Doubly Robust Estimates for Binary Longitudinal Data Analysis with Missing Response and Missing Covariates

Summary

Longitudinal studies often feature incomplete response and covariate data. Likelihood-based methods such as the expectation–maximization algorithm give consistent estimators for model parameters when data are missing at random (MAR) provided that the response model and the missing covariate model are correctly specified; however, we do not need to specify the missing data mechanism. An alternative method is the weighted estimating equation, which gives consistent estimators if the missing data and response models are correctly specified; however, we do not need to specify the distribution of the covariates that have missing values. In this article, we develop a doubly robust estimation method for longitudinal data with missing response and missing covariate when data are MAR. This method is appealing in that it can provide consistent estimators if either the missing data model or the missing covariate model is correctly specified. Simulation studies demonstrate that this method performs well in a variety of situations.

1. Introduction

Incomplete longitudinal data often arise in comparative studies because of difficulties in ascertaining responses at scheduled assessment times, partially completed forms or questionnaires, patients’ refusal to undergo complete examinations, or study subjects failing to attend a scheduled clinic visit. For example, in the Alzheimer’s disease (AD) study introduced in Section 5, at each clinic visit, a patient may have an incomplete response (clinical diagnosis of AD) because the patient failed to attend the clinic, or the patient refused to undergo a complete examination; furthermore, the patient may have other information (covariates) missing because, for example, the patient did not finish the questionnaires or forms completely. Analyses based only on individuals with complete data can lead to invalid inferences if the mechanism leading to the missing data is dependent on the response. Under a missing completely at random (MCAR) mechanism (Little and Rubin, 2002), analyses based on generalized estimating equations (GEE; Liang and Zeger, 1986) yield consistent estimators of the regression parameters. However, when data are missing at random (MAR) or missing not at random (MNAR; Little and Rubin, 2002), analyses based on GEE generally give inconsistent estimators. Robins, Rotnitzky, and Zhao (1995) developed a class of inverse probability weighted generalized estimating equations (IPWGEE), which can yield consistent estimators when data are MAR. The weights are obtained from models for the missing data process, and these models must be correctly specified for the resulting estimators to be consistent. Alternatively, one can use the maximum likelihood method to estimate the parameters, and it gives a consistent estimator if the model is correctly specified, and the parameters are identified.

The literature on methods for missing data has primarily addressed either missing response or missing covariate data (see, e.g., Zhao, Lipsitz, and Lew, 1996; Horton and Laird, 1998; Lipsitz, Ibrahim, and Zhao, 1999; Fitzmaurice et al., 2001; Ibrahim, Lipsitz, and Horton, 2001). In practice, of course, data are often unavailable for both responses and covariates. In this situation, multiple imputation has been used under some parametric model assumptions (e.g., Schafer, 1997; Little and Rubin, 2002; van Buuren, 2007); Chen et al. (2008) provide a careful investigation of likelihood methods for missing response and covariate data via the EM algorithm; Shardell and Miller (2008) propose a marginal modeling approach to estimate the association between a time-dependent covariate and an outcome in longitudinal studies with missing response and missing covariate, but they focus on methods with an assumption that responses are independent; and Chen, Yi, and Cook (2010) develop a marginal method for a general working correlation matrix with missing response and missing covariates.

For the IPWGEE, to obtain a consistent estimator we need to correctly model the missing data process and also need to correctly model the response process given the covariates, but we do not need to specify the distribution of the missing covariates. If the missing data process model is misspecified, it can give biased estimators. That means, the IPWGEE method is sensitive to the misspecification of the missing data model but robust to the misspecification of the covariate process model. For the maximum likelihood method, we do not need to specify the missing data model when data are MAR, but we must correctly specify the joint distribution of the response and the covariates that have missing values. If the distribution of the covariates is misspecified, the maximum likelihood can give inconsistent estimators. That is to say, the maximum likelihood method is sensitive to the misspecification of the covariate model but robust to the misspecification of the missing data model when data are MAR.

A hybrid approach is the doubly robust estimator introduced by Lipsitz et al. (1999), in which they only considered cross-sectional studies with a missing covariate. This is an estimating equation approach with properties similar to the maximum likelihood. To obtain a consistent estimator of the regression parameters, either the missing data model or the distribution of the missing data given the observed data must be correctly specified, which is arguably more robust to the IPWGEE and the maximum likelihood method. The literature for the doubly robust estimator for the cross-sectional study includes Lunceford and Davidian (2004); Carpenter, Kenward, and Vansteelandt (2006); Davidian, Tsiatis, and Leon (2005); and Kang and Schafer (2007), etc. For longitudinal data, Van der Laan and Robins (2003) gave a general theory for the doubly robust estimator with monotone incomplete response; Bang and Robins (2005) showed how to perform doubly robust inference on longitudinal data with dropout; Seaman and Copas (2009) developed a doubly robust estimating equation on longitudinal data with dropout; the other literature includes Robins and Rotnitzky (2001) and Scharfstein, Rotnitzky, and Robins (1999). This literature, however, focuses primarily on monotone missing data patterns; Vansteelandt, Rotnitzky, and Robins (2007) developed regression models for the mean of repeated outcomes under nonignorable nonmonotone nonresponse, but they focus on the problem of missing response when conducting inference about the marginal mean and the conditional mean given the baseline observed covariates. Little work has been devoted to the longitudinal studies with both missing response and missing covariates. This is a challenge because, in most cases, the missingness patterns are not monotone, and the literature methods cannot be implemented. Furthermore, because of the incomplete covariates, the general weighted estimating equations do not work (as we show in Section 3.1), and new estimating equations need to be developed. In this article, we develop a new estimating equation method for binary longitudinal data with both missing response and missing covariates. This approach is appealing in that it can not only deal with the missing response and missing covariate problem with an intermittently missing data pattern but also yields the doubly robust and optimal efficient estimator.

The remainder of this article is organized as follows. In Section 2, we introduce notation and models. In Section 3, we give the forms of the estimating equations and provide details on estimation and inference. Simulation studies are given in Section 4. Data arising from an AD are analyzed in the application in Section 5. Concluding remarks are made in Section 6.

2. Notation and Models

2.1 Response Process

Suppose that n individuals are to be observed, with J_i repeated measurements for subject i, i = 1, …, n. Let Y_i = (Y_i1, Y_i2, …, Y_iJi)^T denote the J_i × 1 binary response vector for subject i that may be missing at some time points. Let X_i = (X_i1, X_i2, …, X _iJi)^T be the covariate vector that may be missing and ${\mathbf{Z}}_i={({\mathbf{Z}}_{i1}^T,{\mathbf{Z}}_{i2}^T,\dots ,{\mathbf{Z}}_{{iJ}_i}^T)}^T$ be the covariate vector that is always observed, where Z_ij is the covariate vector for subject i at time j.

Define μ_ij = E (Y_ij | X_i, Z_i) = P(Y_ij = 1 | X_i, Z_i), and let μ_i = (μ_i1, μ_i2, …, μ_iJi)^T. Provided that the mean structure of Y_ij depends on the covariate vector for subject i at time j (Pepe and Anderson, 1994; Robins, Greenland, and Hu, 1999), we may consider the models for the mean of the form

$$g({\mu }_{ij})=X_{ij}{\beta }_x+{\mathbf{Z}}_{ij}^T{\beta }_z,$$

for j = 1, …, J_i, i = 1, …, n, where g(·) is a known monotone link function, and $\mathit{\beta }={({\beta }_x,{\beta }_z^T)}^T$ is a vector of regression parameters. Here we suppose only one covariate X_ij is potentially missing. Comments on how to deal with the problem when multiple covariates may be missing are given in the discussion. The variance for the response Y_ij is specified as v_ij = Var (Y_ij | X_i, Z_i) = μ_ij (1 − μ_ij), which depends on the regression parameter vector β.

Let $Y_{ij}^{\ast }=(Y_{ij}-{\mu }_{ij})/\sqrt{v_{ij}},{\rho }_{\mathit{ijk}}=E(Y_{ij}^{\ast }{Y}_{ik}^{\ast }),{\rho }_{{ij}_1{j}_2\cdots j_K}=E(Y_{{ij}_1}^{\ast }{Y}_{{ij}_2}^{\ast }\cdots Y_{{ij}_K}^{\ast })$ be the Kth-order correlation among components Y_ij1, Y_ij2, …, Y_ijK of Y_i, and ρ denote all the correlation parameters. For given subject i, the joint probability for a response vector Y_i can be expressed via the Bahadur representation (Bahadur, 1961), which is given by

$$P({\mathbf{Y}}_i={\mathbf{y}}_i\mid {\mathbf{X}}_i,{\mathbf{Z}}_i)=\prod _{j=1}^{J_i}\{{\mu }_{ij}^{y_{ij}}{(1-{\mu }_{ij})}^{1-y_{ij}}\}·\left\{1+\sum _{j<k}{\rho }_{\mathit{ijk}}{y}_{ij}^{\ast }{y}_{ik}^{\ast }+\sum _{j<k<l}{\rho }_{\mathit{ijk}l}{y}_{ij}^{\ast }{y}_{ik}^{\ast }{y}_{il}^{\ast }+\cdots +{\rho }_{i1\cdots J_i}{y}_{i1}^{\ast }\cdots y_{{ij}_i}^{\ast }\right\},$$ (1)

where y_i is a realization of Y_i, and $y_{ij}^{\ast }$ is a realization of $Y_{ij}^{\ast }$. This joint density requires modeling the correlation structures of all orders. In practice, it is often the case that the second order dominates the association structure while the third- and higher-order association is null or nearly null. Under such a circumstance, then the joint density is given by

$$P({\mathbf{Y}}_i={\mathbf{y}}_i\mid {\mathbf{X}}_i,{\mathbf{Z}}_i)=\prod _{j=1}^{J_i}\{{\mu }_{ij}^{y_{ij}}{(1-{\mu }_{ij})}^{1-y_{ij}}\}·\left\{1+\sum _{j<k}{\rho }_{\mathit{ijk}}{y}_{ij}^{\ast }{y}_{ik}^{\ast }\right\}.$$ (2)

In the following, we assume that the third- and higher-order association for Y_i is null, and let C_i (ρ) denote the correlation matrix of Y_i.

