Verifiable identification condition for nonignorable nonresponse data with categorical instrumental variables

There has been a rapidly growing movement to utilize all the available data that may explicitly, even implicitly, contain missing values, such as causal inference (Imbens and Rubin, 2015) and data integration (Yang and Kim, 2020; Hu et al., 2022). For such datasets, appropriate analysis of missing data is indispensable to correct selection bias owing to the missingness. In recent years, analysis of missing data under missing at random (MAR) assumption (Little and Rubin, 2019) has gradually matured (Robins et al., 1994; Kim and Shao, 2021). Although model identifiability is one of the most fundamental conditions in constructing the asymptotic theory, removing the MAR assumption makes statistical inference drastically difficult, especially in model identification (Miao et al., 2016). Estimation with unidentifiable models may provide multiple solutions that have exactly the same model fitting. Several researchers have considered giving sufficient conditions for the model identification under missing not at random (MNAR).

Constructing observed likelihood consists of two distributions: (R) response mechanism and (O) outcome distribution (Kim and Shao, 2021). Miao et al. (2016) considered identification condition with Logistic, Probit, and Robit (cumulative distribution function of $t$-distribution) models for (R) and normal and $t$ (mixture) distributions for (O). Cui et al. (2017) assumed Logistic, Probit, and cLog-log models for (R) and the generalized linear models for (O). These studies depend heavily on the model specification of both (R) and (O). Wang et al. (2014) introduced a covariate called instrument or shadow variable and demonstrated that the use of the instrument could considerably relax conditions on (R) and (O). For example, (O) requires only the monotone-likelihood property, which includes a variety of models, such as the generalized linear model. Tang et al. (2003) and Miao and Tchetgen (2018) derived conditions for model identifiability without postulating any assumptions on (R) with the help of the instrument. Miao et al. (2019) further relaxed the assumption under an assumption referred to as the completeness condition on (R) (D’Haultfœuille, 2010, 2011). For example, the generalized linear model with continuous covariates satisfies the completeness condition. To the best of our knowledge, this combination of an instrument on (R) and completeness on (O) is the most general condition for model identification and has been accepted in numerous studies (Zhao and Ma, 2022; Yang et al., 2019).

Generally, assumptions on (O) rely on the distribution of the complete data, which is untestable from observed data. Recently, modeling (O’) the observed or respondents’ outcome model, instead of (O), has been used to relax the subjective assumption (Miao et al., 2019; Riddles et al., 2016). However, the observed likelihood with (R) and (O’) involves an integration that makes the identification problem intractable. Morikawa and Kim (2021) and Beppu et al. (2021) established that the integration can be computed explicitly with Logistic models for (R) and generalized linear models for (O’) and derived identification condition. For general response mechanisms and respondents’ outcome distributions, the model identification remains an open question. Furthermore, when the instrument is categorical such as smoking history and sex, the completeness condition is not available. For example, Ibrahim et al. (2001) considered a study on the mental health of children in Connecticut and used the parents’ report of the psychopathology of the child as the binary instrument.

In this paper, we consider an identification problem with an instrument for (R) and (O’) that satisfies the monotone-likelihood ratio property. Note that although our model setup is similar to Wang et al. (2014), we can check the validity of (O’) with observed data, for example, by using the information criteria such as AIC and BIC. Furthermore, we can use semiparametric/nonparametric methods for modeling both (O’) and (R).

The rest of this paper is organized as follows. Section 2 introduces the notation and defines model identifiability. Section 3 derives the proposed identification condition. We demonstrate the effects of identifiability via a limited numerical study in Section 4. Moreover, application to real data is presented in Section 5. Finally, concluding remarks are summarized in Section 6. All the technical proofs are relegated to the Appendix.

