Bias due to Selection on Live Births in Studies of Environmental Exposures during Pregnancy: A Simulation Study

Abstract

Background: Studies of the effects of prenatal environmental exposures on postnatal outcomes are particularly vulnerable to live birth bias; i.e., the bias that arises from the necessary restriction of the analysis to live births when that is influenced by both the exposure under study $A$ and unmeasured factors $U$ that also affect the outcome.

Objectives: In the context of a recent publication of nitrogen dioxide (${\mathrm{NO}}_2$) and autism spectrum disorder (ASD) that found an odds ratio (OR) of 0.77 per $5.85\text{ppb}$ ${\mathrm{NO}}_2$ during pregnancy, we aimed to examine what parameters would be needed to account for this protective association through live birth bias.

Methods: We simulated the magnitude of bias under two selection mechanisms and when both mechanisms co-occur, assuming a true null effect. Simulation input parameters were based on characteristics of the original study and a range of plausible values for the prevalence of unmeasured factor $U$ and the ORs for the selection effects (i.e., the effects of ${\mathrm{NO}}_2$ and $U$ on loss and of $U$ on ASD). Each scenario was simulated 1,000 times.

Results: We found that the magnitude of bias was small when ${\mathrm{NO}}_2$ and $U$ independently influenced pregnancy loss (collider-stratification without interaction), was stronger when ${\mathrm{NO}}_2\text{-induced}$ loss preferentially occurred in $U=1$ (depletion of susceptibles), and was strongest when both mechanisms worked together. For example, ORs of 3.0 for ${\mathrm{NO}}_2\text{-loss}$, $U$-loss, $U$-ASD, and U $\text{prevalence}=0.75$ yielded ${\mathrm{NO}}_2\text{-ASD}$ ORs per $5.85\text{ppb}$ ${\mathrm{NO}}_2$ of 0.95, 0.89, and 0.75 for the three scenarios, respectively. The bias is amplified with multiple $Us$, yielding ORs as low as 0.51.

Discussion: Our simulations illustrate that live birth bias may lead to exposure-outcome associations that are biased downward, where the extent of the bias depends on the fetal selection mechanism, the strength of that selection, and the prevalence of U. https://doi.org/10.1289/EHP7961.

Figure 1.

DAG of the structure of live birth bias. Nitrogen dioxide (${\mathrm{NO}}_2$) exposure $A$ affects live births $S$ is also affected by an independent unmeasured risk factor $U$ for ASD (outcome $Y$). Arrows are direct causal effects, and the dashed line is a spurious association induced between $A$ and $U$ after selection on live births (i.e., conditioning on $S=1$). This DAG has the same structure as the birthweight paradox (Hernández-Díaz et al. 2006). Note: ASD, autism spectrum disorder; DAG, directed acyclic graph.

Figure 2.

Live birth bias of $O{R}_{\mathit{AY}}$ under different selection effects. Average odds ratios for the association between nitrogen dioxide (${\mathrm{NO}}_2$; exposure $A$) and ASD (outcome $Y$) among live births $S=1$ ($O{R}_{AY|S=1}$) with varying simulation inputs for the prevalence of the unmeasured risk factor $U$ (${\mathrm{\pi }}_U$) and the magnitude of selection effects ($O{R}_S$) under two selection mechanisms (collider-stratification without interaction, and depletion of susceptibles) and when they both co-occur with a single $U$, assuming a true null effect of ${\mathrm{NO}}_2$ on ASD. Collider-stratification without interaction (Mechanism 1) occurs when $A$ and $U$ have independent causal effects on fetal loss, but with no interaction on the multiplicative scale ($O{R}_{\{AU\}S}=1$, and $O{R}_{AS}=O{R}_{US}=O{R}_{UY}=O{R}_S$). Depletion of susceptibles (Mechanism 2) occurs when $A$ has a causal effect on fetal loss only in the subset of susceptible fetuses ($U=1$), but neither $A$ or $U$ have independent causal effects on fetal loss ($O{R}_{AS}=O{R}_{US}=1$, and $O{R}_{\{AU\}S}=O{R}_{UY}=O{R}_S$). Both mechanisms occur when $A$ and $U$ have independent causal effects on fetal loss, and with interaction on the multiplicative scale ($O{R}_{AS}=O{R}_{US}=O{R}_{\{AU\}S}=O{R}_{UY}=O{R}_S$). Each scenario was simulated 1,000 times. Points represent the mean $O{R}_{AY|S=1}$ in each scenario. Dashed lines indicate the true null effect of ${\mathrm{NO}}_2$ on ASD ($O{R}_{AY}=1$) in the absence of live birth bias, where deviations from 1.0 quantify the magnitude of live birth bias. See Table S1 for corresponding numeric data, including 95% SI. Note: ASD, autism spectrum disorder; OR, odds ratio; SI, simulation intervals.

