Complete-Case GEE Under MCAR

Why the Missingness Pattern Matters

GEE uses a marginal mean model while allowing responses within a cluster to be correlated. When some outcomes are missing, the estimating equation must be built from the observed components. Whether this still targets the intended mean depends on how observation is related to the data. Under missing completely at random (MCAR), the observed pattern does not favor particular covariates or outcomes, so a correctly specified mean model can remain the target of complete-case GEE. The price may be reduced precision. If observation depends on the unobserved outcome itself, however, the retained responses can have a different mean from the population responses.

Here, a cluster is the independent sampling unit, such as a patient with repeated visits. “Complete-case GEE” will mean that the estimating equation uses the observed outcomes within each cluster and excludes missing outcome components. Some analyses instead retain only clusters with every outcome observed. That stricter whole-cluster deletion is a special case; its precision cost is especially easy to quantify.

Observed-Pattern GEE and MCAR

Definition: For cluster \(i\), let \(Y_i\) be the vector of all scheduled outcomes, \(X_i\) the covariates, and \(R_i\) the indicator pattern specifying which outcomes are observed. MCAR means \(R_i\) is independent of the full data \((X_i,Y_i)\). Write \(O_i\) for the observed components. The complete-case GEE contribution is \(U_i(\beta)=D_{i,O_i}(\beta)^T V_{i,O_i}(\beta,\alpha)^{-1}\{Y_{i,O_i}-\mu_{i,O_i}(\beta)\}\). Here, \(\mu_i(\beta)=E(Y_i\mid X_i)\), \(D_i(\beta)=\partial\mu_i(\beta)/\partial\beta^T\), and \(V_{i,O_i}\) is the submatrix of the working covariance corresponding to the observed outcomes. The equation is \(\sum_{i=1}^m U_i(\beta)=0\). The inverse is taken after selecting the observed submatrix; it is not generally obtained by selecting entries from \(V_i^{-1}\).

Assume the marginal mean model is correct at \(\beta_0\): \(E(Y_i\mid X_i)=\mu_i(\beta_0)\). Under MCAR, conditional on \(X_i\) and any observation pattern with positive probability, the conditional distribution of \(Y_i\) is unchanged. Thus \(E(Y_{i,O_i}-\mu_{i,O_i}(\beta_0)\mid X_i,R_i)=0\). The matrix multiplying that residual may depend on \(X_i\) and \(R_i\), but not on the realized outcomes. This is the key reason the population estimating equation remains centered at \(\beta_0\). The working covariance need not equal the true covariance for this centering argument.

Worked Example: A Two-Visit Cluster With Partial Observation

Suppose each cluster has two outcomes with common marginal mean \(\beta\), and use the working covariance \(V=\begin{pmatrix}1&0.5\\0.5&1\end{pmatrix}\). Assume the mean model is correct and the observed pattern is MCAR. Find the GEE contribution when both visits are observed and when only visit 1 is observed, and verify that each has expectation zero at \(\beta_0\).

When both visits are observed, \(V^{-1}=\frac{4}{3}\begin{pmatrix}1&-0.5\\-0.5&1\end{pmatrix}\), and the derivative of the mean vector with respect to \(\beta\) is \(D=(1,1)^T\). Therefore \(D^TV^{-1}=(2/3,2/3)\), giving \(U_i(\beta)=\frac{2}{3}\{(Y_{i1}-\beta)+(Y_{i2}-\beta)\}\). At \(\beta_0\), each residual has conditional expectation zero given \(X_i\) and the observation pattern, so \(E\{U_i(\beta_0)\mid X_i,R_i\}=0\).

If only visit 1 is observed, the observed covariance is the one-by-one matrix \((1)\), and the observed derivative is \((1)\). The contribution is \(U_i(\beta)=Y_{i1}-\beta\), whose conditional expectation at \(\beta_0\) is also zero. The same reasoning applies to a visit-2-only pattern. These contributions need not have the same variance, but each pattern is centered at the target under MCAR.

Consistency: The Population Equation

Theorem: Suppose clusters are independent and identically distributed; cluster sizes and the number of possible observation patterns are bounded; the marginal mean model is correct at \(\beta_0\); and \(R_i\) is independent of \((X_i,Y_i)\). Suppose the observed working covariance matrices are nonsingular, the estimating functions have finite first moments, and the population map \(E\{U_i(\beta)\}\) is continuous on a compact parameter space \(\Theta\) containing \(\beta_0\), with \(\beta_0\) as its unique zero on \(\Theta\). Suppose also that the sample estimating equations satisfy a uniform law of large numbers over \(\Theta\), and that \(\hat\beta\in\Theta\) is a root of the sample equation. Then the complete-case GEE estimator is consistent: \(\hat\beta\xrightarrow{p}\beta_0\). These conditions also require that observation patterns needed to identify the parameters occur with positive probability.