We present the effects of identifiability in numerical experiments by comparing weak and strong identifiable models. We prepared four Scenarios S1–S4:

• S1:

  (Outcome: Normal, Response: Logistic)
  $[y\mid u,z,\delta=1]\sim N(\kappa_{0}+\kappa_{1}u+\kappa_{2}z,\sigma^{2})$, $\text{logit}\{P(\delta=1\mid u,y;\bm{\alpha},\beta)\}=\alpha_{0}+\alpha_{1}u+\beta y$, $u\sim N(0,1^{2})$, and $z\sim B(1,0.5)$, where $(\kappa_{0},\kappa_{1},\sigma^{2})^{\top}=(0.3,0.4,1/{\sqrt{2}^{2}})^{\top}$ and $(\alpha_{0},\alpha_{1},\beta)^{\top}=(0.7,-0.2,0.29)^{\top}$.

• S2:

  (Outcome: Normal, Response: Cauchy)
  $[y\mid u,z,\delta=1]\sim N(\kappa_{0}+\kappa_{1}u+\kappa_{2}z,\sigma^{2})$, $P(\delta=1\mid u,y;\bm{\alpha},\beta)=\Psi(\alpha_{0}+\alpha_{1}u+\beta y)$, $u\sim\mathrm{Unif}(-1,1)$, and $z\sim B(1,0.7)$, where $(\kappa_{0},\kappa_{1},\sigma^{2})^{\top}=(-0.36,0.59,1/{\sqrt{2}^{2}})^{\top}$, $(\alpha_{0},\alpha_{1},\beta)^{\top}=(0.24,-0.1,0.42)^{\top}$, and $\Psi$ is the cumulative distribution function of the Cauchy distribution.

• S3:

  (Outcome: Bernoulli, Response: Probit)
  $[y\mid u,z,\delta=1]\sim B(1,p(u,z;\bm{\kappa})\})$, $P(\delta=1\mid u,y;\bm{\alpha},\beta)=\Psi(\alpha_{0}+\alpha_{1}u+\beta y)$, $u\sim N(0,1^{2})$, and $z\sim N(0,1^{2})$, where $p(u,z;\bm{\kappa})=1/\{1+\exp(-\kappa_{0}-\kappa_{1}u-\kappa_{2}z)$, $(\kappa_{0},\kappa_{1},\kappa_{2})^{\top}=(-0.21,3.8,1.0)^{\top}$, $(\alpha_{0},\alpha_{1},\beta)^{\top}=(0.4,0.39,0.3)^{\top}$, and $\Psi$ is the cumulative distribution function of the standard normal.

• S4:

  (Outcome: Normal+nonlinear mean structure, Response: Cauchy or Logistic)
  $[y\mid u,z,\delta=1]\sim N(\mu(\bm{x}),0.5^{2})$, $P(\delta=1\mid u,y;\bm{\alpha},\beta)=\Psi(\alpha_{0}+\alpha_{1}u+\beta y)$, $u\sim\mathrm{Unif}(-1,1)$, and $z\sim B(1,0.5)$, where $\mu(\bm{x})=z+\cos(2\pi u)+\exp(z+u)$, $(\alpha_{0},\alpha_{1},\beta)^{\top}=(0.1,-0.2,0.3)^{\top}$, and $\Psi$ is the cumulative distribution function of the Cauchy or Logistic distribution.

In S1 and S2, the strength of the identification can be adjusted by changing the parameter $\kappa_{2}$ because $\kappa_{2}=0$ indicates that the model is unidentifiable by Example 3.1. On the other hand, we can verify that the models in S3 and S4 are identifiable by Theorem 3.5. For example, in S4, we can see that checking (C3) and (C4) is straightforward to the setting, while (C5) and (C6) hold from Example 3.6 and Proposition 3.8, respectively. From S3 and S4, we can confirm the successful inference even in the case of discrete outcome and complex mean structures, respectively.

