Causal Inference with Unmeasured Confounding from Nonignorable Missing Outcomes

In this paper, we consider the situation where $\left\{\left(y_{i},r_{i},a_{i},z_{i}\right):i=1,\ldots,n\right\}$ is an independent and identically distributed sample from $\left(Y,R,A,Z\right)$. Vectors are assumed to be column vectors, unless explicitly transposed.

$Y$ is the outcome of interest subject to missingness. We consider the outcome $Y$ a binary variable. We let $R$ denote the missing indicator of $Y$: $R=1$ if Y is observed and $R=0$ otherwise. The observed data include $\left(A,Z\right)$ for all samples and $Y$ only for those $R=1$. We use lower-case letters for realized values of the corresponding variables, for example, $y$ for a value of the outcome variable $Y$. We use $f$ to denote a probability density or mass function. Suppose the observed data are $n$ independent and identically distributed samples.

We use potential outcome model to define causal effects (Neyman,, 1923; Rubin,, 1974). For each unit $i$, We consider $Z$ is a binary instrument variable, where $Z_{i}=1$ indicates the unit $i$ is assigned to the treatment group and $Z_{i}=0$ indicates the unit $i$ is assigned to the control group. Let $A_{i}(z)$ denote the potential outcome of the unit $i$ that the instrument variable $Z$ was set to level $z$. $A_{i}(z)=1$ indicates that unit $i$ would receive the treatment if assigned $z$, and $A_{i}(z)=0$ indicates that unit $i$ would receive the control if assigned $z$. Similar to the definition of $A_{i}(z)$, we let $Y_{i}(z,A_{i}(z))$ denote the potential outcome for an unit $i$ if exposed to treatment $A_{i}(z)$ after $Z$ was set to level $z$. For simplicity, we let $Y_{i}(z)$ and $R_{i}(z)$ denote the potential outcomes after $Z$ was set to level $z$, respectively. Let $Z_{i}$, $A_{i}$, $Y_{i}$, $R_{i}$ denote their observations for $i=1,\dots,n$.

The following assumption is standard in causal inference with observational studies (Rubin,, 1980; Angrist et al.,, 1996).

Under SUTVA assumption, the definition of the potential outcome is reasonable. But when $Z_{i}\neq A_{i}$, there exists noncompliance. Under the principal stratification framework (Angrist et al.,, 1996; Frangakis and Rubin,, 2002), we define $U_{i}$ as the compliance status variable of unit $i$:

Because $A_{i}(1)$ and $A_{i}(0)$ each can take two values, the compliance status variable $U_{i}$ has four different values, $nt$ for never takers, $at$ for always takers, $cp$ for compliers, and $df$ for defiers. Because we can not observe $A_{i}(1)$ and $A_{i}(0)$ jointly, the compliance behavior of an unit is unknown, so $U_{i}$ is an observed variable and it can be viewed as an unmeasured confounder.

Because for the compliers, $Z_{i}=A_{i}$, so CACE is a subgroup causal effect for the compliers, with incompletely observed compliance status.

Randomization means that $Z\perp\!\!\!\perp(Y(0),Y(1),A(0),A(1))$.

Monotonicity assumption means there is no defiers in the population. In some studies, the monotonicity assumption is plausible when the treatment assignment has a nonnegative effect on the treatment received for each unit.

Assumption 5 means that the instrument variable $Z$ only affects the outcome through treatment and has no direct effect.

If the outcome $Y$ can be fully observed, CACE can be identified and estimated by Angrist et al., (1996). With the nonignorable missing outcome, some assumptions about the missing data mechanisms are needed to estimate CACE. We will compare some assumptions about the missing data mechanisms in the next subsection.

Before illustrating the missing data mechanisms, Assumption 2 and Assumption 5 are replaced by Assumption 6 and Assumption 7 in some previous work, respectivitly. Assumption 6 and Assumption 7 are usually combined with the missing outcomes mechanisms in the previous work. However, we only need Assumption 2 and Assumption 5 to estimate CACE in this paper, which are the weak version of Assumption 6 and Assumption 7.

