Sensitivity analysis for causal decomposition analysis: Assessing robustness toward omitted variable bias
-
Soojin Park
,
Suyeon Kang
Abstract
A key objective of decomposition analysis is to identify a factor (the “mediator”) contributing to disparities in an outcome between social groups. In decomposition analysis, a scholarly interest often centers on estimating how much the disparity (e.g., health disparities between Black women and White men) would be reduced/remain if we set the mediator (e.g., education) distribution of one social group equal to another. However, causally identifying disparity reduction and remaining depends on the no omitted mediator–outcome confounding assumption, which is not empirically testable. Therefore, we propose a set of sensitivity analyses to assess the robustness of disparity reduction to possible unobserved confounding. We derived general bias formulas for disparity reduction, which can be used beyond a particular statistical model and do not require any functional assumptions. Moreover, the same bias formulas apply with unobserved confounding measured before and after the group status. On the basis of the formulas, we provide sensitivity analysis techniques based on regression coefficients and
1 Introduction
Decomposition analysis aims to identify factors that may close the observed gap in social, psychological, behavioral, or health outcomes between groups defined by social-demographic factors, such as gender/sex, race/ethnicity, socioeconomic status (SES), etc. Such factors are called “mediators” because they are believed to lie between the exposure (one’s social position) and the outcomes. Traditional decomposition analysis based on the difference-in-coefficients approach [1,2] provides a straightforward way to estimate the degree to which the observed disparity is reduced or remains after controlling for the mediator. However, the traditional method is limited to a specific statistical model that cannot be readily generalizable to discrete mediators or outcomes and nonlinear relationships [3]. Recently, several researchers developed and applied decomposition analysis within the counterfactual framework of causal inference, namely, “causal decomposition analysis,” that overcomes the limitations of traditional decomposition analysis. VanderWeele and Ronbinson [4] and Jackson and VanderWeele [5] conceptualized decomposition analysis within the counterfactual framework, and many articles have appeared on this topic in the past 5 years [6–11].
A central goal of causal decomposition analysis is to estimate the degree to which an observed disparity would be reduced or remain if we equalize the mediator distribution across social groups. For example, how much would health disparities decrease if we equalize the education level between Black women and White men? The causal identification of the disparity reduction and remaining hinges on the strong assumption of no unobserved confounding in the mediator–outcome relationship. One way to assess the robustness of findings against possible violations of this assumption is to conduct a sensitivity analysis, yet few sensitivity analysis techniques are available in the causal decomposition framework. Previously, Park et al. [10] developed a preliminary sensitivity analysis that assesses the robustness of disparity reduction and remaining estimates. However, Park et al. method is restricted to a certain setting that requires conditional independence between unobserved and observed intermediate confounders (i.e., the effects of social groups confounding the mediator–outcome relationship). Another limitation is that the interpretation of the sensitivity parameter is not straightforward since the prevalence difference in an unobserved confounder (e.g., being discriminated), comparing individuals in different groups, is conditioned on the mediator (e.g., education), which is a descendant of (a variable affected by) the group status.
In this study, we propose a set of sensitivity analyses for causal decomposition analysis, consisting of sensitivity parameters that are easy to interpret without making the restrictive assumption. After introducing our motivating example using data from Midlife Development in the U.S. (MIDUS) (Section 2), we review literature on causal decomposition analysis as a statistical framework that identifies contributing factors to disparities (Section 3). This review highlights how causal decomposition analysis differs from the related definitions in the causal mediation literature, including natural direct and indirect effects [12,13]. As a result, we show that sensitivity analysis developed for natural indirect effects can assess the robustness of disparity reduction when no intermediate confounders exist, which is unlikely in many studies that investigate contributing factors to disparities.
Therefore, we develop a set of sensitivity analyses for disparity reduction and remaining that incorporate “observed” intermediate confounders. We begin by deriving general bias formulas for disparity reduction and remaining which are the basis of our proposed sensitivity analyses (Section 4). Since general bias formulas do not rely on any statistical models, they apply to various situations, including linear and nonlinear relationships as well as different types of mediator, outcome, or omitted confounder variables. We show that same bias formulas apply for “unobserved” confounding measured before and after the group status. Second, we provide simplified bias formulas given linear models specified for the outcome and the unobserved confounder (Section 5). The simplified bias formulas offer a sensitivity analysis that is straightforward to use if the linearity assumption is met. Finally, we reparameterize the regression-based sensitivity analysis to
In Section 7, we conclude with a discussion. Our sensitivity analysis is implemented in the “causal.decomp” R package. Code to replicate all analyses can be found at https://github.com/soojinpark33/Sensitivity-Analysis-for-CDA/blob/main/README.md.