The statement allows the working covariance to be misspecified, provided its observed submatrices are well-defined and the resulting population equation identifies \(\beta_0\). If a working-correlation parameter is estimated, it must converge to a finite limiting value and the score must vary regularly with that parameter; the argument applies to the limiting estimating function. A correct mean model alone is not enough if the observed patterns fail to identify its parameters.

Condition on the observed pattern.
Under MCAR, \(Y_i\) and \(R_i\) are independent. Correct specification gives \(E\{Y_i-\mu_i(\beta_0)\mid X_i,R_i\}=0\), and hence the same zero residual mean for the observed components.

Take the conditional expectation of the score.
\(D_{i,O_i}^TV_{i,O_i}^{-1}\) is determined by \(X_i\) and \(R_i\). Therefore \(E\{U_i(\beta_0)\mid X_i,R_i\}=D_{i,O_i}^TV_{i,O_i}^{-1}0=0\), and taking expectations gives \(E\{U_i(\beta_0)\}=0\).

Apply the estimating-equation consistency argument.
Continuity and the unique zero on the compact parameter space imply that, for every \(\epsilon>0\), the norm of \(E\{U_i(\beta)\}\) is bounded away from zero on \(\{\beta\in\Theta:\|\beta-\beta_0\|\ge\epsilon\}\). Uniform convergence then rules out a sample-equation root in that set with probability tending to one, so \(\hat\beta\xrightarrow{p}\beta_0\). \(\square\)

This establishes consistency, not full efficiency. The working covariance affects how residuals are weighted and can affect the estimator’s asymptotic variance. Inference should account for clustering and the missingness pattern; the robust sandwich covariance used for GEE does not itself correct bias from an observation mechanism that shifts the population estimating equation.

Precision Loss From Independent Whole-Cluster Deletion

A simple exact comparison is available when each cluster is either fully retained or entirely discarded. Let \(R_i\) be 1 if cluster \(i\) is retained and 0 otherwise, with \(P(R_i=1)=q>0\), independently of the full data and independently across clusters. Let \(U_i(\beta)\) be the full-data cluster contribution, with \(E\{U_i(\beta_0)\}=0\). Define \(A=-E\{\partial U_i(\beta_0)/\partial\beta^T\}\) and \(B=\operatorname{Var}\{U_i(\beta_0)\}\), with \(A\) nonsingular.

Theorem: Under the usual differentiability and finite-moment conditions for the GEE sandwich limit, the asymptotic covariance of \(\sqrt m(\hat\beta-\beta_0)\) with independent whole-cluster retention is \(q^{-1}A^{-1}BA^{-T}\). Thus, relative to using all \(m\) clusters, the asymptotic covariance is multiplied by \(1/q\), and standard errors are multiplied by \(1/\sqrt q\).

To derive the result, the retained-data equation has contribution \(R_iU_i(\beta)\). Independence gives \(E\{R_iU_i(\beta_0)\}=qE\{U_i(\beta_0)\}=0\). Its negative expected derivative is \(qA\), while its score variance is \(E(R_i^2)B=qB\), since \(R_i^2=R_i\) and the score has mean zero. The sandwich covariance is therefore \((qA)^{-1}(qB)(qA)^{-T}=q^{-1}A^{-1}BA^{-T}\). This compares analyses based on the same planned number \(m\) of clusters. Equivalently, roughly \(qm\) clusters remain, so the increase in uncertainty is the usual loss from having fewer independent clusters. \(\square\)

Worked Example: Retaining 81 Percent of Clusters

Suppose whole clusters are independently retained with probability \(q=0.81\). If the full-data standard error for a coefficient, based on the planned sample size, is \(0.09\), calculate the asymptotic standard error and variance multipliers under complete-case deletion.