Complete Randomization means that $Z\perp\!\!\!\perp\{A(1),A(0),Y(1),Y(0),R(1),R(0)\}$.

Different from traditional Assumption 5, Frangakis and Rubin, (1999) extended it to the compound exclusion restrictions. Assumption 7 is stronger than Assumption 5. Under Assumption 6, Assumption 7 is equivalent to $f(Y,R\mid Z=1,U=nt)=f(Y,R\mid Z=0,U=nt)$ and $f(Y,R\mid Z=1,U=at)=f(Y,R\mid Z=$ $0,U=at)$.

In previous work, some authors proposed the Latent ignorability (LI) assumption (Frangakis and Rubin,, 1999; Osius,, 2004; Zhou and Li,, 2006):

Latent ignorability implies that for given the each principal stratum, the potential outcomes are independent of the missing indicator, which means the missing data mechanism does not depend on the missing outcome. We gave a graph model to illustrate the LI missing data mechanism.

Chen et al., (2009) and Chen et al., (2015) proposed an Outcome-dependent nonignorable (ODN) missing data mechanism:

Under the completely randomization Assumption 6, Assumption 9 becomes $P(R=1\mid Y=y,A=a,U=u,Z=z)=P(R=1\mid Y=y,Z=z)$ and $P(R=1\mid Y=y,Z=1)=P(R=1\mid Y=y,Z=0)$. Outcome-dependent nonignorable assumption means that $R$ depends on $Y$, but is independent of $(Z,A,U)$ given $Y$. Under the ODN missing data mechanism, the missing data indicator depends on the possibly missing outcome Y, which may be more reasonable than LI missing data assumption in some applications. We gave a graph model to illustrate ODN missing data mechanism.

More examples and details about LI assumption and ODN assumption can be found in Chen et al., (2009) and Chen et al., (2015).

In this paper, we are interested in CACE with unmeasured confounding $U$ from the nonignorable missing outcomes $Y$.

In fact, LI assumption and ODN assumption usually need the Assumption 6 and Assumption 7, which are stronger than Assumption 2 and Assumption 5. In this paper, we only need the weak randomization Assumption 2 and exclusion restriction Assumption 5 to estimate CACE. Different from the above two missing outcomes mechanisms, we adapt the shadow variable assumption to guarantee the identification under the nonignorable missing outcomes in this paper.

We suppose that an auxiliary variable $Z$ called a shadow variable is fully observed if it statisfies the following assumption (Miao and Tchetgen Tchetgen,, 2016; Miao et al.,, 2019).

Assumption 10 implies that the shadow variable is associated with the outcome when the outcome is observed, but it is independent of the missing mechanism conditional on fully observed variables and possibly unobserved outcome (d’Haultfoeuille,, 2010; Kott,, 2014; Wang et al.,, 2014; Zhao and Shao,, 2015). Therefore, Assumption 10 allows for the data missing not at random.

In this paper, we suppose an auxiliary variable $Z$ satisfies both Assumptions 1-5 and Assumption 10, which plays two roles of the instrument variable and the shadow variable simultaneously. Different from the ODN missing data mechanism 2, we allow the arrow from $A$ to $R$ in Figure 3, which means the treatment would affect the missing indicator. It is more reasonable in some applications. We present a example to help readers to understand the framework of Fiugre 3. Figure 3 illustrates the framework of this paper.

In the next section, we state that the key of identification of $f(y,r\mid a,z)$ is Assumption 10. If Assumption 10 is violated, which means the shadow variable is not available, the identification of $f(y,r\mid a,z)$ is not guaranteed, even if the parametric missingness mechanisms may not be identifiable (Miao et al.,, 2016; Wang et al.,, 2014). We also illustrate extra conditions to guarantee the identification.