Figure 3.

Bias parameters that drive live birth bias of $O{R}_{\mathit{AY}}$ under different selection effects. Average bias parameters in the selected population with varying simulation inputs for the prevalence of the unmeasured risk factor $U$ (${\mathrm{\pi }}_U$) and the magnitude of selection effects ($O{R}_S$) under two selection mechanisms and when they both co-occur with a single $U$, assuming a true null effect of nitrogen dioxide (${\mathrm{NO}}_2$; exposure $A$) on ASD (outcome $Y$). In the selected population (live births), $O{R}_{AU|S=1}$ is the association between $A$ and $U$, and ${\mathrm{\pi }}_{U|S=1}$ is the prevalence of $U$. Collider-stratification without interaction (Mechanism 1) occurs when $A$ and $U$ have independent causal effects on fetal loss, but with no interaction on the multiplicative scale ($O{R}_{\{AU\}S}=1$, and $O{R}_{AS}=O{R}_{US}=O{R}_{UY}=O{R}_S$). Depletion of susceptibles (Mechanism 2) occurs when $A$ has a causal effect on fetal loss only in the subset of susceptible fetuses ($U=1$), but neither $A$ or $U$ have independent causal effects on fetal loss ($O{R}_{AS}=O{R}_{US}=1$, and $O{R}_{\{AU\}S}=O{R}_{UY}=O{R}_S$). Both mechanisms occur when $A$ and $U$ have independent causal effects on fetal loss, and with interaction on the multiplicative scale ($O{R}_{AS}=O{R}_{US}=O{R}_{\{AU\}S}=O{R}_{UY}=O{R}_S$). Each scenario was simulated 1000 times. Points represent the mean value of the bias parameter in each scenario. Dashed lines indicate the expected values (in the absence of live birth bias) for ${\mathrm{\pi }}_{U|S=1}$ (${\mathrm{\pi }}_{U|S=1}={\mathrm{\pi }}_U$), and $O{R}_{AU|S=1}$($O{R}_{AU|S=1}=O{R}_{AU}=1$), which are the parameters in the selected population that drive the strength of live birth bias of $O{R}_{AY}$. See Table S2 for corresponding numeric data, including 95% SI. Note: ASD, autism spectrum disorder; OR, odds ratio; SI, simulation intervals.

Figure 4.

Live birth bias of $O{R}_{\mathit{AY}}$ under different selection effects and different numbers of unmeasured risk factors for selection and the outcome. Average odds ratios for the association between nitrogen dioxide (${\mathrm{NO}}_2$; exposure $A$) and ASD (outcome $Y$) among live births $S=1$ ($O{R}_{AY|S=1}$) with varying simulation inputs for the prevalence of the unmeasured risk factor $U$ (${\mathrm{\pi }}_U$) and the magnitude of selection effects ($O{R}_S$) under two selection mechanisms and when they both co-occur with one, two or three $U$s, assuming a true null effect of ${\mathrm{NO}}_2$ on ASD. $U$ is a vector that consists of $\le 3$ unmeasured factors ($U_1,U_2$, $U_3$), where input parameters were applied equally for each unmeasured factor; thus, all references to $U$ henceforth applies to each of the unmeasured factors $U_1,U_2$, $U_3$. Collider-stratification without interaction (Mechanism 1) occurs when $A$ and $U$ have independent causal effects on fetal loss, but with no interaction on the multiplicative scale ($O{R}_{\{AU\}S}=1$, and $O{R}_{AS}=O{R}_{US}=O{R}_{UY}=O{R}_S$); Depletion of susceptibles (Mechanism 2) occurs when $A$ has a causal effect on fetal loss only in the subset of susceptible fetuses ($U=1$), but neither $A$ or $U$ have independent causal effects on fetal loss ($O{R}_{AS}=O{R}_{US}=1$, and $O{R}_{\{AU\}S}=O{R}_{UY}=O{R}_S$); Both mechanisms occur when $A$ and $U$ have independent causal effects on fetal loss, and with interaction on the multiplicative scale ($O{R}_{AS}=O{R}_{US}=O{R}_{\{AU\}S}=O{R}_{UY}=O{R}_S$). Each scenario was simulated 1000 times. Points represent the mean $O{R}_{AY|S=1}$ in each scenario. Dashed lines indicate the true null effect of ${\mathrm{NO}}_2$ on ASD ($O{R}_{AY}=1$) in the absence of live birth bias, where deviations from 1.0 quantify the magnitude of live birth bias. See Table S3 for corresponding numeric data, including 95% SI. Note: ASD, autism spectrum disorder; OR, odds ratio; SI, simulation intervals.