Infant mortality (number of deaths per 1,000 live births) among women with singleton pregnancies in the 1997 cohortlinked birth certificate infant mortality files from the National Center for Health Statistics, by birth weight and smoking status
1. Introduction
A central problem in natural science is identifying general laws of cause and effect. Medical science is devoted to revealing causal relationships in humans [1]. The framework for causal inference applied in epidemiology can contribute substantially to clearly specifying and testing causal hypotheses. In some situations, conditioning on an intermediate, which may be between the cause (exposure) and effect (outcome), is of concern for biomedical researchers and public health practitioners [24]. In particular, there is a conflict in the perinatal epidemiology literature between the desire to obtain birthweightspecific associations [57] and increasing awareness that conditioning on this variable can give rise to severe biases [811]. The difficulty arises because birth weight may be on a pathway from the exposure of interest to the perinatal outcome. For example, if the exposure is maternal smoking and the outcome is infant mortality, maternal smoking may partly affect infant mortality through its effects on fetal growth or on the timing of delivery, thereby potentially through the intermediate, birth weight. In an analysis conditioned on an intermediate, without controlling for the common causes of the intermediate and the outcome, biased results and paradoxical findings can emerge [24,12].
It has been reported that maternal smoking appears to have a protective effect against infant mortality among infants with low birth weight [1315]. This perplexing association is often referred to as the






Birth weight ≥ 2,500 g^{a} 
Smoker  353,335  1,729  4.9 
Nonsmoker  2,453,633  5,838  2.4  
Birth weight < 2,500 g^{b} 
Smoker  40,383  2,192  51.5 
Nonsmoker  137,154  9,387  64.1  
Overall^{c}  Smoker  393,830  3,950  9.9 
Nonsmoker  2,591,452  15,384  5.9  
Total  3,749,676  23,693  6.3 
The apparent protective effect of maternal smoking among lowbirthweight infants is an artifact of conditioning on an intermediate without adequate control for intermediateoutcome confounding [8]. In the birthweight paradox, in addition to maternal smoking, birth defects can be a cause of both low birth weight and infant mortality (Figure 1), but birth defects were not considered in the analysis. For mothers who are smokers and have lowbirthweight infants, the low birth weight could either be a consequence of smoking or a birth defect. For mothers who are nonsmokers and have lowbirthweight infants, the low birth weight cannot be a consequence of smoking, and some other cause must be operating [8]. Thus, a comparison of smoking and nonsmoking mothers without controlling for birth defects will artificially bias the comparison. For this group of lowbirthweight infants, no smoking and low birth weight occurring together is more likely to be associated with the presence of a birth defect. This form of bias is sometimes referred to as colliderstratification bias [1720] because, on the path
A considerable volume of literature highlights the hazards of conditioning on an intermediate [9,10,21], but the only solution offered to date is to abandon conditioning on the intermediate altogether. Simply not conditioning on an intermediate will often be the correct way to proceed with an analysis. When the total effect of the exposure on the outcome is of interest, there is no reason to condition on an intermediate. In general, conditioning on an intermediate will be a concern only when other types of effect, such as the direct effect of the exposure on the outcome (not acting through the intermediate), are considered.
In this chapter, we discuss two analytical approaches to help draw inferences when the effect of interest may be obtained by conditioning on an intermediate. The two approaches include (i) the principal stratification approach to assess the
This chapter is organized as follows. Section 2 introduces the notation used throughout the chapter. Section 3 defines the PSE and presents a simple method for sensitivity analysis. The method is illustrated using the NCHS data. In a similar manner, the NDE is discussed in Section 4. In Section 5, the relationship between these two causal effects is briefly discussed, although this may be somewhat theoretical. Finally, Section 6 offers some concluding remarks.
2. Notation and concepts
We let
Unfortunately we do not have access to the full NCHS dataset and can only use the published data [16]. Therefore, we cannot implement any analyses in which the confounders are taken into account.
. Nevertheless, even in cases in which confounders exist between these variables, theories presented in this chapter hold conditional on the confounders (Using the above notation, a comparison of the infant mortality risks between smoking and nonsmoking mothers with infants of low birth weight can be described as follows:
where E[






1  1  1  Always birth death 
2  1  0  Birth death only with maternal smoking 
3  0  1  Birth death only with maternal nonsmoking 
4  0  0  Never birth death 
To illustrate the ideas, we use the concept of potential (or counterfactual) outcomes [22,23]. We let
In other words, the above measure quantifies the total effect of
In the NCHS data, this measure is calculated as 9.9 – 5.9 = 4.0 per 1,000 live births.
Likewise, we let