The variance multiplier is \(1/q=1/0.81=1.234567\ldots\), which rounds to \(1.235\). The standard-error multiplier is \(1/\sqrt q=1/\sqrt{0.81}=1/0.9=1.111111\ldots\), which rounds to \(1.111\). Thus the standard error is approximately \(0.09(1.111111)=0.10\). Checking by variance, the full-data variance is \(0.09^2=0.0081\), and the retained-data variance is \(0.0081/0.81=0.0100\), whose square root is \(0.10\). The variance multiplier is not obtained by squaring the rounded standard-error multiplier: \(1.111^2=1.234321\), whereas the exact multiplier squared is \(1/0.81\approx1.235\).

For partial visit-level missingness, there is generally no single retention probability that captures the whole precision change. Different observation patterns contribute different amounts of information, and within-cluster correlation affects how much information a missing visit removes. MCAR can preserve the target while still producing a substantial efficiency loss.

When Outcome-Dependent Observation Changes the Target

MCAR is sufficient for the conditional residual-centering argument above. If observation depends on the outcome, conditioning on being observed can change the outcome distribution, and the complete-case estimating equation need not have its root at the intended parameter. This is not a claim that every non-MCAR mechanism necessarily causes bias; rather, without additional assumptions, complete-case GEE is not protected by the MCAR argument.

Worked Example: A Binary Outcome More Likely to Be Observed When Positive

Let \(Y\) be Bernoulli with \(P(Y=1)=0.5\), and let \(R=1\) indicate that \(Y\) is observed. Suppose \(P(R=1\mid Y=1)=0.9\) and \(P(R=1\mid Y=0)=0.5\). A complete-case intercept model estimates the mean by the observed proportion. Find its population target and compare it with the full-data mean.

The joint probabilities of being observed and having each outcome are \(P(R=1,Y=1)=0.5(0.9)=0.45\) and \(P(R=1,Y=0)=0.5(0.5)=0.25\). Hence \(P(R=1)=0.45+0.25=0.70\), and \(E(Y\mid R=1)=P(Y=1\mid R=1)=0.45/0.70=9/14\approx0.643\). The full-data mean is \(E(Y)=0.5\).

The complete-case estimating equation for an intercept \(p\) is \(E\{R(Y-p)\}=0\). Its population left side is \(0.45-0.70p\), so its root is \(p=0.45/0.70=9/14\), not \(0.5\). The observed cases therefore target the selected mean in this example. A robust sandwich standard error around the complete-case estimate would not remove this shift in the estimating-equation root.

The diagnostic question is not merely how many outcomes are missing, but whether the observation mechanism can select on the outcomes after accounting for the variables in the analysis. MCAR rules out selection related to both outcomes and covariates. A mechanism that depends only on recorded covariates is different: in some correctly specified conditional mean models, complete-case estimation can still identify a regression parameter, provided the retained covariate distribution supplies adequate support. That possibility does not make MCAR irrelevant; it requires a separate argument about the estimand and model. In particular, it should not be confused with the unconditional guarantee proved here.

Common Mistakes and Practical Interpretation

Assuming a robust sandwich fixes selection bias. The sandwich estimates uncertainty around the root of the estimating equation. If outcome-dependent observation moves that root, robust standard errors do not restore the full-data target.
Equating MCAR with no precision loss. MCAR can preserve the target while reducing the number of observations and changing the observed within-cluster patterns. Precision is generally lower.
Using a full-data inverse covariance after dropping outcomes. For each observed pattern, select the corresponding covariance submatrix and then invert it. The inverse of a submatrix is generally not the corresponding submatrix of the full inverse.
Calling every complete-case analysis the same procedure. Deleting only missing outcome components and deleting every cluster with any missing outcome are different analyses. The whole-cluster variance multiplier \(1/q\) applies to the independent all-or-none retention setup derived above, not automatically to arbitrary visit-level missingness.

Key takeaway: Under MCAR, correct marginal mean specification, adequate identification, and standard estimating-equation regularity conditions, observed-pattern GEE remains centered at the intended parameter. Independent deletion can inflate variance; observation depending on the outcome can instead change the target itself.

Check Your Understanding

Use the definitions and derivations above to answer these questions.

Why does \(E\{U_i(\beta_0)\mid X_i,R_i\}=0\) under MCAR even when different clusters have different observed patterns?
In the whole-cluster retention result, what are the multipliers for variance and standard error when \(q=0.64\)?
Why must the observed covariance submatrix be inverted after selecting the observed outcomes?
In the binary-outcome example, calculate the complete-case estimating-equation root directly from \(E\{R(Y-p)\}=0\).
Explain why a sandwich covariance estimate can be valid for uncertainty around a biased complete-case target.

What You'll Learn