2 Running example
To motivate the concepts and methods that we present, we rely on an epidemiological example; studies have consistently observed that racial and gender minorities, particularly Black women, show poorer cardiovascular health (CVH) than other race–gender groups. SES which is a fundamental cause and a key determinant of access to resources, may operate via many mechanisms to affect multiple disease outcomes [15], including cardiovascular disease [16]. Educational attainment plays a key role in explaining racial and gender disparities in health, and education also affects other subsequent SES measures, such as income and wealth. Therefore, we hypothesize that the observed disparity in CVH between race–gender groups would decrease if we equalize the education levels between the groups. Causal decomposition analysis can be used to test this hypothesis.
From the hypothesis, we define four social groups: White men (
We encode our understanding of data-generation process involving these variables in Figure 1(a). We assume racial and gendered disparities in CVH arise through childhood abuse (
![Figure 1
Directed acyclic graph [19]: (a) when no unobserved confounder exists and (b) when unobserved confounder
U
U
exists. Note. (1) Diagram represents the relationship between race and gender intersectional status
R
R
, cardiovascular health
Y
Y
, and education
M
M
, as well as history
H
H
, age
C
1
{C}_{1}
, genetic vulnerability
C
2
{C}_{2}
, childhood SES
X
1
{X}_{1}
, and childhood abuse
X
2
{X}_{2}
. (2) Placing a box around the conditioning variables implies that a disparity is considered within levels of these variables. (3) Dotted lines indicate intervening on
M
M
given baseline covariates. (4) In figure b,
X
2
{X}_{2}
and
U
U
are separate nodes but presented together because the result remains valid irrespective of the direction of causality between
X
2
{X}_{2}
and
U
U
, or the absence thereof.](/document/doi/10.1515/jci-2022-0031/asset/graphic/j_jci-2022-0031_fig_001.jpg)
Directed acyclic graph [19]: (a) when no unobserved confounder exists and (b) when unobserved confounder
However, one may argue that the relationship between education and later CVH could still be confounded by unobserved confounders (e.g., discrimination based on race, gender, and social class) even after controlling for the observed variables. Figure 1(b) shows a scenario in which an unobserved variable
How can we validate our conclusions on disparity reduction and remaining to the possible omission of unobserved variables? In Section 4, we present a set of sensitivity analyses that applied researchers can use to validate their findings to possible omitted variables. To proceed, investigators should further clarify baseline covariates and intermediate confounders, in addition to identifying the group status, the mediator (education), and the outcome (CVH). Baseline covariates characterize demographics through which CVH or education differences are considered equitable [8], which are age or genetic vulnerability in our example. Intermediate confounders represent the effects of race–gender status that confound education and CVH, which are childhood SES and abuse in the example. We use the following notation to represent baseline covariates and intermediate confounders. Intermediate confounders are denoted as
3 Review and implications on existing sensitivity analyses
This section provides an overview of causal decomposition analysis based on interventional analogues of natural direct and indirect effects (interventional effects [21,22]). We then compare this approach with the related definitions in the causal mediation literature, including natural direct and indirect effects. Finally, we show the conditions under which sensitivity analyses developed for natural indirect effects can be used to assess the sensitivity of disparity reduction.
3.1 Causal decomposition analysis
Causal decomposition analysis does not contribute to the argument of whether socially defined characteristics such as race and gender can be given a causal interpretation [5]. Rather, it focuses on estimating a causal effect of potentially manipulable factors (mediators) in reducing the observed disparity. Therefore, causal decomposition analysis aims to estimate how much the disparity would be reduced by intervening to equalize the mediator distribution between social groups. In the motivating example, this intervention implies increasing the Black women’s education to the level of White men among those with the same baseline covariates. To illustrate, we use the example of comparing Black women (comparison group) and White men (reference group). Statistically, it requires imputing each Black women’s mediator with a randomly drawn value from the mediator distribution of White men among those with the same level of baseline covariates.