1  1  1  Always low birth weight 
2  1  0  Low birth weight only with maternal smoking 
3  0  1  Low birth weight only with maternal nonsmoking 
4  0  0  Never low birth weight 
Finally, we let
3. Principal stratification approach
As an alternative to the crude measure, we introduce the principal stratification approach. We define the PSE in Section 3.1 and present a simple method for the sensitivity analysis under the assumption in Section 3.2. In Section 3.3, the method is illustrated using the NCHS data. In Section 3.4, we present a sensitivity analysis formula by relaxing the assumption used in Section 3.2, although the form may not be simple. The derivations of the equations and inequalities presented in this section are given in Appendix 1.
3.1. Principal strata effect
One approach to making a fair comparison involves assessing the effect of the exposure on the outcome among the subpopulation for which the intermediate would be present irrespective of the exposure status. For example, we might be interested in the effect among the subpopulation for which infants would have a low birth weight irrespective of maternal smoking status. This subpopulation for which the intermediate will occur irrespective of the exposure is sometimes referred to as a principal stratum [26]. More generally, a principal stratum is a subpopulation defined by the joint potential intermediates (
Those for which the intermediate will occur irrespective of exposure status: always low birth weight (response type 1 in Table 3);
Those for which the intermediate will occur with exposure but not without exposure: low birth weight only with maternal smoking (response type 2 in Table 3);
Those for which the intermediate will occur without exposure but not with exposure: low birth weight only with maternal nonsmoking or defiers (response type 3 in Table 3); and
Those for which the intermediate will not occur irrespective of exposure status: never low birth weight (response type 4 in Table 3).
If we are interested in whether maternal smoking has a protective effect among lowbirthweight infants, one potentially relevant question within the context of principal stratification is whether maternal smoking has a protective effect among the subpopulation in which infants would have a low birth weight irrespective of maternal smoking status (
This measure quantifies the total effect of
3.2. Sensitivity analysis method
To derive a simple sensitivity analysis formula, we require the following assumption, which is sometimes referred to as a monotonicity assumption [33]:
This assumption implies that there are no individuals for whom the intermediate would occur without exposure but not with exposure (
where the sensitivity parameter
The interpretation of this sensitivity parameter is the difference in infant mortality risks under maternal smoking for two subpopulations: the subpopulation with smoking mothers whose infants had a low birth weight, and the subpopulation with nonsmoking mothers whose infants had a low birth weight. The parameter is not identified from the observed data.
The sensitivity analysis can be easily conducted. The sensitivity parameter
As it may be troublesome to determine the range of
This assumption will hold if the subpopulation with (
Assumption 2 with
which implies that the infant mortality risk in the subpopulation consisting of those who would never have a lowbirthweight infant is not more than that in the subpopulation consisting of those who would have a lowbirthweight infant only with a smoking mother. Under the scenario in which the mother is a smoker, this would indeed be the case. Thus, Assumption 2 with
which implies that the infant mortality risk in the subpopulation consisting of those who would have a lowbirthweight infant only with a smoking mother is not more than that in the subpopulation consisting of those who would always have a lowbirthweight infant.
When Assumption 2 holds in addition to Assumption 1, the range of
where
The second assumption is sometimes referred to as the assumption of monotone treatment response [34,36,37] and is formalized as follows in the current setting
The assumption of monotone treatment response was originally given as an assumption for all individuals;
In the context of the smokingbirth weight example, this assumption implies that in the subpopulation with (
When it is considered that, in addition to Assumption 1, both Assumptions 2 and 3 hold, we can use a narrower range derived under these two assumptions.
3.3. Illustration
We now apply the principal stratification approach to the NCHS data shown in Table 1. As noted in Section 1, the crude difference in the mortality risk of lowbirthweight infants between smoking and nonsmoking mothers was –12.6 (95% CI: –15.0, –10.1) per 1,000 live births, suggesting that maternal smoking has a protective effect against infant mortality for lowbirthweight infants.
To calculate the PSE defined in Section 3.1, we adjust this crude estimate by the sensitivity parameter
The result of this sensitivity analysis for the PSE suggests that maternal smoking has a harmful effect on the subpopulation of infants who would have a low birth weight irrespective of maternal smoking, although the lower limit of the 95% confidence interval for the PSE was still smaller than 0 when
3.4. Sensitivity analysis without the monotonicity assumption
The monotonicity assumption (Assumption 1) is a strict assumption because the inequality (
In the context of the smokingbirth weight example, the parameter
Using these three sensitivity parameters, the PSE can be expressed as follows [38]:
It will be more difficult to determine the values or ranges of these three sensitivity parameters (
4. Interventionbased approach
As another alternative to the crude measure, we introduce the interventionbased approach. In Section 4.1, we define two types of direct effects, the controlled direct effect (CDE) and the NDE, both of which are based on interventions on the intermediate [2,39]. We mainly focus our discussion on the NDE. A simple method for the sensitivity analysis is presented in Section 4.2 and is illustrated in Section 4.3 using the NCHS data. The derivations of equations and inequalities presented in this section are given in Appendix 2.
4.1. Controlled and natural direct effects
The CDE captures the effect of exposure
Contrary to the PSE, the CDE is a causal effect concerning the whole population. In the context of the smokingbirth weight example, the CDE with
The NDE differs from the CDE in that the intermediate
We can also define the NDE under the exposure level of
which compares the effect of an exposure on the outcome if the intermediate were set to what it would have been when exposure
which compares the effect of the intermediate at levels
This decomposition may help in understanding the meaning of the NDE,
If there is no interaction between the effects of exposure