2.2 Missing Data Process

To indicate the availability of data we let R_ij = 0 if Y_ij and X_ij are missing, R_ij = 1 if Y_ij is missing and X_ij is observed, R_ij = 2 if Y_ij is observed and X_ij is missing, and R_ij = 3 if Y_ij and X_ij are observed. Let R_i = (R_i1, R_i2, …, R_iJi)^T, and R̄_{i j} = {R_i1, …, R_{i, j −1}}.

Instead of modeling the joint probability P (R_i = r_i | Y_i, X_i, Z_i) for R_i directly, because we are focusing on the longitudinal setting we restrict attention to conditional models of the form P (R_{i j} = r_{i j} | R̄_{i j}, Y_i, X_i, Z_i), which reflect the dynamic nature of the observation process over time; we can then obtain P (R_i = r_i | Y_i, X_i, Z_i) through $\prod _{j=2}^{J_i}P(R_{ij}=r_{ij}\mid {\overline{\mathbf{R}}}_{ij},{\mathbf{Y}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i)·P(R_{i1}=r_{i1}\mid {\mathbf{Y}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i)$. Let λ_{i j k} = P (R_{i j} = k | R̄_{i j}, Y_i, X_i, Z_i) denote the conditional probability, where k = 0, 1, 2, 3. We write these probabilities as conditional on the previous missing data indicators for the response and covariate, as well as the full vector of responses and covariates. The formulation thus far encompasses MCAR, MAR, and MNAR mechanisms because we have written the missing data model at assessment j as depending on the full vector of responses Y_i and covariates X_i. For MAR mechanisms we require

$$P({\mathbf{R}}_i={\mathbf{r}}_i\mid {\mathbf{Y}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i)=P({\mathbf{R}}_i={\mathbf{r}}_i\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i)$$ (3)

where ${\mathbf{Y}}_i^o$ and ${\mathbf{X}}_i^o$ represent the observed components of Y_i and X_i, respectively. However, in the longitudinal setting with our conditional formulation it is very natural to make the further assumption that

$$P(R_{ij}=r_{ij}\mid {\overline{\mathbf{R}}}_{ij},{\mathbf{Y}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i)=P(R_{ij}=r_{ij}\mid {\overline{\mathbf{R}}}_{ij},{\overline{\mathbf{Y}}}_{ij}^o,{\overline{\mathbf{X}}}_{ij}^o,{\mathbf{Z}}_i),$$ (4)

for each time point j, where ${\overline{\mathbf{Y}}}_{ij}^o$ and ${\overline{\mathbf{X}}}_{ij}^o$ denote the history of observed responses and covariates until time j − 1. It can be seen that (4) implies (3), but not vice versa. Moreover, while mechanism (3) covers a larger class of MAR models than (4), models under (4) are easier to formulate and interpret. Finally, many useful models can be embedded into the class characterized by (4), and this approach has been commonly used to model missing data processes with a MAR mechanism (e.g., Robins et al., 1995; Chen et al., 2010). For intermittently MAR data, it is often convenient to adopt the further assumption that the missing data indicators at time j depend only on the previously observed outcomes and covariates.

To model λ_ijk, we use the generalized logistic link, with λ_ij0 as a reference:

$$log\left(\frac{{\lambda }_{\mathit{ijk}}}{{\lambda }_{ij0}}\right)={\mathbf{u}}_{\mathit{ijk}}^T{\alpha }_k,k=1,2,3,$$ (5)

where u_ijk may be a subset of {R̄_{i j}, ${\overline{\mathbf{Y}}}_{ij}^o,{\overline{\mathbf{X}}}_{ij}^o$, Z_i}. Let $\mathit{\alpha }={\left({\alpha }_1^T,{\alpha }_2^T,{\alpha }_3^T\right)}^T$.

It is noted that (5) may include too many parameters, which may increase the likelihood of misspecification. This makes the proposed method more appealing because we have a greater chance of obtaining accurate inference if the covariate model is modeled appropriately although the missing data model is misspecified. However, empirical study demonstrates that using appropriate model selection procedure can obtain a suitable model for the missing data mechanism (Ibrahim et al., 2005), thus reducing the likelihood of misspecification of the missing data model. As suggested by Ibrahim et al. (2005), one can use a step-up approach in constructing the missing data model by initially including only the main effects, then adding additional terms sequentially. One can then use the likelihood ratio or Akaike information criterion to evaluate the fit of each model. In many cases, the main effects model will be an adequate approximation to the missing data mechanism. One must be very careful to not build too large a model for the missing data mechanism because the model can easily become nonidentifiable.

Let π_ij = P (R_ij = 3 | Y_i, X_i, Z_i) be the marginal probability of observing both Y and X at time j, given the entire vectors of responses and covariates; it is given by

$${\pi }_{ij}=\sum _{r_{i1},\dots ,r_{i,j-1}}P(R_{ij}=3,R_{i,j-1}=r_{i,j-1},\dots ,R_{i1}=r_{i1}\mid {\mathbf{Y}}_i,{\mathbf{Z}}_i,{\mathbf{X}}_i).$$

This marginal probability can be expressed in terms of the marginal (conditional) probabilities, λ_ijk’s.

2.3 Missing Covariate Model

Because subjects can have X_i missing, we must consider the density of X_i in some situations to obtain valid analysis, where we assume the joint density of X_i depends on the observed response ${\mathbf{Y}}_i^o$ and covariate Z_i. In practice, this joint density can be expressed as

$$P({\mathbf{X}}_i={\mathbf{x}}_i\mid {\mathbf{Y}}_i^o,{\mathbf{Z}}_i;\gamma )=\prod _{j=2}^{J_i}P(X_{ij}=x_{ij}\mid {\overline{\mathbf{X}}}_{ij},{\overline{\mathbf{Y}}}_{ij}^o,{\mathbf{Z}}_i;\gamma )·P(X_{i1}=x_{i1}\mid {\mathbf{Z}}_i;\gamma ),$$ (6)

where X̄_{i j} = {X_i1, …, X_{i, j −1}} is the history of the covariate X_ij until time j − 1, and γ is the corresponding coefficient vector. It is noted that we made an assumption that $P(X_{ij}=x_{ij}\mid {\overline{\mathbf{X}}}_{ij},{\mathbf{Y}}_i^o,{\mathbf{Z}}_i;\gamma )=P(X_{ij}=x_{ij}\mid {\overline{\mathbf{X}}}_{ij},{\overline{\mathbf{Y}}}_{ij}^o,{\mathbf{Z}}_i;\gamma )$.

3. Methods of Estimation

We denote the vector of all the parameters as θ = (β^T, γ^T, α^T)^T. Our main interest is in estimation of β, with γ and α viewed as nuisance parameters.

3.1 Weighted Estimating Equation for the Response Parameters

Following the spirit of the IPWGEE approach of Robins et al. (1995), we introduce a weight matrix ${\mathbf{\Delta }}_i^{\ast }(\alpha )$ into the usual GEE to adjust for the effects of incomplete responses and covariates. That is, if we let ${\mathbf{\Delta }}_i^{\ast }(\alpha )=\text{diag}(I(R_{ij}=3)/{\pi }_{ij},1\le j\le J_i)$, then the product ${\mathbf{\Delta }}_i^{\ast }({\mathbf{Y}}_i-{\mu }_i)$ yields an adjusted contribution from subject i, which involves the observed data alone. Moreover, this element has expectation zero, and hence unbiased estimating equations for β can be obtained as

$${\mathbf{U}}^{\ast }(\beta ,\alpha )=\sum _{i=1}^n{\mathbf{U}}_i^{\ast }(\beta ,\alpha )=0,$$ (7)

where ${\mathbf{U}}_i^{\ast }(\beta ,\alpha )={\mathbf{D}}_i{\mathbf{V}}_i^{-1}{\mathbf{\Delta }}_i^{\ast }(\alpha )({\mathbf{Y}}_i-{\mu }_i)$ with ${\mathbf{D}}_i=\partial {\mu }_i^T/\partial \beta$ being a p × J_i derivative matrix, and V_i the working covariance matrix for the response Y_i.

In practice, the covariance matrix V_i is often expressed as ${\mathbf{V}}_i={\mathbf{F}}_i^{1/2}{\mathbf{C}}_i{\mathbf{F}}_i^{1/2}$, where C_i is a working correlation matrix, and F_i = diag (v_ij, j = 1, …, J_i), which is assumed to only depend on the marginal mean μ_i. When the working correlation matrix C_i is the identity matrix, (7) is computable. However, when a working independence assumption is not adopted, (7) may not be computable because elements of ${\mathbf{D}}_i{\mathbf{V}}_i^{-1}$ associated with the observed pairs (Y_ij, X_ij) may be unknown because they involve other missing covariates X_{i j′} (j′ ≠ j). Here we modify (7) to incorporate general working correlation matrices.

We define Δ_i = [δ_{i j j′}]_{Ji × Ji}, j = 1, …, J_i, where δ_{i j j′} = {I (R_ij = 1, R_{i j′} = 3) + I (R_{i j} = 3, R_{i j′} = 3)}/π_{i j j′} for j ≠ j′, δ_ijj = I (R_ij = 3)/π_ij, and π_{i j j′} = P (R_{i j} = 1, R_{i j′} = 3 | Y_i, X_i, Z_i) + P (R_{i j} = 3, R_{i j′} = 3 | Y_i, X_i, Z_i). Let ${\mathbf{M}}_i={\mathbf{F}}_i^{-1/2}({\mathbf{C}}_i^{-1}•{\mathbf{\Delta }}_i){\mathbf{F}}_i^{-1/2}$, where A • B = [a_ij · b_ij] denotes the Hadamard product of J_i × J_i matrices A = [a_ij] and B = [b_ij]. By introducing the weight matrix Δ_i (α), we ensure that all required elements of M_i can be computed.