We consider the situation where $Z$ is an auxiliary variable taking values of $l=0,1$. The first step of the estimation is to recover the mean value of the outcome which is missing not at random. We want to estimate the population mean $\mu=E\left[Y\right]$ from the observed data. After the identification is guaranteed, the joint distribution of $Y$ and $R$ given $\left(A,Z\right)$ is determined by $f(y\mid a,z)$ and the missing mechanism $f(r\mid y,a,z)$.

The missing mechanism $\pi(y,a,z)=f(r=1\mid y,a,z)$ is the key in the process of the estimation. The conditional probability $\pi(y,a,z)$ is also called the nonresponse mechanism or the propensity of missing data in some literatures (Wang et al.,, 2014; Shao and Wang,, 2016; Zhang et al.,, 2018). For the outcome $Y$ missing not at random, the propensity depends on both observed data and missing data. Some authors imposed some parametric assumptions on both propensity and outcome model to establish some likelihood methods (Greenlees et al.,, 1982; Baker and Laird,, 1988).But it is sensitive for the estimators under the fully parametric models. Some authors proposed some semiparametric approaches. Qin et al., (2002) imposed a parametric model on the propensity , which is difficult to verify under nonignorable missingness. Some authors considerd some weak assumption to the model. Kim and Yu, (2011) proposed a semiparametric logistic regression model for the propensity . Tang et al., (2003) proposed a pseudo-likelihood method, imposing a parametric model for propensity but an unspecified propensity . Zhao and Shao, (2015) consider a generalized linear model for the estimation allowing for a nonparametric missing mechanism. Some authors proposed an empirical likelihood method to estimate the parameters in the missing mechanism with nonignorable missing data (Zhao et al.,, 2013; Tang et al.,, 2014; Niu et al.,, 2014).

Wang et al., (2014) applied the generalized method of moments to estimate the parameters of the missing mechanism. Shao and Wang, (2016) imposed an exponential tilting model on the propensity and estimated the tilting parameter and population mean in two steps. Zhang et al., (2018) proposed an approach that the parameters of interest and the tilting parameter can be estimated simultaneously with generalized method of moments and kernel regression.

In this paper, we apply the generalized method of moments (GMM) to estimate the parameters of the missing mechanism(Hansen,, 1982; Hall,, 2005). For estimation, we consider a parametric model for the propensity $\pi(y,a,z)$ and we don’t impose any parametric assumption on the outcome model $f(y\mid a,z)$, which means the $f(y\mid a,z)$ is nonparametric.

Here we assume the propensity $\pi(y,a,z)=f(r=1\mid y,a,z)$ satisfies the following conditions:

Similar to the discussion of Wang et al., (2014) , we set a parametric propensity in which the nonresponse instrument $Z$ does not appear. However once the identification is guaranteed in Section 3, the Condition (C2) in Wang et al., (2014) and Zhang et al., (2018) for the identiability conditions is not essential. The next step is to estimate these parameters using the observed data.

For applying the GMM, we need to construct a set of $q$ estimating equations:

where $\Theta$ is the parameter space containing the true parameter value $\bm{\theta}_{0}=\left(\alpha_{0},\beta_{0},\gamma_{0}\right)^{\mathrm{T}}$, $\bm{\theta}$ is a $p$-dimensional vector of the parameters that we want to estimate. The estimating equations need to satisfy $E\left[g_{m}\left(y,r,a,z,\bm{\theta}_{0})\right)\right]=0$,  $m=1,\cdots,q$ and $q\geq p$. Let

$\widehat{G}_{n}(\bm{\theta})$ is the sample version of the estimating equations (4.3). The number of estimating equations $q$ is often greater than $p$, which implies that there is no solution to

The best we can do is to make it as close as possible to zero by minimizing the quadratic function

where $W$ is a positive semi-definite and symmetric $q\times q$ matrix of weights. We use two-step GMM to estimate the parameters $\bm{\theta}$.

