Unbiased GEE Under Working-Correlation Misspecification

Why a Wrong Working Correlation Can Still Give a Centered Equation

GEE uses a working covariance to weight a cluster’s residual vector. That weighting can be wrong about within-cluster dependence and still yield an estimating function whose expectation is zero at the true regression parameter. The reason is specific: if the marginal mean is correct, the residual vector has conditional expectation zero, and multiplying it by a suitable covariate-dependent matrix preserves that zero.

This statement concerns the estimating function, not whether a finite-sample solution of the estimating equation is itself an unbiased estimator. The distinction matters: a centered population equation is a key property of GEE, but by itself it does not establish that a particular root exists, is unique, or is unbiased.

Cluster Contributions and the Mean Assumption

For cluster $i$, let $Y_i$ be its $m_i$-vector of responses and $X_i$ its observed covariates and design information. Write $\mu_i(\beta)=E[Y_i\mid X_i]$ for the marginal mean vector under parameter $\beta$, and let $D_i(\beta)=\partial\mu_i(\beta)/\partial\beta^T$, an $m_i$-by-$p$ matrix. A GEE cluster contribution is

U_i(\beta;\alpha)=D_i(\beta)^T V_i(\beta,\alpha)^{-1}\{Y_i-\mu_i(\beta)\}.

Here $V_i$ is the working covariance matrix, possibly constructed from working variances and a working correlation parameter $\alpha$. The full estimating function is $U(\beta;\alpha)=\sum_{i=1}^n U_i(\beta;\alpha)$. This tutorial treats $\alpha$ as fixed, or as a fixed nuisance value at which the population equation is evaluated.

Definition: The marginal mean is correctly specified at $\beta_0$ if $E[Y_i\mid X_i]=\mu_i(\beta_0)$ almost surely for every cluster $i$. The GEE is unbiased at $\beta_0$ (or centered there) if $E[U(\beta_0;\alpha)]=0$. This definition does not say that a solution $\hat\beta$ has expectation $\beta_0$.

The result needs a few conditions. Clusters are independent, and the conditional mean model holds cluster by cluster. The matrices $D_i(\beta_0)$ and $V_i(\beta_0,\alpha)$ are measurable functions of $X_i$ (and fixed model quantities), not functions of that cluster’s observed residuals. Each $V_i$ is finite and nonsingular almost surely, and the contribution $U_i(\beta_0;\alpha)$ is integrable so its expectation exists. These conditions allow the working covariance to be wrong; they do not allow it to be undefined or to use the outcome in an uncontrolled way.

The Centering Result

Theorem: Under the conditions above, for every cluster $i$, $$ E[U_i(\beta_0;\alpha)\mid X_i]=0 \quad\text{almost surely}, $$ and consequently $E[U_i(\beta_0;\alpha)]=0$. This holds for any fixed working covariance $V_i$ satisfying the measurability, nonsingularity, and integrability conditions, whether or not it is the true covariance.

Proof. At $\beta_0$, the matrices $D_i(\beta_0)^T V_i(\beta_0,\alpha)^{-1}$ are functions of $X_i$. The conditional mean assumption gives

E[Y_i-\mu_i(\beta_0)\mid X_i] =E[Y_i\mid X_i]-\mu_i(\beta_0)=0.

For an integrable random vector $Z$ and an $X_i$-measurable matrix $A$ for which the products are integrable, conditional expectation satisfies $E[AZ\mid X_i]=A E[Z\mid X_i]$, entry by entry. Apply this identity with $A=D_i(\beta_0)^T V_i(\beta_0,\alpha)^{-1}$ and $Z=Y_i-\mu_i(\beta_0)$:

\begin{aligned} E[U_i(\beta_0;\alpha)\mid X_i] &=D_i(\beta_0)^T V_i(\beta_0,\alpha)^{-1} E[Y_i-\mu_i(\beta_0)\mid X_i]\\ &=D_i(\beta_0)^T V_i(\beta_0,\alpha)^{-1}0=0. \end{aligned}