3.1.1 Definitions
Here, we fix the reference group to
Using this notation, disparity reduction is defined as, given baseline covariates, the difference between the average CVH of a comparison group and the average counterfactual CVH if their education level was the same as the reference group among those with the same baseline covariates levels. Formally,
Likewise, the disparity remaining is defined as, given baseline covariates, the difference between the average counterfactual CVH of a comparison group after the hypothetical intervention and the average CVH outcome of the reference group. Formally,
The observed disparity is equal to the sum of disparity reduction and disparity remaining due to hypothetically intervening on the mediator as
3.1.2 Identification assumptions and results
Defining disparity reduction and remaining requires an unobservable quantity, i.e.,
A1: Conditional independence:
A2: Positivity:
A3: Consistency: if
These three assumptions are all strong, and the plausibility of the assumptions depends on the context of the study. This study aims to address the possible violation of conditional independence.
Given the assumptions, the disparity reduction and remaining are nonparametrically identified as follows:
where
3.1.3 Estimation
If the assumptions hold, there are many ways to estimate disparity reduction and disparity remaining including regression-based estimators [5,24][1], weighting-based estimators [8], imputation-based estimators [10,25][2], and doubly robust estimators [11]. Although limited due to its modeling assumptions, regression-based methods are perhaps the most straightforward way to estimate disparity reduction and remaining. Consider the following models fitted to the intermediate confounders, mediator, and outcome as follows:
where
Then, the disparity reduction is estimated as
3.2 Comparison with the related definitions in the literature
In this section, we compare disparity reduction and remaining defined in equations (1) and (2) with the related definitions in the literature: conditional average treatment effects (CATE), controlled direct effects (CDEs), natural in/direct effects, and interventional in/direct effects.
CATE and CDE. Defining disparity reduction and remaining requires stochastic interventions on the mediator to follow an alternative distribution. Instead of stochastic interventions, we could intervene to fix the mediator to a prespecified value. Then, disparity reduction and remaining will similarly correspond to CATE and CDE, respectively. The CATE is the effect of an intervention (
Natural in/direct effects. First, we argue that defining natural indirect effects is not straightforward in disparities research. In the example, the natural indirect effect defined between Black women and White men in the motivating example is the expected difference, comparing each Black woman’s actual and potential CVH after setting her education level to a value that would have naturally been observed had she been born a White man. Given that gender and race are essentially nonmodifiable, it is somewhat strange to consider fixing each individual’s mediator to a value that would have naturally been observed had the individual been born a White man [28]. In contrast, interpreting disparity reduction defined in equation (1) is straightforward because we (1) do not define counterfactual outcomes with respect to social groups such as race and gender and (2) assign Black women’s education to a randomly drawn value from the observed distribution of White men’s education, rather than assigning education to the value that would have been realized in a counterfactual world in which she was a White man. Disparity reduction compares the average Black woman’s CVH to the average counterfactual CVH outcome of Black women after equalizing their education level to that of White men as a group.
Next, we compare the conditional independence assumption (A1) with the related assumptions required to identify natural indirect effects. We don’t compare assumptions regarding the exposure (treatment ignorability) because causal decomposition analysis does not attempt to estimate the causal effect of social groups (race and gender) and thus does not make any assumption regarding the exposure (group status). Pearl [19] made the following assumption to identify natural indirect and direct effects:
Alternatively, Robins [13] made the following assumption:
Instead of the no interaction assumption, Petersen et al. [30] require the following assumption:
Interventional in/direct effects. Even when these additional assumptions are not met, previous studies [30,31] have noted that the natural direct effect still estimates an interesting causal parameter: the average of CDEs with respect to the conditional distribution of
By applying the interventional analogs, VanderWeele and Robinson [4] investigated contributing factors to racial disparities. They suggested focusing on the causal effect of manipulable factors (mediators) rather than the causal effect of race (exposure), thereby defining counterfactual outcomes with respect to the mediators, not the exposure. Following their approach, Jackson and VanderWeele [5] proposed formal definitions of disparity reduction and disparity remaining, one of which is shown in equations (1) and (2). The causal estimands were further extended by other scholars, including Lundberg [11] who allowed intervening to fix the mediator to a prespecified value or the mediator to follow various distributions (rather than the observed distribution of the reference group), and Park et al. [10] who accommodated the case of intervening on multiple mediators.