The generalized estimating functions for β are given by

$$\mathbf{U}(\beta ,\alpha )=\sum _{i=1}^n{\mathbf{U}}_i(\beta ,\alpha )=0,$$ (8)

where U_i (β, α) = D_i M_i (Y_i − μ_i). It is easy to see that estimating function (8) depends on the observed data and the parameters only, and hence is computable.

For the estimating equations (8), to obtain a consistent estimator, we need to correctly specify the missing data model. If the missing data model is misspecified, it can yield biased estimators. Under a MAR mechanism, Robins et al. (1995), Scharfstein et al. (1999), and Van der Laan and Robins (2003) proposed methods to improve the robustness of the inverse probability weighted estimators. The idea is to modify these inverse weighted equations by adding an appended function in the a tangent space of the conditional distribution of R_i, yielding an augmented estimating function that remains unbiased. With suitable choice of the appended function, we can get the doubly robust estimators. This approach has, to our knowledge, only been investigated to address the missingness with either incomplete response or covariate processes, but not both. Now, we describe ways for the double robustness for the general missingness patterns when either the covariates model or missing data model is correctly specified.

Following the same spirit of Van der Laan and Robins (2003), the general form of the augmented estimating functions for the general missingness patterns can be written as

$$\sum _{i=1}^n[{\mathbf{U}}_i+{\phi }_i]=0,$$ (9)

where φ_i is a function in the tangent space of the conditional distribution of R_i with mean zero. The optimal φ_i,opt is chosen as the projection of U_i onto the tangent space of the conditional distribution of R_i. It is not hard to show that, in Hilbert space, ${\phi }_{i,\mathit{opt}}=E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i,{\mathbf{R}}_i)}\{{\mathbf{D}}_i{\mathbf{N}}_i({\mathbf{Y}}_i-{\mu }_i)\}$ with , where is a vector a 1’s with length J_i, and ${\mathbf{Y}}_i^m$ and ${\mathbf{X}}_i^m$denote the missing part of Y_i and X_i, respectively. We can then solve estimating equations,

$${\mathbf{S}}_1(\theta )=\sum _{i=1}^n{\mathbf{S}}_{1i}(\theta )=\sum _{i=1}^n\left[{\mathbf{D}}_i{\mathbf{M}}_i({\mathbf{Y}}_i-{\mu }_i)+E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i,{\mathbf{R}}_i)}\{{\mathbf{D}}_i{\mathbf{N}}_i({\mathbf{Y}}_i-{\mu }_i)\}\right]=0,$$

to obtain the estimator of β. It can be shown that the resulting estimator for β is robust to the misspecification of either the missing data model or the covariates model. The proof is given in the Appendix.

As commented by a referee, if Y and X are partially observed, and we assume an independence working correlation matrix, observations with Y unobserved contribute nothing to the estimation. However, when we allow correlations then we do need the individuals with Y missing but X observed for the correlation terms. Further, if we assume independence, then estimation is much simpler: the iterative process in Section 3.3 is not needed. Thus a simple way to analyze the data is to use doubly robust independence equations, and then an appropriate (i.e., sandwich) standard error. However, this estimator may not be efficient, which is demonstrated in Section 4 (see Table 3) if the true correlation matrix is not independent.

Table 3.

Empirical bias, standard deviation, and coverage probabilities for the proposed approaches to estimation and inference with incomplete covariate and response data with independence working correlation matrix (γ₁ = 2 and α₂ = −2 : missing proportion is about 40%)

	β0			β1			β2
Method	Bias%	SD	CP%	Bias%	SD	CP%	Bias%	SD	CP%
ρ = 0.6
DREE(x+, r+)	2.1	0.138	94.7	0.1	0.133	95.1	−1.0	0.131	94.5
DREE(x+, r−)	1.6	0.146	94.4	1.7	0.134	94.3	−1.4	0.110	94.9
DREE(x−, r+)	−1.4	0.114	94.6	−1.0	0.099	95.0	−0.9	0.094	94.8
DREE(x−, r−)	−9.2	0.108	91.7	−6.9	0.097	92.1	−4.0	0.093	93.5
ρ = 0.3
DREE(x+, r+)	0.9	0.105	94.8	1.0	0.103	95.2	0.1	0.083	95.0
DREE(x+, r−)	−0.4	0.119	94.7	0.1	0.114	94.9	0.1	0.091	95.3
DREE(x−, r+)	−1.5	0.100	94.4	−1.7	0.095	94.4	−0.8	0.085	94.6
DREE(x−, r−)	−7.4	0.096	92.8	−6.0	0.097	93.2	2.0	0.080	94.0
ρ = 0
DREE(x+, r+)	0.5	0.090	95.1	0.1	0.095	95.0	−0.1	0.080	94.6
DREE(x+, r−)	0.2	0.089	95.2	0.4	0.099	94.6	1.0	0.081	94.4
DREE(x−, r+)	1.0	0.087	94.9	0.2	0.096	95.2	0.1	0.078	94.9
DREE(x−, r−)	0.8	0.090	94.5	0.2	0.097	94.7	−0.4	0.080	94.5

In practice, the parameters γ and α are unknown, and one must replace γ and α with a consistent estimator. We describe how to obtain an estimator in the next subsection.

3.2 Estimation for the Nuisance Parameters

Because we are assuming that the covariate is MAR, we can obtain the estimator of γ through maximizing the observed likelihood function

$$L_2(\gamma )=\prod _{i=1}^n\int P({\mathbf{X}}_i\mid {\mathbf{Z}}_i,{\mathbf{Y}}_i^o)d{\mathbf{X}}_i^m,$$

which is equivalent to solve the estimating equation ${\mathbf{S}}_2(\gamma )=\partial log{L}_2(\gamma )/\partial {\gamma }^T={\sum }_{i=1}^n{\mathbf{S}}_{2i}(\gamma )=\mathbf{0}$, where ${\mathbf{S}}_{2i}(\gamma )=\partial log\int P({\mathbf{X}}_i\mid {\mathbf{Z}}_i,{\mathbf{Y}}_i^o)d{\mathbf{X}}_i^m/\partial {\gamma }^T$.

For the estimation of the missing data parameter α, we can also use maximum likelihood estimator. Note that the log likelihood for α is given by

$$\ell (\alpha )=\sum _{i=1}^n{\ell }_i(\alpha )=\sum _{i=1}^n\sum _{j=1}^{J_i}\sum _{k=0}^{3}I(R_{ij}=k)log({\lambda }_{\mathit{ijk}}),$$

and the score function is

$${\mathbf{S}}_3(\alpha )=\sum _{i=1}^n{\mathbf{S}}_{3i}(\alpha )=\sum _{i=1}^n\sum _{j=1}^{J_i}\sum _{k=0}^3\frac{I(R_{ij}=k)}{{\lambda }_{\mathit{ijk}}}·\frac{\partial {\lambda }_{\mathit{ijk}}}{\partial {\alpha }^T}.$$

Solving the estimating equation S₃(α) = 0 leads to the maximum likelihood estimator α̂.

3.3 Estimation and Inferences

In this section, we give details on the estimation and inference for the parameter β. Plugging in the estimated parameters γ̂ and α̂, we then solve the following estimating equations for the parameter β:

$${\mathbf{S}}_i(\beta ,\widehat{\gamma },\widehat{\alpha })=\sum _{i=1}^n{\mathbf{S}}_{1i}(\beta ,\widehat{\gamma },\widehat{\alpha })=\mathbf{0}.$$ (10)

For simplicity, we use notation S₁(β) and S_1i (β) to denote S₁(β, γ̂, α̂) and S_1i (β, γ̂, α̂) in the following. It can be shown that, provided that the response model, p(y_i | x_i, z_i), is correctly specified, either the correct specification of the missing data model, p(r_i | x_i, y_i, z_i), or the correct specification of the covariate model, $p({\mathbf{x}}_i\mid {\mathbf{z}}_i,{\mathbf{y}}_i^o)$, leads to the asymptotically unbiased estimator of β. The details of proof are given in the Appendix.

To solve estimating equations (10), we employ an expectation–maximization (EM)-type algorithm (Lipsitz et al., 1999). The key is that we need to calculate the conditional expectation in S₁.

When X is discrete, then the second part in S_1i can be written as

$$E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i,{\mathbf{R}}_i)}\{{\mathbf{D}}_i{\mathbf{N}}_i({\mathbf{Y}}_i-{\mu }_i)\}=\sum _{({\mathbf{y}}_i^m,{\mathbf{x}}_i^m)}{w}_{\mathit{ixy}}(\beta )\{{\mathbf{D}}_i{\mathbf{N}}_i({\mathbf{Y}}_i-{\mu }_i)\},$$

where

$$\begin{array}{l}{w}_{\mathit{ixy}}(\beta )=P({\mathbf{Y}}_i^m={\mathbf{y}}_i^m,{\mathbf{X}}_i^m={\mathbf{x}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i,{\mathbf{R}}_i;\beta ,\widehat{\gamma })\\ =P\left({\mathbf{Y}}_i^m={\mathbf{y}}_i^m,{\mathbf{X}}_i^m={\mathbf{x}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i;\beta ,\widehat{\gamma }\right)\\ =P\left({\mathbf{Y}}_i^m={\mathbf{y}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i={\mathbf{x}}_i,{\mathbf{Z}}_i;\beta \right)\times P\left({\mathbf{X}}_i^m={\mathbf{x}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i,\widehat{\gamma }\right)\\ =\frac{P({\mathbf{Y}}_i={\mathbf{y}}_i\mid {\mathbf{X}}_i={\mathbf{x}}_i,{\mathbf{Z}}_i;\beta )}{{\displaystyle \sum _{{\mathbf{y}}_i^m}}P({\mathbf{Y}}_i={\mathbf{y}}_i\mid {\mathbf{X}}_i={\mathbf{x}}_i,{\mathbf{Z}}_i;\beta )}·\frac{P({\mathbf{X}}_i={\mathbf{x}}_i\mid {\mathbf{Y}}_i^o,{\mathbf{Z}}_i,\widehat{\gamma })}{{\displaystyle \sum _{{\mathbf{x}}_i^m}}P({\mathbf{X}}_i={\mathbf{x}}_i\mid {\mathbf{Y}}_i^o,{\mathbf{Z}}_i,\widehat{\gamma })}\end{array}$$

can be regarded as a weight. The distribution of P (Y_i = y_i | X_i = x_i, Z_i) and $P({\mathbf{X}}_i={\mathbf{x}}_i\mid {\mathbf{Z}}_i,{\mathbf{Y}}_i^o)$ can be obtained from (2) and (6), respectively.

