2.2 Nonmonotonicity
Consider the causal graph to the left in Figure 1. This section characterizes a case where E [Y|A, C] is not monotone in C and, thus, E [Y|A, D] is not monotone in D by Theorem 1, and yet RDobs lies between RDtrue and RDcrude. Specifically, let A, D and Y represent three diseases, and C a risk factor for the three of them. Suppose that suffering A affects the risk of suffering Y. Suppose that half of the population is exposed to the risk factor C, i.e. p(c) = 0.5. Suppose also that the exposure to C affects the risk of suffering A and D as p(a|c) =
Theorem 3
Consider the causal graph to the left in Figure 1. Let p(c) = 0.5 and
Proof. We start by proving some auxiliary facts.
Fact 1. Recall from the proof of Theorem 1 that
where the last equality follows from the assumption that p(c) = 0.5, and the fact that A and D are conditionally independent given C due to the causal graph under consideration. Note that the previous equation implies that
Fact 2. Note that
where the second equality follows from the fact that Y and D are conditionally independent given A and C due to the causal graph under consideration, and the fact that
since
Fact 3. Note that
by the assumptions that p(c) = 0.5 and
since A and D are conditionally independent given C due to the causal graph under consideration, and
because p(d|c) = p(c|d) as shown above, whereas p(a|c) = p(d|c) by assumption. The last equation implies that
We now prove the theorem. Note that the assumption that p(c) = 0.5 implies that
Note also that p(d) = 0.5 by Fact 3 and, thus,
since Y and D are conditionally independent given A and C due to the causal graph under consideration. Note that
by Equation 7 and the fact that the term in penultimate line above is nonnegative. The latter follows from the assumption that
Having proven that RDobs ≥ RDtrue, it only remains to prove that RDcrude ≥ RDobs. Let β = p(d|a) − 0.5. Note that
by Equation 8 and the fact that the term in the penultimate line above is nonnegative. The latter follows from the fact that
Theorem 4
Consider the causal graph to the left in Figure 1. Let p(c) = 0.5 and
Proof. The proof is analogous to that of the previous theorem.
Consider now replacing the assumption that
Theorem 5
Consider the causal graph to the left in Figure 1. Let
Proof. We start by proving that RDcrude ≥ RDtrue. Recall from the proof of Theorem 1 that
where the second equality follows from the assumption that p(c) = 0.5. Likewise,
Note also that p(c|a) ≥ 0.5 and
which together with the fact that σ() and ln() are increasing functions imply that
The results in the previous paragraph allow us to write p(c|a) = 0.5+α and
because α ≥ β ≥ 0 as shown above,
due to the assumption that p(c) = 0.5.
We continue by proving that RDobs ≥ RDtrue. First, recall again from the proof of Theorem 1 that
where the second equality follows from the assumption that p(c) = 0.5, and the fact that A and D are conditionally independent given C due to the causal graph under consideration. Likewise,
Therefore,
by the assumption that
by the assumption that
because
by the assumptions that
which implies that
Then, consider the expression
Therefore,
by the assumption that
as shown in Equation 9. Third, σ() and ln() are increasing functions. We can analogously prove that
because p(d) = 0.5 as shown above.
Finally, the results in the previous paragraphs allow us to write
where the third equality follows from the fact that Y and D are conditionally independent given A and C due to the causal graph under consideration. Then,
by Equation 10, the above shown fact that α ≥ β ≥ 0, and the assumption that
One can analogously prove the following result.
Theorem 6
Consider the causal graph to the left in Figure 1. Let
3 On a Driver of the Confounder
Consider the causal graph to the right in Figure 1. Note that D is now a driver rather than a proxy of C, i.e. D causes C. The graph entails the following factorization:
We show next that our previous results also apply to the new causal graph under consideration.
Theorem 7
Consider the causal graph to the right in Figure 1. If E [Y|A, D] is monotone in D, then E [Y|A, C] is monotone in C.