The first step: Let $W=I_{q\times q}$ in (4.6), $I_{q\times q}$ is a $q\times q$ identity matrix, then we obtain $\tilde{\bm{\theta}}$ by minimizing ${Q}_{n}(\bm{\theta})=[\widehat{G}_{n}(\bm{\theta})]^{\mathrm{T}}[\widehat{G}_{n}(\bm{\theta})]$, which means $\tilde{\bm{\theta}}=\mathop{\arg\min}\limits_{\bm{\theta}\in\Theta}Q_{n}(\bm{\theta})$.

The second step: Then we obtain $\widehat{W}=W(\tilde{\bm{\theta}})$, plugging in $\widehat{W}$ in (4.6). Finally, we obtain the GMM estimator $\widehat{\bm{\theta}}$ by minimizing $\widehat{Q}_{n}(\bm{\theta})=[\widehat{G}_{n}(\bm{\theta})]^{\mathrm{T}}\widehat{W}[\widehat{G}_{n}(\bm{\theta})]$ over $\bm{\theta}\in\Theta$, which means $\widehat{\bm{\theta}}=\mathop{\arg\min}\limits_{\bm{\theta}\in\Theta}\widehat{Q}_{n}(\bm{\theta})$.

The next step is to construct the estimating equations to estimate $\bm{\theta}=\left(\alpha,\beta,\gamma\right)^{\mathrm{T}}$. $Z$ is a discrete variable taking values of $l=0,1$. We let $\bm{k}$ be a $L$-dimensional column vector whose $l$-th component is $I(z=l)$ $(L=2)$, where $I(\cdot)$ is the indicator function. Similar to $\bm{k}$, we let $\bm{j}$ also be a $L$-dimensional column vector whose $l$-th component is $I(a=l)$ .

We now consider the general situation. The estimating equations in (4.3) can be constructed by the following $q$ functions

where $\pi(y,a,z,\bm{\theta})=f(r=1\mid y,a,z)=f(r=1\mid y,a)=\psi\left(\alpha+\beta y+\gamma a\right)$ in (4.2).

For example, if $z$ is a constant and there is no other auxiliary variables, and (4.7) is insufficient for estimating unknown $\bm{\theta}=\left(\alpha,\beta,\gamma\right)^{\mathrm{T}}$. For using the GMM, we need $q\geq p$. If there is only a discrete variable $Z$, the requirement $L\geq 2$ is satisfied if $Z$ is not a constant.

The estimating functions $G(y,r,a,z,\bm{\theta})$ are motivated by the following equations, if $\bm{\theta}_{0}=(\alpha_{0},\beta_{0},\gamma_{0})^{\mathrm{T}}$ is the true parameter value,

where $\bm{\eta}=\left(\bm{k}^{\mathrm{T}},\bm{j}^{\mathrm{T}}\right)^{\mathrm{T}}$ is a $q$-dimensional vector. Let $G(y,r,a,z,\bm{\theta})$ in (4.7) be the estimating equations in (4.3), $\widehat{G}_{n}(\bm{\theta})$ in (4.4) is the sample version. $\widehat{W}$ is given by the two-step method on the above. Then we will obtain $\widehat{\bm{\theta}}=\left(\hat{\alpha},\hat{\beta},\hat{\gamma}\right)^{\mathrm{T}}$ as the two-step GMM estimator of $\bm{\theta}=\left(\alpha,\beta,\gamma\right)^{\mathrm{T}}$.

The proof of the Theorem 2 and more details about GMM can be found in the Supplementary Materials. Let $\nabla_{\bm{\theta}}(\cdot)$ and $\nabla_{{\bm{\theta}}{\bm{\theta}}}(\cdot)$ denote the first- and second-order derivatives with respect to $\bm{\theta}$. The asymptotic normality of $\widehat{\bm{\theta}}$ is established by Theorem 3 under the following additional Condition 3.