We now introduce the EM-type algorithm to solve S₁(β̂) = 0 as follows:

1. Obtain an initial value of the parameter β = β⁽⁰⁾;

2. At the tth step, we have β^(t), and calculate $w_{\mathit{ixy}}^{(t)}=w_{\mathit{ixy}}({\beta }^{(t)})$;

3. Treating $w_{\mathit{ixy}}^{(t)}$ as fixed, solve ${\mathbf{S}}_1({\beta }^{(t+1)}\mid {\beta }^{(t)})={\sum }_{i=1}^n{\mathbf{S}}_{1i}({\beta }^{(t+1)}\mid {\beta }^{(t)})=\mathbf{0}$ for β^(t+1), where

   $${\mathbf{S}}_{1i}({\beta }^{(t+1)}\mid {\beta }^{(t)})={\mathbf{D}}_i{\mathbf{M}}_i({\beta }^{(t)})({\mathbf{Y}}_i-{\mu }_i)+\sum _{({\mathbf{y}}_i^m,{\mathbf{x}}_i^m)}{w}_{\mathit{ixy}}^{(t)}\{{\mathbf{D}}_i{\mathbf{N}}_i({\beta }^{(t)})({\mathbf{Y}}_i-{\mu }_i)\};$$ (11)

4. Iterate until convergence to, say β̂, which gives the solution to S₁(β̂) = 0.

When X is continuous, (11) becomes

$${\mathbf{S}}_{1i}({\beta }^{(t+1)}\mid {\beta }^{(t)})={\mathbf{D}}_i{\mathbf{M}}_i({\beta }^{(t)})({\mathbf{Y}}_i-{\mu }_i)+{\int }_{({\mathbf{y}}_i^m,{\mathbf{x}}_i^m)}\times w_{\mathit{ixy}}^{(t)}\{{\mathbf{D}}_i{\mathbf{N}}_i({\beta }^{(t)})({\mathbf{Y}}_i-{\mu }_i)\}d{\mathbf{Y}}_i^{m}d{\mathbf{X}}_i^m=\mathbf{0},$$ (12)

where the weights

$$\begin{array}{l}{w}_{\mathit{ixy}}^{(t)}=P({\mathbf{Y}}_i^m={\mathbf{y}}_i^m,{\mathbf{X}}_i^m={\mathbf{x}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i;{\beta }^{(t)},\widehat{\gamma })\\ =P({\mathbf{Y}}_i^m={\mathbf{y}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i={\mathbf{x}}_i,{\mathbf{Z}}_i;{\beta }^{(t)})\times P({\mathbf{X}}_i^m={\mathbf{x}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i,\widehat{\gamma })\\ =\frac{P({\mathbf{Y}}_i={\mathbf{y}}_i\mid {\mathbf{X}}_i={\mathbf{x}}_i,{\mathbf{Z}}_i;{\beta }^{(t)})}{{\displaystyle \sum _{{\mathbf{y}}_i^m}}P({\mathbf{Y}}_i={\mathbf{y}}_i\mid {\mathbf{X}}_i={\mathbf{x}}_i,{\mathbf{Z}}_i;{\beta }^{(t)})}·\frac{P({\mathbf{X}}_i={\mathbf{x}}_i\mid {\mathbf{Y}}_i^o,{\mathbf{Z}}_i,\widehat{\gamma })}{{\int }_{{\mathbf{x}}_i^m}P({\mathbf{X}}_i={\mathbf{x}}_i\mid {\mathbf{Y}}_i^o,{\mathbf{Z}}_i,\widehat{\gamma })d{\mathbf{X}}_i^m}.\end{array}$$

To solve (12), we need the integrations. In this case, rather than use numerical integration, we would suggest using rejection sampling (Gilks and Wild, 1992). Adopting this approach we would sample ( $y_i^m,x_i^m$) from the conditional density $w_{\mathit{ixy}}^{(t)}$. Repeat this L times, with the lth draw of ( ${\mathbf{y}}_i^m,{\mathbf{x}}_i^m$) denoted by ( ${\mathbf{y}}_i^{ml(t)},{\mathbf{x}}_i^{ml(t)}$). Then

$${\int }_{({\mathbf{y}}_i^m,{\mathbf{x}}_i^m)}{w}_{\mathit{ixy}}^{(t)}\{{\mathbf{D}}_i{\mathbf{N}}_i({\beta }^{(t)})({\mathbf{Y}}_i-{\mu }_i)\}d{\mathbf{Y}}_i^{m}d{\mathbf{X}}_i^m\approx {\frac{1}{L}\sum _{l=1}^L\{{\mathbf{D}}_i{\mathbf{N}}_i({\beta }^{(t)})({\mathbf{Y}}_i-{\mu }_i)\}|}_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m)=({\mathbf{y}}_i^{ml(t)},{\mathbf{x}}_i^{ml(t)})}.$$

We can similarly introduce an iterative algorithm as is done when X is discrete.

To state the asymptotic properties of β̂, we define $\mathbf{\Gamma }(\beta ,\gamma ,\alpha )=E\{\partial {\mathbf{S}}_{1i}^T(\beta ,\gamma ,\alpha )/\partial \beta \},{\mathbf{I}}_{12}(\beta ,\gamma ,\alpha )=E\{\partial {\mathbf{S}}_{1i}^T(\beta ,\gamma ,\alpha )/\partial \gamma \},{\mathbf{I}}_{13}(\beta ,\gamma ,\alpha )=E\{\partial {\mathbf{S}}_{1i}^T(\beta ,\gamma ,\alpha )/\partial \alpha \},{\mathbf{I}}_2(\gamma )=E\{\partial {\mathbf{S}}_{2i}^T(\gamma )/\partial \gamma \},{\mathbf{I}}_3(\alpha )=E\{\partial {\mathbf{S}}_{3i}^T(\alpha )/\partial \alpha \}$, and ${\mathbf{Q}}_i(\beta ,\gamma ,\alpha )={\mathbf{S}}_{1i}(\beta ,\gamma ,\alpha )-{\mathbf{I}}_{12}(\beta ,\gamma ,\alpha ){\mathbf{I}}_2^{-1}(\gamma ){\mathbf{S}}_{2i}(\gamma )-{\mathbf{I}}_{13}(\beta ,\gamma ,\alpha ){\mathbf{I}}_3^{-1}(\alpha ){\mathbf{S}}_{3i}(\alpha )$.

Theorem 1

Suppose that the regularity conditions stated in the Appendix hold, if either the missing data model or the covariate model is correctly specified, we have

$$n^{1/2}(\widehat{\beta }-{\beta }_0)\to N(\mathbf{0},{\mathbf{\Gamma }}^{-1}({\beta }_0,{\gamma }_0,{\alpha }_0)\mathbf{\sum }{\{{\mathbf{\Gamma }}^{-1}({\beta }_0,{\gamma }_0,{\alpha }_0)\}}^T),$$

where β₀ is the true value of β, γ₀ and α₀ are the probability limits of γ̂ and α̂, and $\mathbf{\sum }=E\{{\mathbf{Q}}_i({\beta }_0,{\gamma }_0,{\alpha }_0){\mathbf{Q}}_i^T({\beta }_0,{\gamma }_0,{\alpha }_0)\}$. The proof is given in the Appendix. To make inferences, we consistently estimate the matrix Γ by $\widehat{\mathbf{\Gamma }}=n^{-1}{\sum }_{i=1}^n\{\partial {\mathbf{S}}_{1i}^T(\widehat{\theta })/\partial \beta \}$, and Σ by $\widehat{\mathbf{\sum }}=n^{-1}{\sum }_{i=1}^n\{{\widehat{\mathbf{Q}}}_i{\widehat{\mathbf{Q}}}_i^T\}$, where θ̂ = (β̂^T, γ̂^T, α̂^T)^T, ${\widehat{\mathbf{Q}}}_i={\mathbf{S}}_{1i}(\widehat{\theta })-{\widehat{\mathbf{I}}}_{12}(\widehat{\theta }){\widehat{\mathbf{I}}}_2^{-1}(\widehat{\gamma }){\mathbf{S}}_{2i}(\widehat{\gamma })-{\widehat{\mathbf{I}}}_{13}(\widehat{\theta }){\widehat{\mathbf{I}}}_3^{-1}(\widehat{\alpha }){\mathbf{S}}_{3i}(\widehat{\alpha }),{\widehat{\mathbf{I}}}_{12}(\widehat{\theta })=n^{-1}{\sum }_{i=1}^n\{\partial {\mathbf{S}}_{1i}^T(\widehat{\theta })/\partial \gamma \},{\widehat{\mathbf{I}}}_{13}(\widehat{\theta })=n^{-1}{\sum }_{i=1}^n\{\partial {\mathbf{S}}_{1i}^T(\widehat{\theta })/\partial \alpha \},{\widehat{\mathbf{I}}}_2(\widehat{\gamma })=n^{-1}{\sum }_{i=1}^n\{\partial {\mathbf{S}}_{2i}^T(\widehat{\gamma })/\partial \gamma \}$, and ${\widehat{\mathbf{I}}}_3(\widehat{\alpha })=n^{-1}{\sum }_{i=1}^n\{\partial {\mathbf{S}}_{3i}^T(\widehat{\alpha })/\partial \alpha \}$.

