A Risk-Ratio-Based Marginal Sensitivity Model for Causal Effects in Observational Studies

Randomized controlled trials (RCTs) are considered to be the gold standard for estimating the causal effect of a treatment on an outcome of interest. However, researchers are often limited to investigating causal relationships using only observational data when conducting randomized experiments is not feasible. In the absence of randomized treatment assignments, the estimation of causal effects in observational studies is usually based on a set of identifiability assumptions. One of the most important such assumptions is the strong ignorability or the no unmeasured confounding (NUC) assumption, which implies that we can observe all relevant confounders in an observational study. Since the NUC assumption is essentially unverifiable using observed data, it is necessary to perform sensitivity analyses that assess the robustness of the causal conclusions obtained from observational studies in the presence of unmeasured confounding.

Sensitivity analysis is recognized as an essential step in the process of causal inference from observational studies. It refers to the approaches that investigate how causal conclusions are impacted in the presence of unmeasured confounding in observational studies. Bind & Rubin (2021), in their guidelines for the statistical reporting of observational studies, listed sensitivity analysis as one of the five major steps that need to be carried out before reporting causal conclusions elicited from observational studies. Many sponsors and regulatory bodies also nowadays require researchers to conduct some form of sensitivity analysis while drawing causal inferences from observational studies (for example, see standard MD-4 of the Patient-Centered Outcome Research Institute (PCORI) methodology standards for handling missing data at https://tinyurl.com/kcbkvfwv).

Cornfield et al. (1959) conducted the first formal sensitivity analysis to investigate the impact of unmeasured confounding on the causal relationship between smoking and lung cancer. However, this initial sensitivity analysis framework was applicable to only binary outcomes, and it did not account for the sampling variation. Rosenbaum and colleagues overcame these limitations and greatly expanded the theory and methods for sensitivity analysis in a series of pioneering works (Rosenbaum, 1987; Gastwirth et al., 1998; Rosenbaum et al., 2002). Recently, sensitivity analysis has gained a lot of research interest including Bonvini et al. (2022); Dorn et al. (2021); Carnegie et al. (2016); VanderWeele & Ding (2017); Zhao et al. (2019); Kallus et al. (2019); Yadlowsky et al. (2022), among others. Most of these proposed sensitivity analysis frameworks deal with only binary treatments.

Zhao et al. (2019) recently proposed a sensitivity analysis framework for smooth estimators of causal effects, such as the inverse probability weighting (IPW) estimators. Their proposed marginal sensitivity model is a natural modification of Rosenbaum’s sensitivity model (Rosenbaum, 2002) for matched observational studies, and it quantifies the magnitude of unmeasured confounding by the odds ratio between the conditional probability of being treated given the measured confounders (observed propensity score) and conditional probability of being treated given both the measured and unmeasured confounders (true propensity score). It is well known that risk ratios are more consistent with the general intuition than odds ratios, and hence, are easier to interpret. Therefore, it is desirable to develop sensitivity analysis frameworks that interpret the sensitivity analysis results using risk ratios instead of odds ratios.

In this paper, we first modified the odds-ratio-based sensitivity analysis framework of Zhao et al. (2019) to a risk-ratio-based framework. In particular, we proposed a modified marginal sensitivity model that measures the violation of the NUC assumption using the risk ratio between the true propensity score and the observed propensity score (see Section \arabicsection.\arabicsubsection for the exact definition). The proposed modified marginal sensitivity model introduces a new implicit sensitivity parameter that restricts the true propensity scores within its valid range for a given sensitivity model. We then extended the proposed modified marginal sensitivity model to the multivalued treatment setting based on the work of Basit et al. (2023). We showed that the estimation of causal effects for binary and multivalued treatments can be performed efficiently under the proposed sensitivity analysis framework.