3.3 Implications on existing sensitivity analyses
3.3.1 Under no intermediate confounders
If no intermediate confounders exist, interventional direct and indirect effects coincide with natural direct and indirect effects, respectively. Although it may be a rare situation, suppose that the intermediate confounders (e.g., childhood SES and abuse) do not exist. Then, one can use a nonparametric sensitivity analysis [36,37] or parametric sensitivity analysis [3] developed for natural direct and indirect effects to assess the robustness of estimated disparity reduction and remaining to possible unobserved pre-exposure confounding.
3.3.2 Under no interaction between
R
and
M
Now, we consider a situation where intermediate confounders exist, but no interaction exists between social groups and the mediator. This situation is perhaps plausible in some studies that examine disparities depending on a type of mediator. Then, the estimator of disparity reduction
In conclusion, a new sensitivity analysis is necessary for disparity reduction and remaining that incorporates existing intermediate confounders and possible interaction effects.
3.4 An application to MIDUS
We estimate disparity reduction and remaining with varying assumptions. Table 1 shows the estimated quantities of interest between Black women and White men. Other comparisons are available, but we only present the results between Black women and White men for simplicity. The initial disparity between Black women and White men is
Estimates of the disparity reduction and disparity remaining
| Estimate (95% CI) | |||
|---|---|---|---|
| R–M interaction | No | Yes | Yes |
| Intermediate confounders | Yes | No | Yes |
| Initial disparity (
|
|
|
|
| (95% CI) |
|
|
|
| Disparity remaining (
|
|
|
|
| (95% CI) |
|
|
|
| Disparity reduction (
|
|
|
|
| (95% CI) |
|
|
|
| % Reduction | 41.6% | 50.2% | 37.3% |
Note: Black women:
First, we estimate disparity reduction and remaining, assuming no interaction between education and race–gender status (first column in Table 1). These estimators are consistent for natural direct and indirect effects defined by Robins [13]. Disparity reduction is negative (
We compare this result after relaxing the no interaction assumption (third column), which is consistent with disparity reduction and remaining defined in equation (3). The difference in disparity reduction and remaining estimates is small. Disparity reduction changes from
Finally, we examine the result after relaxing the no interaction assumption but without intermediate confounders (second column), which is consistent with the natural indirect effect defined by Pearl [19]. Not controlling for intermediate confounders results in an overestimation or underestimation of disparity reduction and remaining when compared to controlling for intermediate confounders (third column). After including observed intermediate confounders, disparity reduction changes from
4 General bias formulas
We begin by calculating the bias for disparity reduction and remaining when the unobserved confounder
The conditions required to calculate the bias are as follows: (1) no omitted confounding exists in the mediator and outcome relationship given group status (
Suppose that we only had observed data. Then, researchers would often estimate the disparity reduction using observed data,
Note that we used
Then, for a particular value of
A proof is given in Appendix A. These general bias formulas do not require any assumptions regarding functional forms, variable types of the mediator, outcome, or unobserved confounders. While these bias formulas can be used in general settings, their applicability may be limited due to too many moving parts (i.e., sensitivity parameters). Therefore, we provide simplified bias formulas under linear models specified for the outcome and the unobserved confounder that are straightforward to use in Section 6. We will discuss the applicability of the general bias formulas later.
Note that the same bias formulas apply with pre-exposure unobserved confounding under the additional assumption of
This equation leads to the same expression shown in equation (8) since
5 Sensitivity analysis using regression coefficients
Unlike the general bias formulas, the proposed sensitivity analyses in this study are developed under a particular statistical model that assumes linearity. However, we show that an extension is possible when linearity is violated.