Proof. The proof of Theorem 1 also applies when D is a driver of C.
Corollary 8
Consider the causal graph to the right in Figure 1. If E [Y|A, D] is monotone in D, then RDobs lies between RDtrue and RDcrude.
Proof. Note that every probability distribution that is representable by the causal graph to the right in Figure 1 can be represented by the causal graph to the left in Figure 1: Simply, let p L(A|C) = p R(A|C) and p L(Y|A, C) = pR(Y|A, C) where the subscript L or R indicates whether we refer to Equation 1 or 11, respectively. Moreover, let
and
Therefore, RDcrude, RDobs and RDtrue are the same whether they are computed from the graph to the right or to the left in Figure 1. Likewise, if E [Y|A, D] is monotone in D for the graph to the right in Figure 1, then it is also monotone in D for the graph to the left, which implies that RDobs lies between RDtrue and RDcrude by Corollary 2.
VanderWeele et al. [8, Result 1] prove that (i) if E[Y|A, C] and E [A|C] are both nondecreasing or both non-increasing in C, then RDobs ≥ RDtrue, and (ii) if E[Y|A, C] and E [A|C] are one nondecreasing and the other nonincreasing in C, then RDobs ≤ RDtrue. The antecedents of these rules cannot be verified empirically, because C is unobserved. Therefore, one must rely on substantive knowledge to apply the rules. Luckily, the rules also hold for D and, thus, the antecedents can be verified empirically. The following theorem proves it.
Theorem 9
Consider the causal graph to the right in Figure 1. If E [Y|A, D] and E [A|D] are both nondecreasing or both nonincreasing in D, then E [Y|A, C] and E[A|C] are both nondecreasing or both nonincreasing in C. If E[Y|A, D] and E[A|D] are one nondecreasing and the other nonincreasing in D, then E[Y|A, C] and E[A|C] are one nondecreasing and the other nonincreasing in C.
Proof. We prove the result when E [Y|A, D] and E[A|D] are both nondecreasing in D. The proofs for the rest of the cases are similar. Then, we have that (i)
Likewise, (ii) and (iv) imply that
As shown in the proof of Theorem 1, this contradicts the fact that C and D are dependent. Therefore, either the assumption (iii) or (iv) or both are false. In the latter case, we get a similar contradiction. So, either the assumption (iii) or (iv) is false. We reach a similar contradiction if replace a with a in the assumptions (i) and (iii). This together with the fact that E [Y|A, C] and E [A|C] are both monotone in C by Theorem 1 prove the result.
Corollary 10
Consider the causal graph to the right in Figure 1. If E [Y|A, D] and E [A|D] are both nondecreasing or both nonincreasing in D, then RDobs ≥ RDtrue. If E[Y|A, D] and E[A|D] are one nondecreasing and the other nonincreasing in D, then RDobs ≤ RDtrue.
Proof. The result follows directly from Theorem 9 and VanderWeele et al. [8, Result 1].
For completeness, we show below that the converse of Theorem 9 also holds.
Theorem 11
Consider the causal graph to the right in Figure 1. If E [Y|A, C] and E [A|C] are both nondecreasing or both nonincreasing in C, then E [Y|A, D] and E [A|D] are both nondecreasing or both nonincreasing in D. If E[Y|A, C] and E[A|C] are one nondecreasing and the other nonincreasing in C, then E[Y|A, D] and E[A|D] are one nondecreasing and the other nonincreasing in D.
Proof. As shown in the proof of Corollary 8, every probability distribution that is representable by the causal graph to the right in Figure 1 can be represented by the causal graph to the left in Figure 1. Therefore, if E [Y|A, C] and E [A|C] are monotone in C for the right graph, then they are so for the left graph as well. Then, E [Y|A, D] and E [A|D] are monotone in D for the left graph [4, Lemma 1] and, thus, they are so for the right graph as well. The result follows now from the contrapositive formulation of Theorem 9.