We generated 1,000 independent Monte Carlo samples and computed two estimators for $E[y]$ and $\beta$ with two methods: fractional imputation (FI) and complete case (CC) estimators, which use only completely observed data. The estimator for $E[y]$ is computed by the standard inverse probability weighting method with estimated response models (Riddles et al., 2016). We used correctly specified models for Scenarios S1–S3 but used nonparametric models for Scenario S4 because it is unrealistic to assume that the complicated mean structure is known. The R package ‘crs’ specialized in nonparametric spline regression on the mixture of categorical and continuous covariates (Nie and Racine, 2012) is used to estimate the respondents’ outcome model. Response models are estimated by using the method discussed in Section 2.2.

Bias, root mean squared error (RMSE), and coverage rate for 95% confidence intervals in S1–S4 are reported in Table 1. In all the Scenarios, CC estimators have a significant bias, and the coverage rates are far from 95%, while FI estimators work well when the model is surely identifiable. When $\kappa_{2}$ is small in S1 and S2, the performance of variance estimation with FI is poor, as expected, although that of point estimates is acceptable. The results in S4 indicate that the model is identifiable even if we use a nonparametric mean structure, and the estimates are almost the same between the two response models.

We analyzed a dataset of 2139 HIV-positive patients enrolled in AIDS Clinical Trials Group Study 175 (ACTG175; Hammer et al. (1996)). In this analysis, we specify 532 patients for analysis who received zidovudine (ZDV) monotherapy. Let each $y$, $x_{1}$, and $x_{2}$ be the CD4 cell count at $96\pm 5$ weeks, at the baseline, and at $20\pm 5$ weeks, $x_{3}$ be the CD8 cell count at the baseline, and $z$ be sex. The outcome was subject to missingness with a 60.34% observation rate, while all covariates were observed. To make estimation stable and easy, we standardized all the data. We expect that $z$ (sex) is a reasonable choice for an instrument variable because the information is a biological value, which affects the value of CD4, but has little effect on the response probability.

Patients who are suffering from a mild illness of HIV tend to have higher CD4 cell count; thus, one may consider that missingness of the outcome relates to serious conditions and may expect that the missing value of the outcome would be a lower CD4 cell count than the respondent. We therefore considered five different MNAR response models:

where $\Psi$ represents either the Logistic function or the distribution functions of the Cauchy or $t$ distribution with degrees of freedom $v\,(=2,5,10)$. Theorem 3.5 and Proposition 3.8 ensure that all the models with these five response models are identifiable, even when the instrumental variable $z$ is discrete. From the above conjecture on missing values, the sign of $\beta$ is expected to be negative. We assumed that the respondent’s outcome is a normal distribution with a nonparametric mean structure and estimated by the ‘crs’ R package as considered in Scenario S4 in Section 4. The residual plots shown in Figure 1 and the computed $R^{2}$-value $(=0.453)$ signify the assumed distribution on the respondents’ outcome fit well. Table 2 reports the estimated parameters and their estimated standard errors calculated by 1,000 bootstrap samples. The results of the five response models were almost similar. This suggests that the response mechanism is robust to the choice of response models. Although we cannot determine whether it is MNAR or MAR because the estimated standard error for $\beta$ is large, the point estimate is negative, as we expected. This result is consistent with the result in Zhao et al. (2021).

In this paper, we proposed a new identification condition for models using respondents’ outcome and response models. Although our method requires the specification of the two models, the model can be very general with the help of an instrument. As considered in Scenario S4 in Section 4, the mean function in the respondents’ outcome model can be nonparametric, and the response model can be any strictly monotone function, other than Logistic models. Our condition guarantees model identifiability even when instruments are categorical, which is not covered by previous conditions. Another advantage of using our method is the identification condition is easy to verify with observed data.

However, our method has some limitations. First, respondents’ outcome models need to have the monotone-likelihood property by Condition (C5). For example, we cannot deal with mixture models in our framework. Second, the specification of instruments is necessary in advance. To date, some studies on finding the instruments have been proposed (Zhao et al., 2021), but there are still no gold standard methods.

Research by the second author was supported by MEXT Project for Seismology toward Research Innovation with Data of Earthquake (STAR-E) Grant Number JPJ010217.