5.1 Under the linearity assumption
The general bias formulas in equation (9) can be simplified under linear models specified for the outcome and the unobserved confounder as follows:
for
where an unbiased estimate of
We offer several remarks about these bias formulas. First, the bias for disparity reduction and remaining is the same except for the sign. This is because the initial disparity is an observed quantity conditional on specified covariates, and hence, no bias exists due to the unobserved mediator–outcome confounder.[3]
Second, the bias is zero when either
Third, the unobserved variable confounds the intermediate confounders–outcome and intermediate confounders–mediator relationships (
Finally, the intermediate confounder and outcome models specified in equation (10) imply that the effect of unobserved confounder (
The bias formulas expressed in regression coefficients give us an intuitive idea of sensitivity analysis. The basic idea is to find a combination of
5.2 When the linearity assumption is violated
When the linearity assumption made in equation (10) is violated, one can either (1) derive their bias formulas based on modified models for
5.3 Illustration using regression-based sensitivity analysis
This section uses the example provided in Section 3.4 to describe and interpret the sensitivity analysis using regression coefficients. In Section 3.4, we estimated the disparity reduction and disparity remaining between Black women and White men. We here focus on the case after assuming differential effects of education by group (R–M interaction) and accounting for existing intermediate confounders. As shown in Table 1, the initial disparity would be reduced by about 37.3% if Black women’s education level was the same as that of White men among those with the same age and genetic vulnerability. This result can be given a causal interpretation under the assumptions described in A1–A3. Suppose that conditional independence (A1) is violated because an unobserved variable, that is, whether each individual perceives to be discriminated, confounds education and CVH. How strongly does this unobserved confounder have to be to explain away the estimates?
In Figure 2, hypothetical values for
Sensitivity contour plots using regression coefficients. (a) Disparity reduction. (b) Disparity remaining. Note. (1) Bold lines represent the points at which the estimates become zero. (2) Standard lines represent the points at which the estimates become the respective value (e.g.,
6 Sensitivity analysis using
R
2
values
The regression-based sensitivity analysis has often been used with a binary unobserved confounder, such as whether discriminated. It is perhaps because the interpretation of sensitivity parameters with a binary unobserved confounder is straightforward. Specifically,
In this section, we re-express the simplified bias formulas shown in equation (11) using scale-free
This reparameterization to
6.1 Under the linearity assumption
6.1.1 Point estimates
Let the partial
for
6.1.2 Standard errors
Some investigators might also be interested in quantifying the amount of confounding that would change the significance of the effects. The standard error for disparity reduction for
where
We can obtain an unbiased estimate of
Note that the standard error of disparity reduction shown in equation (15) is the same as the sample standard deviation of the disparity reduction estimate (
Calculating the standard error for disparity remaining is more complex. Therefore, we use
where
Again, the standard error of disparity remaining shown in equation (16) is the same as the sample standard deviation of the remaining disparity estimate (
This bias formulas in equation (14) and standard errors for disparity reduction and remaining in equations (15) and (16) can be used as sensitivity analysis that depends on two sensitivity parameters
We conduct a Monte Carlo simulation study to evaluate how well the proposed standard error estimators perform in varying sample sizes and effect sizes of sensitivity parameters. In this simulation, we find that the 95% CI coverage rate exceeds 0.91 regardless of sample sizes and effect sizes of sensitivity parameters. The coverage rate tends to be closer to the nominal level (0.95) with a sample size of 500 or larger. The coverage rate is lower than expected (0.916) with a small sample size (e.g., 100) and a small effect size of sensitivity parameters (e.g., 0.02). The results and details of the simulation study are reported in Appendix E. In addition, we provide a standard error estimator for a percent reduction in Appendix F.
6.2 When the linearity assumption is violated
The expression of point estimates and standard errors becomes more cumbersome when the interaction effect exists in the group–mediator relationships in the outcome model. In such a case, the bias for disparity reduction and remaining estimates is the same as equation (14), except that
Perhaps, a more straightforward way to address differential effects is to use an algorithm to change the reference group to
However, this algorithm does not address other nonlinear effects, such as the interaction effect in the mediator and intermediate confounder relationship. In this case, one should consider using the general bias formulas directly.
6.3 RVs
Despite the development of numerous sensitivity analyses, not many social or medical scientists have used sensitivity analysis to assess the robustness of their findings. Recent literature on sensitivity analysis [14,42] emphasized the use of standard way of reporting the robustness of research findings, which is expected to facilitate the discussions regarding how credible the estimated effect is to possible violations of no omitted confounding. For example, Ding and VanderWeele [42] advanced the E-value that reports the robustness of research findings measured in the risk ratio; Cinelli and Hazlett [14] advanced the RV that reports the robustness of research findings derived from linear regressions. In this section, we extend the RV computed for the treatment effect to disparity reduction and remaining.