We note that the preceding development treats the working correlation matrix C_i as known. In applications, C_i may contain an unknown parameter vector, say ρ, that is functionally independent of the β, γ, and α parameters but must be estimated. Following the spirit of Liang and Zeger (1986), we suggest an iterative fitting procedure by which the current values of β̂ and α̂ are used to compute a weighted moment-type estimator obtained as a function of the weighted Pearson residual $e_{ij}={\delta }_{ij}(y_{ij}-{\mu }_{ij})/\sqrt{v_{ij}}$, where δ_ij = I (R_ij = 3)/π_ij. The expression for the estimator of ρ depends on the correlation structure. For example, for an unstructured correlation matrix where Corr(Y_ij, Y_{i j′}) = ρ_{j j′} for j ≠ j′, ρ_{j j}′ is estimated by ${\widehat{\rho }}_{{jj}^{\prime }}=n^{-1}{\sum }_{i=1}^n{e}_{ij}{e}_{{ij}^{\prime }}·({\pi }_{ij}{\pi }_{{ij}^{\prime }}/{\pi }_{{\mathit{ijj}}^{\prime }}^{\ast })$. Here ${\pi }_{{\mathit{ijj}}^{\prime }}^{\ast }=P(R_{ij}=3,R_{{ij}^{\prime }}=3\mid Y_i,X_i,Z_i)$ is calculated in the same way as π_ij is calculated above. It is easy to show that ρ̂_{j j′} is an unbiased estimator.

To obtain a n^1/2-consistent estimator in Theorem 1, we require that the probability π_ij is bounded away from 0. In practice, if ${\pi }_{ij}^{\prime }$s are relatively too small for some subjects, the inverse probability weights will inflate the effects of these subjects and lead to great biases. Thus, an investigator should examine the weight distribution carefully before implementing the proposed method. Some protection using truncation or downgrading may be an option for controlling weights properly, if unreasonable influences are detected (Bang, 2005).

4. Numerical Studies

4.1 Performance of the Proposed Estimators

In this subsection, we evaluate the performance of the proposed method compared to other methods commonly used in practice through simulation studies. In the simulation studies, we focus on a setting where J_i = J = 3 and n = 500. We simulate the longitudinal binary responses from a model with

$$\text{logit}{\mu }_{ij}={\beta }_0+{\beta }_1{X}_{ij}+{\beta }_2{Z}_{ij},$$ (13)

where Z_ij is a time-varying binary covariate generated from Bin(1, 0.5), and X_ij is a time-varying binary covariate, which may be missing at some time points and is generated from the model

$$\text{logit}{\omega }_{ij}={\gamma }_0+{\gamma }_1{X}_{i,j-1}+{\gamma }_2{Z}_{ij},$$ (14)

where ω_{i j} = P(X_{i j} = 1 | X̄_{i j}, Z_{i j}). We take β₀ = log (1.5), β₁ = log(0.5), β₂ = log(2), γ₀ = log(1), γ₂ = 2, and let γ₁ vary from −2 to 2. The correlation matrix of the response vector Y_i is exchangeable with correlation coefficient ρ.

For the missing data process, we take

$$log\left(\frac{{\lambda }_{\mathit{ijk}}}{{\lambda }_{ij0}}\right)={\alpha }_{0k}+{\alpha }_{1k1}I(R_{i,j-1}=1)+{\alpha }_{1k2}I(R_{i,j-1}=2)+{\alpha }_{1k3}I(R_{i,j-1}=3)+{\alpha }_{2k}{y}_{i,j-1}^{†}+{\alpha }_{3k}{x}_{i,j-1}^{†},$$ (15)

for k = 1, 2, 3, where y†_{i,j −1} = y_{i,j −1} if y_{i,j −1} is observed and 0 otherwise, x†_{i,j −1} = x_{i,j −1} if x_{i,j −1} is observed and 0 otherwise. The true values are taken as α_0k = log (1.5), α_1k1 = log (1.5), α_1k2 = log (1.3), α_1k3 = log (1.1), α_3k = −2, and α_2k = α₂, varying from −2 to 2.

In the simulations, we always assume that the model for (Y_i | x_i, z_i), equation (13), is correctly specified. We consider the following 10 methods: (1) The proposed method, i.e., the douldy robust estimating equation (DREE) method that both the missing data model (15) and covariate model (14) are correctly specified, which we denote DREE(x+, r+). (2) The proposed method that the missing data model is correctly specified, but the model for ω is misspecified as

$$\text{logit}{\omega }_{ij}={\gamma }_0^{\ast }+{\gamma }_2^{\ast }{Z}_{ij},$$ (16)

which we denote DREE(x−, r+). (3) The proposed method that the model for ω is correctly specified, but the missing data model is misspecified as

$$log\left(\frac{{\lambda }_{\mathit{ijk}}}{{\lambda }_{ij0}}\right)={\alpha }_{0k}^{\ast }+{\alpha }_{1k1}^{\ast }I(R_{i,j-1}=1)+{\alpha }_{1k2}^{\ast }I(R_{i,j-1}=2)+{\alpha }_{1k3}^{\ast }I(R_{i,j-1}=3)+{\alpha }_{3k}^{\ast }{x}_{i,j-1}^{†},$$ (17)

which we denote DREE(x+, r−). (4) The proposed method that the model for ω is misspecified as (16), and the missing data model is misspecified as (17), which we denote DREE(x−, r−). (5) The maximum likelihood method via the EM algorithm with the covariate model incorrectly specified as (16), which we denote EM(x−). (6) The simple weighted GEE (not the robust method) method with the missing data model incorrectly specified as (17), which we denote GEE(r−). (7) The complete case (CC) analysis using the GEE method, which we denote cc. (8) The augmented weighted GEE analysis based on Chen et al. (2010) with correct missing data model (denoted as AIP(r+)), which solves the following estimating equation:

$$\sum _{i=1}^n\{{\mathbf{U}}_i(\beta ,\alpha )-\eta {\mathbf{A}}_i(\alpha )\}=0,$$

where A_i (α) is a function of the observed data that is free of β with expectation 0, and η is the regression coefficient of A_i in the population regression of U_i regress A_i and S_3i. Here we choose A_i, in the same spirit of Chen et al. (2010), as ${\mathbf{A}}_i(\alpha )={({\mathbf{A}}_{i1}^T,{\mathbf{A}}_{i2}^T,{\mathbf{A}}_{i3}^T)}^T$, where

$$\begin{array}{l}{\mathbf{A}}_{i1}=\left[\left\{\frac{I(R_{ij}=2)}{P(R_{ij}=2\mid {\mathbf{X}}_i,{\mathbf{Y}}_i)}{\pi }_{ij}^y-1\right\}{R}_{ij}^y{Y}_{ij},j=1,2,3\right],\\ {\mathbf{A}}_{i2}=\left[\left\{\frac{I(R_{ij}=1)}{P(R_{ij}=1\mid {\mathbf{X}}_i,{\mathbf{Y}}_i)}{\pi }_{ij}^x-1\right\}{R}_{ij}^x{X}_{ij},j=1,2,3\right],\\ {\mathbf{A}}_{i3}=\left[\left\{\frac{I(R_{ij}=0)}{P(R_{ij}=0\mid {\mathbf{X}}_i,{\mathbf{Y}}_i)}-1\right\}{Z}_{ij},j=1,2,3\right],\end{array}$$

$R_{ij}^y=I(R_{ij}=2)+I(R_{ij}=3),R_{ij}^x=I(R_{ij}=1)+I(R_{ij}=3),{\pi }_{ij}^y=P(R_{ij}=2\mid {\mathbf{X}}_i,{\mathbf{Y}}_i)+P(R_{ij}=3\mid {\mathbf{X}}_i,{\mathbf{Y}}_i)$, and ${\pi }_{ij}^x=P(R_{ij}=1\mid {\mathbf{X}}_i,{\mathbf{Y}}_i)+P(R_{ij}=3\mid {\mathbf{X}}_i,{\mathbf{Y}}_i)$. (9) The simple weighted GEE (not the robust method) method with the missing data model correctly specified, which we denote GEE(r+). (10) The maximum likelihood method via the EM algorithm to the subgroup with either no missing data or Y observed and X missing, and the covariate model is correctly specified, which we denote EM ^*(x+). In each setting, we perform 2000 simulations.

The results are reported in Table 1, where the bias is the percentage relative bias, SD is the standard deviation for the 2000 simulations, and CP represents the empirical coverage probability for 95% confidence intervals. It is seen that the DREE(x−, r−), EM(x−), EM*(x+), GEE(r−), and cc approaches yield larger biases and poor coverage probabilities; as the response correlation coefficient increases, the performance becomes worse. The DREE(x+, r+), DREE(x+, r−), DREE(x−, r+), GEE(r+), and AIP(r+) methods provide ignorable finite sample biases and good coverage probabilities. The DREE(x+, r+), DREE(x+, r−), and DREE(x−, r+) estimators are more efficient than the AIP(r+) estimator, and the AIP(r+) estimator is more efficient than the GEE(r+) estimator. To clarify for readers the effect of the missing data model, we report the expected missing value patterns in Table 2 when ρ = 0.3, γ₁ = 2, and α₂ = −2.

Table 1.

