Missingness and Marginal GEE Estimands

Why Missingness Changes the Question

A marginal GEE describes a population mean, such as the mean outcome at each visit conditional on covariates. When some repeated outcomes are missing, the outcomes that enter the estimating equation may no longer represent that population. The central issue is therefore not only how many measurements are absent, but whether response depends on the outcomes themselves.

It helps to separate two questions. The first is the estimand: which population regression parameter is intended? The second is the estimating procedure: do the observed outcomes provide an estimating equation whose expectation is zero at that parameter? A method can calculate a number from the observed data without that number estimating the intended full-data target.

Full-Data and Observed-Data Targets

Suppose independent clusters \(i=1,\ldots,n\) have \(m\) scheduled outcomes. Write \(Y_i=(Y_{i1},\ldots,Y_{im})^T\), let \(X_i\) denote the covariates, and let \(R_{ij}=1\) if \(Y_{ij}\) is observed and \(R_{ij}=0\) otherwise. The full-data marginal mean model is

g\{E(Y_{ij}\mid X_i)\}=x_{ij}^T\beta_0,\qquad j=1,\ldots,m.

Here \(g\) is the link function and \(x_{ij}\) is the covariate vector for occasion \(j\). The parameter \(\beta_0\) describes the marginal mean in the target population of clusters, not only among measurements that happen to be observed. Assume for now that this mean model is correctly specified and that \(\beta_0\) is identifiable.

For complete outcomes, a GEE uses \(D_i=\partial\mu_i/\partial\beta^T\), where \(\mu_i=E(Y_i\mid X_i)\), and a working covariance matrix \(V_i\). Its contribution is \(D_i^T V_i^{-1}(Y_i-\mu_i)\). With missing outcomes, the observed-data contribution is formed from the observed components and their corresponding derivative and covariance submatrices. The observed-data estimating equation therefore uses a response-dependent subset of the full vector.

Definition: The full-data marginal estimand is the parameter \(\beta_0\) in the specified marginal mean model for the target population, whether or not every outcome is observed. An observed-data estimand is the parameter solving the population version of a specified estimating equation built from observed outcomes. It need not equal \(\beta_0\).

The observed-data estimand depends on the estimating equation, the response process, and the covariate distribution. If response changes which outcomes contribute in a way related to their unobserved values, the equation can be systematically shifted. In that case, a large sample does not by itself make the ordinary GEE estimate the full-data target.

Missingness Assumptions and Ordinary GEE

Missingness assumptions describe how the response indicators \(R_i=(R_{i1},\ldots,R_{im})\) relate to the complete outcomes and covariates. The definitions below refer to the joint response pattern; they apply whether measurements are missing intermittently or after dropout.

Definition: Missing completely at random (MCAR) means \(R_i\) is independent of \((Y_i,X_i)\). More generally, response may depend on covariates without depending on outcomes: \(R_i\) is independent of \(Y_i\) conditional on \(X_i\). Missing at random (MAR) means the conditional distribution of \(R_i\), given \(Y_i\) and \(X_i\), depends on \(Y_i\) only through the components observed under that response pattern. Missing not at random (MNAR) means response still depends on unobserved outcomes after conditioning on observed outcomes and covariates.

Terminology for covariate-dependent response varies: it is not strict MCAR, but it is outcome-independent conditional on the modeled covariates. That distinction matters. If the conditional mean model is correct given \(X_i\), ordinary GEE can have an unbiased estimating equation under this conditional independence, even when response rates differ across covariate values. By contrast, MAR that depends on previously observed outcomes does not generally preserve the ordinary GEE equation’s mean-zero property.

Result: Suppose clusters are independent and identically distributed, the marginal mean model is correct, and the full-data parameter \(\beta_0\) is identifiable. If, conditional on \(X_i\), the response pattern is independent of \(Y_i\), then the ordinary observed-component GEE contribution has expectation zero at \(\beta_0\), provided its weights and covariance submatrices are functions of \(X_i\) and the response pattern (and are well-defined). Thus, under standard regularity conditions for estimating equations, its population root is \(\beta_0\).

Proof. Condition first on \(X_i\) and on a particular response pattern \(R_i=r\). Let \(S_r\) be the matrix selecting the observed components. The observed residual vector is \(S_r(Y_i-\mu_i)\); the corresponding observed derivative and working covariance are denoted \(D_{ir}\) and \(V_{ir}\). The contribution is

U_i(\beta_0)=D_{ir}^T V_{ir}^{-1}S_r(Y_i-\mu_i).

Conditional independence gives \(E(Y_i\mid X_i,R_i=r)=E(Y_i\mid X_i)=\mu_i\). Since \(D_{ir}\) and \(V_{ir}\) are fixed given \(X_i\) and \(r\),

E\{U_i(\beta_0)\mid X_i,R_i=r\} =D_{ir}^T V_{ir}^{-1}S_r E(Y_i-\mu_i\mid X_i,R_i=r)=0.

Taking expectation over \(R_i\) and \(X_i\) gives \(E\{U_i(\beta_0)\}=0\). This establishes the mean-zero property. Consistency additionally requires the usual conditions: a unique population root, suitable moment and smoothness conditions, and a law of large numbers for the estimating equation. The result does not claim that every response mechanism called MAR makes ordinary GEE valid; it uses the stronger condition that response is outcome-independent after conditioning on the modeled covariates. \(\square\)

Worked Example: Covariate-Dependent Response

Suppose half the target population has \(X=0\) and half has \(X=1\). The binary outcome has conditional means \(E(Y\mid X=0)=0.2\) and \(E(Y\mid X=1)=0.8\). Response probabilities are \(P(R=1\mid X=0)=0.9\) and \(P(R=1\mid X=1)=0.1\), and response is independent of \(Y\) given \(X\).