Similar to the CDE, the NDE is also a causal effect concerning the whole population, and it can be estimated from the data under certain assumptions [42,43]. However, neither the CDE nor the NDE can be identified when an unmeasured confounder exists between the intermediate and the outcome, as in Figure 1. Therefore, sensitivity analysis techniques have been discussed to assess their magnitudes [4,4448]. In the next subsection, a simple sensitivity analysis method for the NDE is presented. Methods for the CDE are found elsewhere [4,44,47].
4.2. Sensitivity analysis method for the natural direct effect
Although the monotonicity assumption (Assumption 1) was necessary to derive a simple sensitivity analysis formula for the PSE, this assumption is not required to derive one for the NDE.
For each possible value of intermediate
The sensitivity parameter
We now consider a weighted mean of the sensitivity parameters
After calculating the crude risk differences E[
The variance of the first term in equation (4) is calculated by the delta method, var(
We can determine the upper limits of
The upper limit of
Assumptions 2* and 3* (2 and 3) relate to monotone treatment selection and monotone treatment response regarding the exposure, respectively. We can also make these types of assumptions about the intermediate [49]:
In the context of the smokingbirth weight example, Assumption 4 implies that infants with a low birth weight, representing the subpopulation with (
While Assumptions 2* and 3* can lead to only the upper limit, Assumptions 4 and 5 can lead to both limits. Furthermore, when Assumption 1 is added, under Assumptions 1 and 5, the lower limit of Γ is improved to:
However, neither the lower nor the upper limit can be derived under only one of the Assumptions 4 and 5.
4.3. Illustration
We now apply this interventionbased approach to the NCHS data shown in Table 1. To calculate the NDE defined by equation (4), we determine a range of values for Γ. Under Assumptions 2* and 3*, the respective upper limits of
The result of this sensitivity analysis for the NDE suggests that maternal smoking has a directly harmful effect on infant mortality. Thus, the birthweight paradox is also resolved in terms of the NDE.
5. Relationship between the principal strata effect and the natural direct effect
We briefly discuss the relationship between the PSE and NDE. Again, we note that the individual NDE is defined as NDE(
To prove this theorem, we consider a probability of
where the first equation is from the decomposition of the total effect to the NDE and NIE, the second is by NDE(
The converse of this theorem does not hold: the absence of a PSE does not imply the absence of a NDE. Nevertheless, from the contraposition of this theorem, when there is a PSE, there must be some individuals for whom there is a NDE. The results of the sensitivity analyses in Sections 3.3 and 4.3 showed that the true PSE was not smaller than 0 and that the true NDE was larger than 2.3. The results do not contradict the contraposition of Theorem 1.
6. Conclusion
In this chapter, we described two approaches related to calculating the effect of an exposure on an outcome that is conditional on potential intermediates or one that does not act through the intermediate. Here, we made an impractical assumption in the observational studies that no confounder exists between the exposure and the outcome or between the exposure and the intermediate. Nevertheless, the methodologies described here also hold conditional on confounders if no unmeasured confounder exists between these variables. We considered a risk difference as the effect measure, but the methodologies can be extended to other effect measures.
Each approach has a unique interpretation and its own strengths and weaknesses. In the principal stratification approach, one conditions on the subpopulation for which the intermediate would occur irrespective of exposure. An advantage of this approach is that the subpopulation is a particularly highrisk group in which the intermediate will necessarily occur. A disadvantage is that we do not know who is in the subpopulation such that the intermediate will occur irrespective of the exposure. In the interventionbased approach, the NDE appears to capture the effect of an exposure on the outcome if the intermediate was set to what it would be when the exposure is set to 0. An advantage of this approach is that it can be used to decompose the total effect into direct and indirect components. A disadvantage is that it is difficult to understand the meaning of the NDE from the form. In addition, it is difficult to interpret the sensitivity parameter for the sensitivity analysis, although this may be avoided by applying a parametric model [46].
In many studies, the total effect of the exposure on the outcome in the whole population may be of central interest, and then none of the approaches described here are required. The approaches described here are of relevance only when the investigators are interested in the direct effect of the exposure not acting through the intermediate or the effect of the exposure on the outcome for certain groups at high risk for the intermediate. In some birth weight settings, the exposure or intervention under study may occur after birth in some cases [6]. In these cases, birth weight becomes a preexposure baseline variable, and the approaches described here are not needed. These settings should be distinguished from those similar to the birthweight paradox. When the approaches described here are of relevance, both the PSE and NDE may be in a consistent direction in some situations, as seen in Sections 3.3 and 4.3. However, it is important to note that the two approaches need not give effect estimates in the same direction. Having effect estimates in different directions with the two approaches is not necessarily an indication that one of the estimates is in the wrong direction. The two approaches estimate two different effects (effects for two different populations), and these may in fact be in different directions. Before these approaches are applied, it is important to be clear about the scientific or policy question.
The approaches described in this chapter are applicable to a number of similar settings in all areas of epidemiological research. As the existing literature has made clear, conditioning on an intermediate can be problematic and can give rise to severe biases. In many contexts, conditioning on an intermediate is not necessary and is best avoided. Nevertheless, there are cases in which conditioning on an intermediate is of scientific or policy interest. We have shown that alternative approaches can be used to draw inferences in such settings. Although these methodological tools are imperfect and need to be interpreted carefully, they can be useful in examining conditional and direct effects.
Acknowledgments
This work was supported partially by GrantinAid for Scientific Research (No. 23700344) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.Appendix
Appendices 1 and 2 outline the derivations of the equations and inequalities presented in Sections 3 and 4, respectively. As noted in Section 2, we assume that no confounder exists between
Appendix 1: Derivations of equations and inequalities in Section 3