Empirical bias, standard deviation, and coverage probabilities for ten approaches to estimation and inference with incomplete covariate and response data (γ₁ = 2 and α₂ = −2 : missing proportion is about 40%)

	β0			β1			β2
Method	Bias%	SD	CP%	Bias%	SD	CP%	Bias%	SD	CP%
ρ = 0.6
DREE(x+, r+)	1.1	0.103	94.5	1.0	0.093	94.6	−0.8	0.086	94.7
DREE(x+, r−)	1.2	0.107	94.6	1.8	0.102	94.7	−1.1	0.094	94.7
DREE(x−, r+)	−1.3	0.105	94.5	−1.2	0.098	94.9	−1.4	0.088	94.6
GEE(r+)	1.3	0.119	94.3	1.5	0.110	94.4	−0.6	0.106	94.6
AIP(r+)	2.1	0.111	94.6	1.2	0.104	94.3	−1.5	0.101	94.3
DREE(x−, r−)	−9.5	0.108	91.4	−7.3	0.099	91.9	−4.3	0.096	94.2
EM(x−)	−25.0	0.140	90.5	−73.3	0.136	80.6	26.3	0.110	88.3
EM*(x+)	12.1	0.142	90.4	−10.6	0.146	92.0	−0.1	0.139	95.2
GEE(r−)	32.0	0.230	81.3	18.8	0.251	80.4	2.6	0.264	94.2
cc	−272.6	0.690	89.0	72.6	0.977	96.0	28.8	0.655	95.2
ρ = 0.3
DREE(x+, r+)	1.4	0.095	95.2	0.9	0.091	94.8	0.1	0.081	94.7
DREE(x+, r−)	0.8	0.109	94.6	1.0	0.101	94.7	0.6	0.083	94.8
DREE(x−, r+)	−0.6	0.097	94.7	−0.9	0.092	94.4	−0.3	0.082	94.5
GEE(r+)	1.0	0.116	94.4	0.5	0.114	94.4	−1.3	0.093	94.5
AIP(r+)	2.3	0.111	94.2	1.9	0.109	94.3	−2.2	0.089	94.3
DREE(x−, r−)	−7.5	0.097	93.8	−6.3	0.095	94.1	0.0	0.083	95.2
EM(x−)	−10.2	0.121	91.7	−86.1	0.122	74.3	21.6	0.099	90.8
EM*(x+)	6.6	0.126	95.4	−4.4	0.145	95.0	−0.3	0.139	95.8
GEE(r−)	15.9	0.215	81.4	11.5	0.252	86.4	−1.3	0.255	94.5
cc	−221.1	0.955	75.8	−26.8	1.112	97.0	−27.7	1.423	93.9
ρ = 0
DREE(x+, r+)	0.1	0.089	94.8	0.2	0.098	94.5	0.1	0.079	94.6
DREE(x+, r−)	−0.4	0.089	94.4	0.9	0.099	94.7	0.9	0.080	94.8
DREE(x−, r+)	−0.3	0.089	94.6	−0.2	0.098	94.2	−0.4	0.080	94.9
GEE(r+)	0.2	0.100	95.2	1.2	0.101	95.3	0.8	0.089	95.2
AIP(r+)	0.5	0.097	94.4	0.8	0.099	94.5	1.4	0.086	94.5
DREE(x−, r−)	−0.0	0.089	94.7	−0.6	0.097	94.8	−0.1	0.081	94.5
EM(x−)	49.1	0.102	84.9	−109.0	0.106	71.5	−12.0	0.079	93.5
EM*(x+)	2.5	0.125	96.4	0.2	0.145	95.2	−1.2	0.138	93.2
GEE(r−)	3.8	0.225	95.2	3.2	0.259	94.3	2.8	0.259	94.2
cc	−157.7	0.828	85.7	−6.8	1.021	93.3	−19.0	1.356	94.5

Table 2.

Missing value patterns when ρ = 0.3, γ₁ = 2, and α₂ = −2 : “+” and “−” represent observed and missing values, respectively

Y1	X1	Y2	X2	Y3	X3	n (%)
+	+	+	+	+	+	1.0
+	+	+	+	+	−	0.8
+	+	+	+	−	+	0.9
+	+	+	+	−	−	7.4
+	+	+	−	+	+	1.9
+	+	+	−	+	−	1.8
+	+	+	−	−	+	1.8
+	+	+	−	−	−	4.7
+	+	−	+	+	+	1.4
+	+	−	+	+	−	1.4
+	+	−	+	−	+	1.3
+	+	−	+	−	−	6.1
+	+	−	−	+	+	17.3
+	+	−	−	+	−	17.6
+	+	−	−	−	+	17.3
+	+	−	−	−	−	17.5

To study the performance of the proposed method with misspecified working correlation matrix, we consider using the independence working correlation matrix. Table 3 lists the results. As expected, the DREE(x−, r−) still yields larger biases when the true correlation coefficient ρ is not equal to 0; DREE(x+, r+), DREE(x+, r−), and DREE(x−, r+) give negligible biases; however, using the independence working correlation matrix yields less efficient estimators than the estimators with correct specification of the correlation matrix when ρ ≠ 0; there is, however, not much difference in the efficiency between the two types of estimators when ρ = 0, which makes sense.

4.2 Impact of Model Misspecification

The validity of the proposed method depends on correct specification of the model for the response process and either the missing data process or the covariate process. Here we investigate the impact of misspecification of the missing data process model and/or the covariate process model.

Let θ̂^† denote the estimator for θ when the missing data process and the covariate process models are misspecified. To characterize the asymptotic bias of θ̂^†, we use the methods of White (1982) to find the value to which θ̂^† converges. In the spirit of Rotnitzky and Wypij (1994); Fitzmaurice, Molenberghs, and Lipsitz (1995); and Cook, Zeng, and Yi (2004), we take the expectation of S_i (θ) with respect to the true distribution of G = (R_i, Y_i, X_i, Z_i) and set it equal to zero. The solution to this equation, denoted θ^†, is the value to which θ̂^† converges in probability. If G is the sample space for G, and P(g; θ) is the true probability of observing the realized value g of G, then solving the equation,

$$\sum _{\mathbf{g}\in \mathbf{G}}{\mathbf{S}}_i({\theta }^{†})·P(\mathbf{g};\theta )=0,$$ (18)

gives the relationship between θ and θ^†, and enables one to characterize the asymptotic bias.

In this study, response measurements are featured by the same model (13) with ρ = 0.3; the true model for missing data indicators is (15), and the true model for the missing covariate model is (14).

Now we consider the misspecification of the missing data model and the covariate model. The misspecified models are (16) and (17). Figures 1 and 2 plot the asymptotic percentage relative biases of β₁ and β₂ against γ₁ as α₂ changes. It is seen that β₁ and β₂ are sensitive to the misspecification of the missing data and covariate models. As the absolute value of α₂ goes to 0, the relative biases decrease; as the absolute value of γ₁ goes to 0, the relative biases decrease. For fixed α₂ that are not very big, the relative biases are flat functions of γ₁, and the biases are small, indicating that the estimator is less sensitive to the misspecification of the covariate model in our model set-up; while for fixed γ₁ that are not very close to 0, the relative biases change rapidly as α₂ changes, indicating that the estimator is more sensitive to the misspecification of the missing data model in our model set-up; however, this may not be generally true.

Figure 1.

Asymptotic percentage relative bias of β₁ with misspecified covariate model and missing data model.

Figure 2.

Asymptotic percentage relative bias of β₂ with misspecified covariate model and missing data model.

In summary, estimation of the response parameters is generally sensitive to the misspecification of the missing data model and/or covariate model, although the degree of the sensitivity could be varying for different kinds of misspecifications.

5. Application to an Alzheimer’s Disease Study

We apply the proposed method to the National Alzheimer’s Coordinating Center Uniform Data Set. One of the goals of the study is to investigate the risk factors that influence the occurrence of AD. The response we are using is the clinical diagnosis for subjects as having AD (Yes/No), because the gold standard for AD is through neuropathological examination, which is not available for the majority of subjects. The covariates that may influence the occurrence of AD include sex, congestive heart failure (CVCHF, yes/no), family history of dementia (FHDEM, yes/no), diabetes (DIABETE, yes/no), depression or dysphoria (DEP., yes/no), hypertension (HYPERT, yes/no), education (EDUC, years), Mini-Mental State Exam (MMSE) score, and age, in which CVCHF, diabetes, DEP., hypertension, and MMSE are time varying, and the others are not. There are 16,223 subjects from 29 AD Centers included at the entry of this study. Follow-up visits for subjects are scheduled at approximately 1-year intervals, with up to four clinical visits until 2008. The median follow-up is 2. However, patients may miss a clinic visit or refuse to undergo a clinical examination during the clinic visit, leading to the incomplete responses; furthermore, patients may partially complete a form or questionnaire that the DEP. covariate is based on, leading to incomplete covariates. We think the MAR mechanism may be reasonable because, for example, a patient’s visit to a clinic may depend on his/her previous observed diagnosis of disease status: if s/he was diagnosed AD last year, s/he may be likely to attend the clinic visit this year to control the disease efficiently. There are 8724 subjects with complete data observed. About 11.9% of observation time points have the response and DEP. missing simultaneously; about 31.2% of observation time points have the responses missing and the DEP. observed; about 3.2% of observation time points have the behavioral assessment missing but the response observed; and about 53.7% of observation time points have both the response and the DEP. observed.

Consider the following regression model for the response process logit

$$\text{logit}{\mu }_{ij}={\mathbf{u}}_{ij}^T\beta ,$$ (19)

where u_ij is the covariate vector at time point j, which includes the function of sex, CVCHF, FHDEM, diabetes, DEP., hypertension, education, MMSE, age, and the indicator variables for centers.

For the missing indicators, we build the following regression models:

$$log\left(\frac{{\lambda }_{\mathit{ijk}}}{{\lambda }_{ij0}}\right)={\mathbf{v}}_{\mathit{ijk}}^T{\alpha }_k,k=1,2,3,$$ (20)

where v_ijk include functions of history of the missing indicators, sex, CVCHF, FHDEM, diabetes, hypertension, education, MMSE, age, previous observed response, and previous observed depression status.