The proof of Theorem 3 can be found in the Supplementary Materials. For asymptotic covariance variance estimation $\widehat{\Delta}$, which can be estimated by substituting estimators for each of $H$, $W$ and $\Omega$. To estimate $\Omega$, we can replace the population moment by a sample average and the true parameter $\bm{\theta}_{0}$ by an estimator $\widehat{\bm{\theta}}$. $\widehat{H}$ is similar to $\widehat{\Omega}$, respectively. To form

where $G_{n}(y_{i},r_{i},a_{i},z_{i},\bm{\theta})=\left(g_{1}\left(y_{i},r_{i},a_{i},z_{i},\bm{\theta}\right),\cdots,g_{q}\left(y_{i},r_{i},a_{i},z_{i},\bm{\theta}\right)\right)^{\mathrm{T}},\quad i=1,\cdots,n,\quad\bm{\theta}\in\Theta$. So that we have the Theorem 4.

The proof of Theorem 4 can be found in the Supplementary Materials. The optimal weight matrix $W_{1}=\Omega^{-1}$, with this choice of $W_{1}$, $\widehat{W}$ can be constructed by the following,

the asymptotic covariance matrix $\Delta$ reduces to $\left(H^{\mathrm{T}}\Omega H\right)^{-1}$ , and the asymptotic covariance variance estimation $\widehat{\Delta}$ reduces $\left(\widehat{H}^{\mathrm{T}}\widehat{\Omega}\widehat{H}\right)^{-1}$.

Once $\widehat{\bm{\theta}}$ is obtained, the mean value of $\mu=E\left[Y\right]$ can be constructed by the inverse probability weighting with the estimated propensity as the weight function,

Actually, if the dimension of the estimating equations $q$ is more than the dimension of the parameters $p$, an additional estimating equation $g_{q+1}=\mu-ry/\pi(y,a,z,\bm{\theta})$ can be added into the estimating equations $G(y,r,a,z,\bm{\theta})$ in (4.7). Then $G(y,r,a,z,\bm{\theta})$ is a set of $q+1=2L+1$ estimating equations , and $\bm{\theta}=\left(\mu,\alpha,\beta,\gamma\right)^{\mathrm{T}}$ is a set of $p+1$ unknown parameters. If $\tilde{\mu}$ is the GMM estimator of the above estimating equations $q+1=2L+1$ by the two-step method, as noted by Wang et al., (2014), the difference between the estimator $\widehat{\mu}$ and $\tilde{\mu}$ is the weighting matrix $\widehat{W}$, the weighting matrix of the esimator $\widehat{\mu}$ is not the optimal and $\tilde{\mu}$ is asymptotically more efficient unless the weighting matrix of the esimator $\widehat{\mu}$ is optimal. But in this paper, we propose $\widehat{\mu}$ to construct the estimator of CACE. The functional delta method can be used to establish the asymptotic results for $\widehat{\mu}$, for simplicity, we omit this part.

Under Assumptions 1-5, we can estimate CACE for the subpopulation of compliers characterized by $U_{i}=cp$, by taking the ratio of the average difference in $Y_{i}$ by instrument and the average difference in $A_{i}$ by instrument (Angrist et al.,, 1996):

If the outcome $Y$ is fully observed, CACE can be estimated by equation (4.10). So, the missing data problem is equivalent to the problem of estimating the mean difference $E(Y_{i}\mid Z_{i}=1)$ and $E(Y_{i}\mid Z_{i}=0)$ from incomplete outcome data.

In equation (4.10), the conditional expectation of the nonignorable missing outcomes $Y$, $E\left[Y_{i}\mid Z_{i}=l\right],l=0,1$ can be estimated by equation (4.9) with the stratification method by the instrument $Z$ ; $E\left[A_{i}\mid Z_{i}=l\right],l=0,1$ can be estimated with the fully observed data. Then the estimator $\widehat{CACE}$ can be obtained by equation (4.9) and (4.10).