We define the RV as the strength of association that will explain the estimated disparity reduction or remaining, assuming an equal association to the mediator and the outcome, as
where
Next, we define
6.4 Illustration using
R
2
-based sensitivity analysis
Section 5.3 presents sensitivity analysis using regression coefficients. This section presents the same sensitivity analysis, but it is parameterized in
Sensitivity contour plots using
The analysis indicates that the disparity reduction would still be negative (i.e., education significantly reduces the CVH gap between Black women and White men) if the unobserved confounder explains less than 10.7% of the variance of the mediator and the outcome after accounting for the existing confounders. The disparity reduction would still be significant at the 95% CI if the unobserved confounder explains less than 6.4% of the variance of the mediator and the outcome after accounting for the existing confounders.
Similarly, the disparity remaining would still be negative if the unobserved confounder explains less than 12.9% of the variance of the mediator and the outcome after accounting for the existing confounders, respectively. The disparity remaining would still be significant at the 95% CI if the unobserved confounder explains less than 5.2% of the variance of the mediator and the outcome after accounting for the existing confounders.
Although the qualitative classifications of effect size depend on the context of studies, we use Cohen’s [43] guideline to judge how large the amount of confounding is. The amount of confounding required to change the disparity reduction (10.7%) and remaining (12.9%) estimates from negative to positive is considered medium. The amount of confounding required to change the significance of the disparity reduction (6.4%) and remaining (5.2%) estimates is considered small. These results indicate that the conclusion regarding disparity reduction and remaining due to the mediator (education) could be changed with an unobserved confounder that has a small effect on the mediator and the outcome after controlling for the existing confounders.
7 Discussion
In this study, we developed a set of sensitivity analyses that assesses the sensitivity of disparity reduction and remaining to possible violations of no-unobserved mediator–outcome confounding. Although we used the example of estimating disparity reduction and remaining after intervening on a mediator, the proposed sensitivity analyses can be used in any settings based on interventional indirect and direct effects when the exposure is randomized (or conditionally ignorable given covariates).
Our study contributes to the fast-growing causal decomposition literature in several ways. First, we compared causal decomposition analysis based on interventional effects with causal mediation analysis based on natural effects. More importantly, we clarified that sensitivity analyses developed for natural indirect effects to possible pre-exposure confounding can only assess the sensitivity of disparity reduction when no intermediate confounding exists. We argued that the assumption (no intermediate confounding) is restrictive and may only be met when the mediator is measured shortly after the group status, for example, childhood poverty. Yet, such an assumption would be unrealistic in many studies that examine life-long health disparities between social groups.
Second, we derived general bias formulas for disparity reduction and remaining, which serve as the basis of our proposed sensitivity analyses. The general bias formulas can be used beyond a particular statistical model and are applicable to any variable type of mediator, outcome, or intermediate confounder and do not require any assumptions. Moreover, the same bias formulas apply with pre-exposure and intermediate unobserved confounding. Within the causal mediation framework based on natural indirect effects, it has long been desired by many researchers to address both pre-exposure and intermediate unobserved confounding [21,38,44]. Researchers can handle both types of confounding by utilizing the proposed sensitivity analyses, as the causal estimand is based on interventional direct and indirect effects. We acknowledge that not every situation is suitable for interventional direct and indirect effects. However, in some cases, such as the motivating example, defining interventional indirect effects makes more sense than natural indirect effects.
Third, we proposed a sensitivity analysis based on regression coefficients and
We also acknowledge the limitations of the proposed sensitivity analyses. First, the
Acknowledgments
We would like to thank the editor and reviewers for their comments and suggestions, which helped us to improve the quality of the manuscript.
-
Funding information: This research was supported by a grant from the American Educational Research Association which receives funds for its ‘AERA Grants Program’ from the National Science Foundation under NSF award NSF-DRL #1749275. Opinions reflect those of the author and do not necessarily reflect those AERA or NSF.
-
Conflict of interest: Authors state no conflict of interest.
-
Data availability statement: The data and code utilized for the current study are accessible in the personal GitHub repository at https://github.com/soojinpark33/Sensitivity-Analysis-for-CDA.
Appendix A General bias formulas of
δ
(
r
,
0
∣
r
)
and
ζ
(
0
∣
r
,
0
)
The bias for disparity reduction for
The second equality holds because of the law of total probability. The third equality is due to Bayes’ theorem.
The last expression of equation (A1) is the general bias formula for
B Sensitivity analysis using regression coefficients
Suppose that the following regression models for
This completes the proof.