Given a sufficiently large sample from p(A, D, Y), we may conclude from it that E [Y|A, D] is monotone in D, which implies that RDobs lies between RDtrue and RDcrude by Corollary 8. We can also estimate RDobs and RDcrude from the sample, which implies that (i) if RDcrude ≤ RDobs then RDobs ≤ RDtrue, and (ii) if RDcrude ≥ RDobs then RDobs ≥ RDtrue. Consequently, Corollary 10 is superfluous when data over (A, D, Y) are available. The following example illustrates that the corollary may be useful when no such data are available.
Example 12
Let A and Y represent a treatment and a disease, respectively. Let D and C represent pre-treatment covariates such as socio-economic and health status, respectively. Say that we have a sample from p1(A, D, Y) and a sample from p2(A, D, Y), i.e. we have two samples from two different populations. We are interested in drawing conclusions about RDtrue for a third population, from which we have no data. We make the following assumptions:
p1(D) ≠ p3(D) ≠ p2(D) because the socio-economic profile of the third population differs from the other populations’ profiles.
p1(C|D) = p2(C|D) = p3(C|D) because this distribution represents psychological and physiological processes shared by the three populations.
p1(Y|A, C) = p3(Y|A, C) ≠ p2(Y|A, C) because these distributions represent psychological and physiological processes shared by the first and third populations but not by the second. Then, E3[Y|A, D] = E1[Y|A, D] which can be estimated from the sample from p1(A, D, Y).
p1(A|C) ≠p2(A|C) = p3(A|C) because the second and third populations share the same treatment policy but the first does not. Then, E3[A|D] = E2[A|D] which can be estimated from the sample from p2(A, D, Y).
Then, we cannot estimate RDcrude for the third population and, thus, we cannot use Corollary 8 as we did before to bound RDtrue. Corollary 10 may, on the other hand, be useful in drawing conclusions. For instance, assume that E3[Y|A, D] and E3[A|D] are both nondecreasing or both nonincreasing in D. Then, RDobs ≥ RDtrue by the corollary. If we are interested in testing whether k ≥ RDtrue for a given constant k, then it may be worth assuming the cost of collecting data from the third population in order to compute RDobs, in the hope that k ≥ RDobs which confirms the hypothesis. If we are interested in testing whether RDtrue ≥ k, then we may also be willing to assume the cost, in the hope that k ≥ RDobs which allows us to reject the hypothesis. In the latter case, we may instead decide to not assume the cost because we can never confirm the hypothesis. Such a seemingly negative result may save us time and money. Similar conclusions can be drawn when E [Y|A, D] and E [A|D] are one nondecreasing and the other nonincreasing in D. On the other hand, no such conclusions can be drawn from Corollary 8 before collecting data.
3.1 Bounds
Causal effects are typically defined in terms of distributions of counterfactuals. For instance, the causal effect on Y of an intervention setting A = a is defined as E [Ya]. It can be rewritten as follows [6, Theorem 3.3.2]:
Since C is unobserved, this effect cannot be computed. It can be approximated by the following quantity:
It can also be approximated by Sa = E [Y|a]. Likewise for the causal effect on Y of an intervention setting
VanderWeele et al. [8, Result 1] prove that (i) if E [Y|A, C] and E [A|C] are both nondecreasing or both nonincreasing in C, then
and we say that it is nonincreasing in D if
Likewise for E [Y|a, D].
Corollary 13
Consider the causal graph to the right in Figure 1. If E [Y|a, D] and E [A|D] are both nondecreasing or both nonincreasing in D, then Sa ≥ E [Ya]. If E [Y|a, D] and E [A|D] are one nondecreasing and the other nonincreasing in D, then Sa ≤ E [Ya]. Likewise for a instead of a replacing ≥ with ≤ and vice versa.