The rest of the paper is structured as follows: Section \arabicsection describes necessary notations and briefly reviews the existing odds-ratio-based marginal sensitivity model; Section \arabicsection introduces the proposed modified marginal sensitivity model for the binary treatment setting; Section \arabicsection extends the proposed sensitivity model to multivalued treatment settings; Section \arabicsection reports results from simulation studies; Section \arabicsection demonstrates sensitivity analysis under the proposed framework using a real data application; Section \arabicsection provides some concluding remarks.

In this section, we extend our proposed sensitivity analysis framework to the multivalued treatment setting. Suppose we observe i.i.d. $(A_{i},\boldsymbol{X}_{i},Y_{i})_{i=1}^{n}$ from an observational study with $J>2$ treatment levels, where $A_{i}\in\mathcal{A}=\{1,2,\ldots,J\}$ with the corresponding set of potential outcomes $\mathcal{Y}_{i}=\{Y_{i}(1),Y_{i}(2),\ldots,Y_{i}(J)\}$, and $\boldsymbol{X}_{i}\in\mathscr{X}\subset\mathbb{R}^{d}$ is a vector of observed confounders for each subject $i\in[n]$. Let us also define a treatment indicator $D_{i}(a)$ ($1$ if $A_{i}=a$, $0$ otherwise) for $a\in\mathcal{A}$.

The identifiability assumptions for observational studies with multivalued treatments are almost equivalent to those in the binary treatment setting. We assume the overlap assumption that implies $\mathbb{P}_{0}(A=a|\boldsymbol{X}=\boldsymbol{x})>0$ for all $a\in\mathcal{A}$. However, instead of the strong ignorability assumption, we assume weak ignorability (Imbens, 2000).

[weak ignorability] $D(a)\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern 2.0mu{\displaystyle\perp}}}{\mathrel{\hbox to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptstyle\perp$\hss}\mkern 2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}Y(a)\thinspace|\thinspace\boldsymbol{X},\thickspace\forall\thinspace a\in\mathcal{A}.$

Imbens (2000) extended the concept of propensity scores from binary treatments to multivalued treatments introducing the generalized propensity score (GPS), which is defined as $r_{a,0}(\boldsymbol{x})=\mathbb{P}_{0}(A=a|\boldsymbol{X}=\boldsymbol{x})$ for $a\in\mathcal{A}$. As in Section \arabicsection, let us further denote the unobserved or true propensity score as $r_{a,0}(\boldsymbol{x},y)=\mathbb{P}_{0}(A=a|\boldsymbol{X}=\boldsymbol{x},Y(a)=y)$.

Based on these assumptions and notations, Basit et al. (2023) recently proposed a sensitivity analysis framework for the multivalued treatment setting extending the framework of Zhao et al. (2019) for binary treatments. Using similar ideas, we propose the following risk-ratio-based modified marginal sensitive model for observational studies with multivalued treatments.

Next, as in section \arabicsection.\arabicsubsection, let us define

and let $l_{a}(\boldsymbol{x},y)=k_{a,\beta_{0}}(\boldsymbol{x})-l_{a}(\boldsymbol{x},y)$ for all $a\in\mathcal{A}$ be the log-scale difference between the observed and the unobserved GPSs under any specified sensitivity model $r_{a}(\boldsymbol{x},y)$. Then, it can be shown that the sensitivity model (\arabicequation) is equivalent to assuming $l_{a}\in\mathcal{L}_{\beta_{0}}(\gamma_{0},\gamma_{1})$, where

$\gamma_{0}=\log(\Gamma_{0})$, and $\gamma_{1}=\log(\Gamma_{1})=\operatorname{max}\big{(}\log(r_{a,\beta_{0}}(\boldsymbol{x})),-\gamma_{0}\big{)}$ are the sensitivity parameters.

In order to define estimands of causal effects in the multivalued treatment setting, we use a general class of additive causal estimands proposed by Basit et al. (2023). This class of estimands is based on inverse probability weighting and is defined as

where $\boldsymbol{c}=(c_{1},\ldots,c_{J})^{\prime}$ is a vector of contrasts, $m(a)=\mathbb{E}_{0}[Y(a)]$ is the average potential outcome for $a\in\mathcal{A}$. These estimands encompass many commonly used causal estimands of interest for multivalued treatments, such as the pairwise average treatment effects (PATEs).