For the covariate, we build the following model:

$$\text{logit}{\omega }_{ij}={\mathbf{w}}_{ij}^T\gamma ,j>1,$$ (21)

where ω_ij is the conditional probability that patient i at time j is depressed given the covariate vector w_ij, which may include functions of history of depression, sex, CVCHF, FHDEM, diabetes, hypertension, education, MMSE, age, and previous observed response.

Here we use four methods to analyze the data. The first method, labeled “EM,” is the maximum likelihood method via the EM algorithm; the second method, labeled “DREE,” is the proposed doubly robust method based on the augmented estimating equations; the third method is the CC analysis via the GEE; the fourth method, labeled “RE,” is the random effects (intercept) model by using the available data. The results are reported in Table 4, where the parameters are (adjusted) log odds ratios (except the intercept term) of diagnosis of AD at time j. The response model we use is (19) with unstructured correlation matrix, and the missing data and covariate models are (20) and (21), respectively. Here we only list the effects for the risk factors and omit the center effects though some of them are highly significant. All methods reveal the same results: sex, CVCHF, FHDEM, and EDUC have no significant effect on the occurrence of AD; depression has a negative effect on the occurrence of AD; MMSE has a positive effect to protect the occurrence of AD; diabetes and hypertension have positive effects to protect the occurrence of AD; and age has a negative effect on the occurrence of AD.

Table 4.

Log odds ratios (except the intercept term) of diagnosis of AD for the National Alzheimer’s Coordinating Center Uniform Dataset (the correlation matrix is unstructured for EM, DREE, and CC methods)

	EM			DREE			CC			RE
Parameter	Est.	SE	p	Est.	SE	p	Est.	SE	p	Est.	SE	p
(Intercept)	−0.831	0.157	<0.001	−0.611	0.138	<0.001	−0.165	0.195	0.398	−0.289	0.207	0.163
SEX(F)	−0.056	0.031	0.075	−0.041	0.028	0.141	−0.031	0.039	0.430	−0.049	0.034	0.150
CVCHF	0.128	0.075	0.087	−0.045	0.070	0.524	−0.062	0.097	0.521	0.068	0.084	0.418
DEP.	0.314	0.028	<0.001	0.415	0.030	<0.001	0.449	0.042	<0.001	0.378	0.031	<0.001
MMSE	−0.003	0.001	<0.001	−0.005	0.001	<0.001	−0.024	0.001	<0.001	−0.019	0.001	<0.001
FHDEM	0.009	0.035	0.788	0.023	0.030	0.459	−0.026	0.043	0.548	0.014	0.038	0.702
DIABETE	−0.139	0.045	0.002	−0.253	0.041	<0.001	−0.163	0.057	<0.005	−0.172	0.049	<0.001
HYPERT	−0.223	0.033	<0.001	−0.236	0.029	<0.001	−0.217	0.041	<0.001	−0.224	0.035	<0.001
EDUC	0.000	0.002	0.858	0.001	0.001	0.418	0.001	0.002	0.549	0.000	0.002	0.873
AGE	0.020	0.002	<0.001	0.018	0.002	<0.001	0.019	0.002	<0.001	0.019	0.002	<0.001

For the missing data model, we carry out standard diagnostic tests for the fit of regression models by comparing a model with an expanded model to do a model selection. Here, we only list the results for the final model without reporting the tables due to limited space. Significance of the previous missing indicator indicates that there exists strong series dependence; sex, CVCHF, MMSE, FHDEM, diabetes, hypertension, education, age, the previous observed response, and the previous observed depression status are also significant in some missing indicator models, indicating that the MCAR are not feasible. Therefore, we can draw the conclusion that the CC analysis can draw a wrong conclusion about a relationship between a risk factor and the occurrence of AD. Our doubly robust method may have a very good chance to reduce the biases of the estimate if the MAR assumption is true. Rejecting the MCAR may not be sufficient to say the missing data mechanism is MAR because it can be MNAR. For the MNAR mechanism, we comment on it in the next section.

6. Discussion

The consistent estimators of longitudinal data analysis with both missing response and missing covariates under MAR generally depend on the correct specification of the missing data model or the covariate model. Likelihood-based methods are robust to the misspecification of the missing data process model, while the weighted estimating equation method is robust to the misspecification of the covariate model. In this article, we develop a doubly robust estimation method, which is robust to the misspecification of either the missing data model or the covariate model, but not both. Simulation studies have shown that, subject to the correct specification of the response model, the estimators are consistent and empirical studies have shown that there is negligible bias in finite samples, when the missing data or the covariate model is correctly specified.

The asymptotic studies have provided insight into the nature of the biases one can expect with different types of model misspecification, which suggests that there is a very good chance that our proposed method will reduce the bias with which the covariate model is misspecified and the missing data model is approximately correct. Use of model diagnostics for the missing data process, perhaps most easily carried out in the MAR setting through model expansion, is warranted. It appears that empirically there is often little price to pay for introducing additional covariates into the missing data regression models. This is comforting because the more comprehensive the missing data model the more plausible it is since there is no residual dependence on the missing response, say.

The doubly robust methods can be unstable if weights are extreme and will be wrong if both models are misspecified. Furthermore, it can also draw the wrong conclusion if the data are not MAR. In many applications, however, data may not be sufficient to distinguish the MAR from MNAR mechanism. Actually, it is generally not possible to check formally for the presence of a MNAR mechanism, so sensitivity analysis is required if this is a serious concern. Scharfstein et al. (1999); Rotnitzky, Robins, and Scharfstein (1998); Scharfstein and Irizarry (2003); Robins, Rotnitzky, and Scharfstein (2000); and Robins and Rotnitzky (2001) each discuss strategies for conducting sensitivity analyses for marginal semiparametric methods for incomplete data. For other approaches, several authors have proposed the use of global and local influence tools to sensitivity analyses in missing data contexts (e.g., Kenward, 1998; Verbeke and Molenberghs, 2000; Molenberghs, Kenward, and Goetghebeur, 2001; Van Steen, Molenberghs, and Thjis, 2001; Verbeke et al., 2001; Zhu and Lee, 2001; Jansen et al., 2003; Molenberghs and Verbeke, 2005). Another route for sensitivity analysis is to consider pattern-mixture models as a complement to selection models. Other approaches have been considered by Copas and Li (1997), Copas and Shi (2000), and Copas and Eguchi (2001). More details on some of these procedures can be found in Molenberghs and Verbeke (2005), and Little and Rubin (2002).

We focused here primarily on estimation and inference regarding one covariate that is subject to missingness. Multiple covariates subject to missing are very common in practice. A future research is to extend this method to the multiple missing covariates problem. The idea is that we build missing data models to construct the weights in the weighted estimating equations, and we also need to build joint models for the covariates that are subject to missingness, which is challenging in practice, especially for missing covariates with both continuous and categorical.

Appendix: Proof of the Doubly Robust Estimation Property and Theorem 1

Proof of the doubly robust estimation property

Let $\mathbf{S}(\theta )={({\mathbf{S}}_1^T(\theta ),{\mathbf{S}}_2^T(\gamma ),{\mathbf{S}}_3^T(\alpha ))}^T$, then using the first Taylor series expansion, it can be shown that

$$n^{1/2}(\widehat{\theta }-\theta )\approx n{\left[-\frac{\partial E\{\mathbf{S}(\theta )\}}{\partial {\theta }^T}\right]}^{-1}{n}^{-1/2}\mathbf{S}(\theta ),$$

which implies that

$$n^{1/2}(\widehat{\beta }-\beta )\approx n{\mathbf{I}}^{11}{n}^{-1/2}{\mathbf{S}}_1(\theta )+n{\mathbf{I}}^{12}{n}^{-1/2}{\mathbf{S}}_2(\gamma )+n{\mathbf{I}}^{13}{n}^{-1/2}{\mathbf{S}}_3(\alpha ),$$ (A1)

where I¹¹, I¹², and I¹³ are the appropriate submatrices of [−∂E{S(θ)}/∂θ^T ]⁻¹.

1. Missing data model is correctly specified

   Suppose the missing data model and p(y_i | x_i, z_i) are correctly specified, but the distribution of (x_i | z_i, ${\mathbf{y}}_i^o$) is misspecified. Then, in (10) we rewrite $E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i,{\mathbf{R}}_i)}[·]$ and $E_{({\mathbf{X}}_i^m\mid {\mathbf{X}}_i^o,{\mathbf{Y}}_i^o,{\mathbf{Z}}_i,{\mathbf{R}}_i)}[·]$ as $E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i)}^{\ast }[·]$ and $E_{({\mathbf{X}}_i^m\mid {\mathbf{X}}_i^o,{\mathbf{Y}}_i^o,{\mathbf{Z}}_i)}^{\ast }[·]$ because of the MAR assumption, where the subscript “*” represents the expectation taken over the wrongly specified distribution for ( ${\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i$) and ${\mathbf{X}}_i^m\mid {\mathbf{X}}_i^o,{\mathbf{Y}}_i^o,{\mathbf{Z}}_i$.

   If the missing data model is correctly specified, we have E(δ_ijk) = 1, and thus E(Δ_i) = , $E({\mathbf{M}}_i)={\mathbf{V}}_i^{-1}$, and E(N_i) = 0. Therefore, we have $E\{{\mathbf{D}}_i{\mathbf{M}}_i({\mathbf{Y}}_i-{\mu }_i)\}=E\{{\mathbf{D}}_i{\mathbf{V}}_i^{-1}({\mathbf{Y}}_i-{\mu }_i)\}=\mathbf{0}$, and $E[E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i,{\mathbf{R}}_i)}^{\ast }\{{\mathbf{D}}_i{\mathbf{N}}_i({\mathbf{Y}}_i-{\mu }_i)\}]=E[E_{({\mathbf{Y}}_i^m,{\mathbf{x}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i)}^{\ast }\{{\mathbf{D}}_i·\mathbf{0}·({\mathbf{Y}}_i-{\mu }_i)\}]=\mathbf{0}$, if the distribution of (y_i | x_i, z_i) is correctly specified. That means E{S₁(θ)} = 0. It is easy to show that E{S₃(α)} = 0.