Using 

where the second equation is by Assumption 1, the third is by the independency and consistency assumptions, the fourth is by using 



The relationship between 

(A1)  
because Pr( 

(A2)  
Similarly,  
(A3)  
(A4)  
(A5)  
As 

where it is assumed that 

(A6)  
(A7)  
(A8)  
(A9)  
The differences between (A6) and (A7) and between (A8) and (A9) lead to, respectively  
Assumption 2 with 



Substituting (A7) into (A6) gives  
(A10)  
and substituting (A8) into (A9) leads to  
(A11)  
Given that these two equations are equal, some algebra yields  
(A12)  
Then, a range of 

Under Assumption 3,  
and similarly 



Using 

and (A3) with 

The difference between these two equations leads to equation (2). 
Appendix 2: Derivations of equations and inequalities in Section 4


Using 

where the first equation is by the independency assumption, the third is by the consistency assumption, and the fourth is by 

The difference between these two equations leads to equation (4).  


A range of 

(A13)  
(A14)  
where 



By the consistency assumption, E[ 

where the first inequality is by Assumption 5 with 

A similar calculation gives the upper limit.  
Equations (A13) and (A14) hold under Assumption 1 and 

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Notes
 Unfortunately we do not have access to the full NCHS dataset and can only use the published data [16]. Therefore, we cannot implement any analyses in which the confounders are taken into account.
 The assumption of monotone treatment response was originally given as an assumption for all individuals; i.e., Y(0)≤Y(1) for all individuals [34]. Therefore, Assumption 3 is a somewhat weaker assumption than the original one.
 We can also define the NDE under the exposure level of A = 1 as NDE ≡ E[Y(1,M(1))] – E[Y(0,M(1))]. The NIE corresponding to this NDE is equal to E[Y(0,M(1))] – E[Y(0,M(0))].