Proof. We prove the result for when E [Y|a, D] and E [A|D] are both nondecreasing in D. The proof is similar for the remaining cases. If E [Y|a, D] is not nondecreasing in D, then make it so by parameterizing
Of course, S a is always an upper or lower bound of E [Ya]. The previous corollary allows us to determine always whether it is the one or the other, because E [Y|a, D] and E [A|D] are always nondecreasing or nonincreasing in D. Likewise for a instead of a. On the other hand, given a random parameterization, there is only 50 % chance that E [Y|a, D] and E [Y|a, D] are both nondecreasing or both nonincreasing in D and, thus, E [Y|A, D] is nondecreasing or nonincreasing in D and, thus, we can apply the combination of Theorem 9 and the result by VanderWeele et al. [8, Result 1] as we did above.
3.2 Transitivity
Consider the causal graph A → B → C. Let E [B|A] and E [C|B] be nondecreasing in A and B, respectively. Unfortunately, there is no guarantee that E [C|A] is nondecreasing in A, i.e. the nondecreasing property is not transitive in general [7, Example 3.2]. However, transitivity does hold when A, B and C are binary random variables [7, p. 119]. For binary random variables, Ogburn and VanderWeele [4, Lemma 1] also implies a sort of transitivity result: If E [C|B] is monotone in B, then E [C|A] is monotone in A. Theorem 7 implies then a sort of inverse transitivity result: If E [C|A] is monotone in A, then E [C|B] is monotone in B.
4 Discussion
We have extended the result in Lemma 1 by [4] stating that if E [Y|A, C] is monotone in C, then RDobs lies between RDtrue and RDcrude.We have done so by showing that the result also holds when E [Y|A, D] is monotone in D. This makes the result much more applicable in practice, as the monotonicity condition in D can be verified empirically. We have also extended along the same lines the results reported in Result 1 by [8].
The monotonicity condition in D is, however, sufficient but not necessary. In fact, we have shown through experiments that 94 %of the random parameterizations of the causal graph studied resulted in RDobs being inside the range of RDtrue and RDcrude. However, the monotonocity condition did not hold for approximately half of them. To shed some light on this question, we have characterized a nonmonotonic case (albeit empirically untestable) where RDobs still lies between RDtrue and RDcrude. In future work, we plan to investigate how to relax the monotonicity condition while keeping it sufficient and empirically testable.
Acknowledgement
We thank the Associate Editor and Reviewers for their comments, which helped us to improve our work. This work was funded by the Swedish Research Council (ref. 2019-00245).
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© 2020 J. M. Peña, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
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Articles in the same Issue
- Research Article
- Unifying Gaussian LWF and AMP Chain Graphs to Model Interference
- A Combinatorial Solution to Causal Compatibility
- Post-randomization Biomarker Effect Modification Analysis in an HIV Vaccine Clinical Trial
- The Inflation Technique Completely Solves the Causal Compatibility Problem
- Averaging causal estimators in high dimensions
- Estimating population average treatment effects from experiments with noncompliance
- A Two-Stage Joint Modeling Method for Causal Mediation Analysis in the Presence of Treatment Noncompliance
- On the Monotonicity of a Nondifferentially Mismeasured Binary Confounder
- Beyond Manipulation: Administrative Sorting in Regression Discontinuity Designs
- Instruments with Heterogeneous Effects: Bias, Monotonicity, and Localness
- When is a Match Sufficient? A Score-based Balance Metric for the Synthetic Control Method
- A note on a sensitivity analysis for unmeasured confounding, and the related E-value
- Estimating Average Treatment Effects Utilizing Fractional Imputation when Confounders are Subject to Missingness
- Identification and Estimation of Intensive Margin Effects by Difference-in-Difference Methods
- Direct Effects under Differential Misclassification in Outcomes, Exposures, and Mediators
- Improved Doubly Robust Estimation in Marginal Mean Models for Dynamic Regimes