The full-population mean is

$$ E(Y)=0.5(0.2)+0.5(0.8)=0.5. $$

Among respondents, the probability of a response is \(0.5(0.9)+0.5(0.1)=0.5\). The respondent mean is

E(Y\mid R=1) =\frac{0.5(0.9)(0.2)+0.5(0.1)(0.8)}{0.5} =\frac{0.09+0.04}{0.5}=0.26.

The unadjusted respondent mean is not the full-population mean: response has changed the covariate composition. But within each covariate group, the respondents still have the specified conditional mean, since response is independent of \(Y\) given \(X\). A correctly specified regression that conditions on \(X\) targets the full-data conditional regression parameter; simply treating the respondent mean as the population mean does not.

Worked Example: MAR Can Shift an Ordinary GEE Contribution

Consider two perfectly correlated binary outcomes, \(Y_1=Y_2=Z\), where \(P(Z=1)=P(Z=0)=0.5\). Let the second outcome be observed with probability \(0.9\) when \(Y_1=1\), and probability \(0.1\) when \(Y_1=0\). Since \(Y_1\) is observed, this response mechanism is MAR. The full-data mean at visit 2 is \(E(Y_2)=0.5\).

The probability that \(Y_2\) is observed is

$$ P(R_2=1)=0.5(0.9)+0.5(0.1)=0.5. $$

Among observed second outcomes, the probability of a value of 1 is

P(Y_2=1\mid R_2=1) =\frac{P(Y_2=1,R_2=1)}{P(R_2=1)} =\frac{0.5(0.9)}{0.5}=0.9.

Thus the observed mean is \(0.9\), not the full-data mean \(0.5\). A mean-only estimating equation for visit 2 has a nonzero expected residual at the full-data mean: \(E\{R_2(Y_2-0.5)\}=0.5(0.9-0.5)=0.2\). This illustrates why MAR alone does not guarantee that ordinary observed-component GEE targets the full-data parameter. The response depends on an observed outcome associated with the missing outcome, so the observed second outcomes are selected.

What Observed Data Can Identify

When response depends on unobserved outcomes, the observed-data distribution may not determine the full-data mean. This is not just a concern about inefficient estimation: different complete-data distributions can produce exactly the same observed data while implying different target means.

Result: Without restrictions on the outcome distribution or the missingness mechanism, the full-data mean is not generally identified from observed data under MNAR.

Proof by construction. Let \(Y\) be binary, and suppose the observed data have probabilities \(P(R=1,Y=1)=0.4\), \(P(R=1,Y=0)=0.1\), and \(P(R=0)=0.5\). The observed response rate is \(0.5\), and among respondents the mean is \(0.4/0.5=0.8\). If all unobserved outcomes are zero, the full-data mean is \(0.4\). If all unobserved outcomes are one, it is \(0.4+0.5=0.9\). Both possibilities reproduce exactly the same observed-data probabilities, because \(Y\) is unrecorded when \(R=0\). The observed data therefore do not determine a unique full-data mean. \(\square\)

Worked Example: Two Full-Data Means, One Observed Distribution

Use the observed probabilities in the proof: 0.4 of all units are observed with \(Y=1\), 0.1 are observed with \(Y=0\), and 0.5 have missing outcomes. In one compatible full-data model, all missing outcomes equal zero. Then \(P(Y=1)=0.4\). In another, all missing outcomes equal one. Then \(P(Y=1)=0.4+0.5=0.9\).

In both models, the observed response rate is \(0.4+0.1=0.5\), and the respondent mean is \(0.4/(0.4+0.1)=0.8\). The data analyst sees the same observed records in either case. Yet the full-data means differ by \(0.9-0.4=0.5\). Consequently, an observed-data calculation alone cannot choose between these target values; additional assumptions or information are needed.

Practical Interpretation and a Common Mistake

Before interpreting a GEE coefficient with incomplete repeated outcomes, state its target in words: for example, the marginal change in mean outcome across visits in the population of eligible clusters, conditional on specified covariates. Then ask whether the response process preserves the estimating equation for that target. The answer depends on how response relates to both covariates and outcomes, not simply on the fraction missing.

Common mistake: “The data are MAR, so ordinary GEE is valid.” MAR permits response to depend on observed outcomes. As the binary example shows, those outcomes can predict missing outcomes and select a different observed mean.
Correction: Check the specific conditional independence needed for the estimating equation. Outcome-independent response given the modeled covariates supports its mean-zero property; MAR based on observed outcomes does not guarantee it.
Another mistake: Calling the respondent mean the population mean when response rates differ by covariate group. The respondent covariate distribution may differ from the target distribution, even if response is independent of outcomes within covariate strata.
Interpretive safeguard: Separate the target population, the marginal mean model, and the response assumptions. A clearly stated estimand makes it possible to assess whether the analysis supports the intended conclusion.

Missingness assumptions are substantive claims, not properties that can generally be verified from observed outcomes alone. They determine whether the ordinary observed-component GEE is estimating the intended full-data marginal parameter or a response-selected alternative.

Check Your Understanding

For each question, identify the target or assumption at issue.

State the difference between a full-data marginal estimand and an observed-data estimand.
Why can ordinary GEE have a mean-zero estimating equation when response depends on covariates but is independent of outcomes conditional on those covariates?
In the two-visit example, why does MAR fail to ensure that the observed mean at visit 2 equals its full-data mean?
Can two full-data distributions have the same observed-data distribution but different outcome means under unrestricted MNAR? Explain.
A study reports that 20% of scheduled outcomes are missing. Is that percentage alone enough to establish that ordinary GEE targets the full-data mean? Why or why not?