Based on the identifiability assumptions defined for multivalued treatments, we define the shifted estimand of the average potential outcome $m(a)$ under a specified sensitivity model $l_{a}$ as

where $r^{(l_{a})}(\boldsymbol{x},y)=\big{[}\exp\big{(}l_{a}(\boldsymbol{x},y)-k_{a,\beta_{0}}(\boldsymbol{x})\big{)}\big{]}^{-1}$ is the shifted GPS. Consequently, under the modified marginal sensitivity model (\arabicequation), the shifted causal estimand becomes

Similar to the estimation of the shifted ATE defined in Equation (\arabicequation) for binary treatments, we can show that the estimation of the causal estimand can be converted to a linear programming problem (LPP) that allows us to efficiently estimate the partially identified point estimate intervals of $\tau(\boldsymbol{c})$ for any $l_{a}\in\mathcal{L}_{\beta_{0}}(\gamma_{0},\gamma_{1})$. The percentile bootstrap approach of Zhao et al. (2019) is also applicable for the computation of $100(1-\alpha)\%$ asymptotic confidence intervals for $\tau(\boldsymbol{c})$ (Basit et al., 2023).

We conducted simulated studies to investigate the performance of the proposed sensitivity analysis framework in the multivalued treatment setting. Our data generating mechanism and simulation settings are equivalent to Basit et al. (2023). We simulate three covariates as $X_{i1}\sim\operatorname{Bernoulli}(0.5)$, $X_{i2}\sim\operatorname{U}(-1,1)$, and $X_{i3}\sim\operatorname{N}(0,0.5)$. For each $i\in[n]$, the covariate vector then becomes $\boldsymbol{X}_{i}=(1,X_{i1},X_{i2},X_{i3})^{T}$. We simulated the treatment assignment mechanism using the following multinomial distribution

where $D_{i}(a)$ is the treatment indicator and

denotes the complete or unobserved GPSs for $a\in\{1,2,3\}$ with $\beta_{1}=(0,0,0,0)^{T}$, $\beta_{2}=k_{2}\times(0,1,1,1)^{T}$, and $\beta_{3}=k_{3}\times(0,1,1,-1)^{T}$. In order to assess the influence of the degree of overlap on the point estimates and confidence intervals of our proposed frameworks, we considered two simulation scenarios. In the first scenario, we set $(k_{2},k_{3})=(0.1,-0.1)$ to simulate a scenario with adequate overlap in the covariates, and in the second scenario, we set $(k_{2},k_{3})=(3,3)$ to induce lack of overlap. The potential outcomes are generated from the following multinomial distribution

where $Y_{i}(a)$ is the potential outcome for treatment level $a$ and

with $\delta_{1}=(1,1,1,1)^{T}$, $\delta_{2}=(1,1,-1,1)^{T}$, and $\delta_{3}=(1,1,1,-1)^{T}$. The observed outcome was then obtained as $Y_{i}=\sum_{a=1}^{J}D_{i}(a)Y_{i}(a)$ for each subject $i\in[n]$. We simulated $1000$ datasets with sample size $n=750$ for each scenario and estimate the interval of point estimates and confidence intervals for the pairwise ATEs using our proposed sensitivity analysis framework. The pairwise ATEs are denoted by $\tau_{i,j}$, for ${i,j}\in\{1,2,3\}$. We considered six values of the sensitivity parameters $\gamma_{0}$, namely, $\gamma_{0}=\{0,0.1,0.2,0.5,1,2\}$. The true partially identified intervals were obtained under each simulation scenario using large scale numerical approximations. Since the treatment under consideration is multivalued with three treatment levels, the observed generalized propensity scores (GPSs) are modeled using the multinomial logit regression model.