   Now if the distribution of (x_i | z_i, ${\mathbf{y}}_i^o$) is incorrectly specified, then E{S₂(γ)} ≠ 0. However, the second term on the right-hand side of (A1) still has expectation 0. Using the theory of partitioned matrices, it can be shown that I¹² = 0 if E{∂S₁(θ)/∂γ^T } = 0. Note that the first term of S₁(θ) does not depend on γ, so the derivative is equal to 0, hence

   $$\begin{array}{l}E\left[\frac{\partial {\mathbf{S}}_{1i}(\theta )}{\partial {\gamma }^T}\right]=E\left[\frac{\partial E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i)}^{\ast }{E}_{({\mathbf{R}}_i\mid {\mathbf{Y}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i)}\{{\mathbf{D}}_i{\mathbf{N}}_i({\mathbf{Y}}_i-{\mu }_i)\}}{\partial {\gamma }^T}\right]\\ =E\left[\frac{\partial E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i)}^{\ast }\{\mathbf{0}\}}{\partial {\gamma }^T}\right]\\ =\mathbf{0}.\end{array}$$

   Thus all the terms on the right-hand side of (A1) have expectation 0, and β̂ is asymptotically unbiased.

2. $p({\mathbf{x}}_i\mid {\mathbf{z}}_i,{\mathbf{y}}_i^o)$ correctly specified

   Suppose that the distribution of (y_i | x_i, z_i), and (x_i | z_i, ${\mathbf{y}}_i^o$) are correctly specified but the missing data indicator model is incorrectly specified. To be specific, suppose that π_ijk is misspecified as ${\pi }_{\mathit{ijk}}^{\ast }$, and π_ij is misspecified as ${\pi }_{ij}^{\ast }$, which are still functions of x_i, y_i, and z_i. We define Δ_i = [δ̃_{i j k} ] with ${\stackrel{\sim }{\delta }}_{\mathit{ijk}}=\{P(R_{ij}=1,R_{ik}=3\mid {\mathbf{Y}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i)+P(R_{ij}=3,R_{ik}=3\mid {\mathbf{Y}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i)\}/{\pi }_{\mathit{ijk}}^{\ast }$ for k ≠ j and ${\stackrel{\sim }{\delta }}_{\mathit{ijj}}=P(R_{ij}=3\mid {\mathbf{Y}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i)/{\pi }_{ij}^{\ast }$. We show that E{S₁(θ)} = 0, E {S₂(γ)} = 0 and I¹³ = 0, implying that each term on the right-hand side of (A1) has 0 expectation and β̂ is asymptotically unbiased. Note that expectation of the first term of S_1i (θ) is

   $$E\{{\mathbf{D}}_i{\mathbf{M}}_i({\mathbf{Y}}_i-{\mu }_i)\}=E\{{\mathbf{D}}_i{\mathbf{F}}_i^{-1/2}({\mathbf{C}}_i^{-1}•{\stackrel{\sim }{\mathbf{\Delta }}}_i){\mathbf{F}}_i^{-1/2}({\mathbf{Y}}_i-{\mu }_i)\}.$$
   If both p(y_i | x_i, z_i) and $p({\mathbf{x}}_i\mid {\mathbf{z}}_i,{\mathbf{y}}_i^o)$ are correctly specified, then the joint probability $p({\mathbf{y}}_i^m,{\mathbf{x}}_i^m\mid {\mathbf{y}}_i^o,{\mathbf{x}}_i^o,{\mathbf{z}}_i)$ is correctly specified, and hence the expectation of the second term of S_1i (θ) is

   
   Thus we have

   $$\begin{array}{l}E\{{\mathbf{S}}_{1i}(\theta )\}=E\{{\mathbf{D}}_i{\mathbf{M}}_i({\mathbf{Y}}_i-{\mu }_i)\}+E\{{\mathbf{D}}_i{\mathbf{N}}_i({\mathbf{Y}}_i-{\mu }_i)\}\\ =E\{{\mathbf{D}}_i{\mathbf{F}}_i^{-1/2}{\mathbf{C}}_i^{-1}{\mathbf{F}}_i^{-1/2}({\mathbf{Y}}_i-{\mu }_i)\}\\ =E\{{\mathbf{D}}_i{\mathbf{V}}_i^{-1}({\mathbf{Y}}_i-{\mu }_i)\}\\ =\mathbf{0},\end{array}$$

   if the distribution of (y_i | x_i, z_i) is correctly specified. Similarly, we can prove that E[S_2i (γ)] = 0.

   If the missing data model is misspecified, then E{S₃(α)} ≠ 0. However, the third term on the right-hand side of (A1) still has expectation 0 if I¹³ = 0. By using the theory of partitioned matrices, we can show that I¹³ = 0 if E{δS₁(θ)/∂αT} = 0. Note that

   $$\begin{array}{l}E\left\{\frac{\partial {\mathbf{S}}_{1i}(\theta )}{\partial {\alpha }_j^T}\right\}=E\left\{{\mathbf{D}}_i{\mathbf{F}}_i^{-1/2}\left({\mathbf{C}}_i^{-1}•\frac{\partial {\mathbf{\Delta }}_i}{\partial {\alpha }_j^T}\right){\mathbf{F}}_i^{-1/2}({\mathbf{Y}}_i-{\mu }_i)\right\}\\ +E\left\{E_{({\mathbf{Y}}_i^m,{\mathbf{X}}_i^m\mid {\mathbf{Y}}_i^o,{\mathbf{X}}_i^o,{\mathbf{Z}}_i)}\left[{\mathbf{F}}_i^{-1/2}\left\{{\mathbf{C}}_i^{-1}•\left(-\frac{\partial {\mathbf{\Delta }}_i}{\partial {\alpha }_j^T}\right)\right\}\times {\mathbf{F}}_i^{-1/2}({\mathbf{Y}}_i-{\mu }_i)\right]\right\}=\mathbf{0},\end{array}$$

   for j = 1, …, p₃, where p₃ = dim(α). Then all the three terms on the right-hand side of (A1) have expectation 0, and β̂ is asymptotically unbiased if the distribution of (y_i | x_i, z_i) and (x_i | z_i, ${\mathbf{y}}_i^o$) are correctly specified.

Proof of Theorem 1

The regularity conditions required in Theorem 1 include standard conditions that are assumed for the estimating function theory, plus the requirement for the missing data processes and covariate process. Specifically, we require P (R_{i j} = 3 | R̄_ij, Y_i, X_i, Z_i) is bounded away from zero. This condition ensures that the estimating functions in (8) are bounded, which is necessary for a $\sqrt{n}$-consistent estimator. Other routine conditions are similar to those in Robins et al. (1995) with a proper modification.

By standard Taylor expansion arguments we have that

$$n^{1/2}(\widehat{\gamma }-{\gamma }_0)=-{\mathbf{I}}_2^{-1}({\gamma }_0)n^{-1/2}\sum _{i=1}^n{\mathbf{S}}_{2i}({\gamma }_0)+o_p(1),$$ (A2)

and

$$n^{1/2}(\widehat{\alpha }-{\alpha }_0)=-{\mathbf{I}}_3^{-1}({\alpha }_0)n^{-1/2}\sum _{i=1}^n{\mathbf{S}}_{3i}({\alpha }_0)+o_p(1).$$ (A3)

Furthermore, based on the proof of the doubly robust properties, we have E[S_1i (β₀, γ₀, α₀)] = 0 if either the missing data model or the covariate model is correctly specified. Thus, another Taylor expansion gives

$$\mathbf{0}=n^{-1/2}\sum _{i=1}^n{\mathbf{S}}_{1i}({\beta }_0,{\gamma }_0,{\alpha }_0)+\mathbf{\Gamma }({\beta }_0,{\gamma }_0,{\alpha }_0)n^{1/2}(\widehat{\beta }-{\beta }_0)+{\mathbf{I}}_{12}({\beta }_0,{\gamma }_0,{\alpha }_0)n^{1/2}(\widehat{\gamma }-{\gamma }_0)+{\mathbf{I}}_{13}({\beta }_0,{\gamma }_0,{\alpha }_0)n^{1/2}(\widehat{\alpha }-{\alpha }_0)+o_p(1).$$

Replacing (A2) and (A3) into (A4), we obtain

$$\mathbf{0}=n^{-1/2}\sum _{i=1}^n{\mathbf{S}}_{1i}({\beta }_0,{\gamma }_0,{\alpha }_0)+\mathbf{\Gamma }({\beta }_0,{\gamma }_0,{\alpha }_0)n^{1/2}(\widehat{\beta }-{\beta }_0)-{\mathbf{I}}_{12}({\beta }_0,{\gamma }_0,{\alpha }_0){\mathbf{I}}_2^{-1}({\gamma }_0)n^{-1/2}\sum _{i=1}^n{\mathbf{S}}_{2i}({\gamma }_0)-{\mathbf{I}}_{13}({\beta }_0,{\gamma }_0,{\alpha }_0){\mathbf{I}}_3^{-1}({\alpha }_0)n^{-1/2}\sum _{i=1}^n{\mathbf{S}}_{3i}({\alpha }_0)+o_p(1).$$

If Γ(β₀, γ₀, α₀) is nonsingular, we have

$$n^{1/2}(\widehat{\beta }-{\beta }_0)=-{\mathbf{\Gamma }}^{-1}({\beta }_0,{\gamma }_0,{\alpha }_0)n^{-1/2}\sum _{i=1}^n{\mathbf{Q}}_i({\beta }_0,{\gamma }_0,{\alpha }_0)+o_p(1).$$

Then the asymptotic distribution of n^1/2(β̂ − β₀) follows by Slutsky’s theorem and the central limit theorem.