C Bias formulas based on the coefficients of determination
This proof is a straightforward extension of Cinelli and Hazlett [14]. By using equation (11), the bias for disparity reduction is given by
where Cov and Cor represent the covariance and correlation, respectively. The third equality is because
Suppose that the partial
The first equality is derived from
D Standard errors of disparity reduction and remaining
We first calculate the standard error for disparity reduction. The standard errors for
The ratio of these standard errors is
because
Using the parametric identification result shown in Section 3.2, the standard error for disparity reduction for
where
Next, we calculate the standard error for disparity remaining. By using
where
E A simulation study
A Monte Carlo simulation study is conducted to investigate the performance of the proposed standard error estimators of disparity reduction and remaining given in equations (15) and (16). We consider varying sample sizes (100, 500, and 1,000) and effect sizes of sensitivity parameters, i.e., partial
For the simulation study, we first generate synthetic population data that have similar characteristics with the MIDUS data in terms of the distributions of variables
Simulation results: 95% CI coverage rates of the standard error estimators of disparity reduction and remaining
| Effect size | Sample size | Disparity reduction | Disparity remaining |
|---|---|---|---|
| Small | 100 | 0.916 | 0.931 |
| 500 | 0.934 | 0.942 | |
| 1,000 | 0.927 | 0.935 | |
| Medium | 100 | 0.921 | 0.930 |
| 500 | 0.946 | 0.940 | |
| 1,000 | 0.941 | 0.941 | |
| Large | 100 | 0.932 | 0.935 |
| 500 | 0.959 | 0.942 | |
| 1,000 | 0.951 | 0.951 |
The 95% CI coverage rate reaches the nominal level (0.95) with a sample size of 500 or larger, yet it reaches more quickly with medium and large effect sizes of sensitivity parameters. With a small effect size and a sample size of 100, the coverage is as low as 0.916 for disparity reduction. This result suggests that the coverage rate could be lower than expected with a small sample size (e.g., 100) and a small effect size of sensitivity parameters (e.g., 0.02).
F Standard errors for a percent reduction
As the variance of a ratio of two random variables with nonzero means can be approximated using the delta method and a first-order Taylor expansion, an uncertainty measure of a percent reduction can be derived in a similar way as
where an unbiased estimate of
G RVs for disparity reduction and remaining
We define the RV as the strength of association that will explain away the estimated disparity reduction (or remaining), assuming an equal association to the mediator and the outcome, as
Dividing both sides by
We define
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Articles in the same Issue
- Research Articles
- Adaptive normalization for IPW estimation
- Matched design for marginal causal effect on restricted mean survival time in observational studies
- Robust inference for matching under rolling enrollment
- Attributable fraction and related measures: Conceptual relations in the counterfactual framework
- Causality and independence in perfectly adapted dynamical systems
- Sensitivity analysis for causal decomposition analysis: Assessing robustness toward omitted variable bias
- Instrumental variable regression via kernel maximum moment loss
- Randomization-based, Bayesian inference of causal effects
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- Double machine learning and automated confounder selection: A cautionary tale
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- Efficient and flexible mediation analysis with time-varying mediators, treatments, and confounders
- Minimally capturing heterogeneous complier effect of endogenous treatment for any outcome variable
- Quantitative probing: Validating causal models with quantitative domain knowledge
- On the dimensional indeterminacy of one-wave factor analysis under causal effects
- Heterogeneous interventional effects with multiple mediators: Semiparametric and nonparametric approaches
- Exploiting neighborhood interference with low-order interactions under unit randomized design
- Robust variance estimation and inference for causal effect estimation
- Bounding the probabilities of benefit and harm through sensitivity parameters and proxies
- Potential outcome and decision theoretic foundations for statistical causality
- 2D score-based estimation of heterogeneous treatment effects
- Identification of in-sample positivity violations using regression trees: The PoRT algorithm
- Model-based regression adjustment with model-free covariates for network interference
- All models are wrong, but which are useful? Comparing parametric and nonparametric estimation of causal effects in finite samples
- Confidence in causal inference under structure uncertainty in linear causal models with equal variances
- Special Issue on Integration of observational studies with randomized trials - Part II
- Personalized decision making – A conceptual introduction
- Precise unbiased estimation in randomized experiments using auxiliary observational data
- Conditional average treatment effect estimation with marginally constrained models
- Testing for treatment effect twice using internal and external controls in clinical trials