Taking expectations on both sides and using the tower property gives $E[U_i(\beta_0;\alpha)]=E\{E[U_i(\beta_0;\alpha)\mid X_i]\}=0$. The calculation used the conditional mean and the fact that the weight matrix is fixed given $X_i$; it did not use the true within-cluster covariance. $\square$

Corollary: For a finite collection of clusters satisfying the theorem’s conditions, $E[U(\beta_0;\alpha)]=0$. If the clusters are independent, their cluster contributions are independent as well, provided each contribution depends only on its own cluster and fixed quantities.

Proof. By linearity of expectation and the theorem,

E[U(\beta_0;\alpha)] =E\left[\sum_{i=1}^n U_i(\beta_0;\alpha)\right] =\sum_{i=1}^n E[U_i(\beta_0;\alpha)] =\sum_{i=1}^n 0=0.

The expectation identity itself does not require independence: linearity suffices. Independence is part of the usual independent-cluster framework and ensures that contributions from separate clusters are independent, a property used in subsequent sampling and asymptotic arguments. $\square$

Worked Examples

Worked Example: Binary Responses with the Wrong Correlation

Suppose each cluster contains two binary responses. Their joint probabilities are $P(Y_1=1,Y_2=1)=0.30$, $P(Y_1=1,Y_2=0)=0.10$, $P(Y_1=0,Y_2=1)=0.30$, and $P(Y_1=0,Y_2=0)=0.30$. Use separate marginal means $p_1,p_2$ as parameters and an independence working covariance. Check that the contribution is centered at the true marginal means despite the wrong working correlation.

The probabilities sum to $1$. The marginal means are $p_1=0.30+0.10=0.40$ and $p_2=0.30+0.30=0.60$. Since $Y_j$ is binary, each marginal variance is $p_j(1-p_j)=0.40(0.60)=0.24$. The true covariance is

\operatorname{Cov}(Y_1,Y_2) =E[Y_1Y_2]-E[Y_1]E[Y_2] =0.30-(0.40)(0.60)=0.06.

Thus the true correlation is $0.06/\sqrt{(0.24)(0.24)}=0.25$, not zero. The independence working covariance is $V=\operatorname{diag}(0.24,0.24)$, while $D=I_2$ for the two direct mean parameters. Therefore

U= \begin{pmatrix} (Y_1-0.40)/0.24\\ (Y_2-0.60)/0.24 \end{pmatrix}, \qquad E[U]= \begin{pmatrix} (0.40-0.40)/0.24\\ (0.60-0.60)/0.24 \end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}.

The working covariance ignores a genuine positive association, but it only rescales each centered residual here. The joint distribution affects the covariance of $U$, not its expectation.

Worked Example: A Poisson Mean Model with an Incorrect Working Correlation

Consider a cluster of two conditionally independent Poisson responses, with means $\mu_j=\exp(\beta_0+\beta_1x_j)$, covariates $x_1=0,x_2=1$, and true parameter $\beta_0=\log 2,\ \beta_1=\log(3/2)$. Thus the means are $2$ and $3$. Use the working covariance with Poisson variances and working correlation $0.25$.

The design matrix has rows $(1,0)$ and $(1,1)$, so the derivative matrix and the working covariance are

D= \begin{pmatrix}2&0\\3&3\end{pmatrix}, \qquad V= \begin{pmatrix} 2&0.25\sqrt{6}\\ 0.25\sqrt{6}&3 \end{pmatrix}.

The true correlation is zero because the responses are conditionally independent, so the working correlation is wrong. The determinant of $V$ is $6-(0.25\sqrt6)^2=6-0.375=5.625>0$, and its diagonal entries are positive; hence it is positive definite. Put $r_1=Y_1-2$ and $r_2=Y_2-3$. Since the determinant is $45/8$, the inverse is

V^{-1}=\frac{8}{45} \begin{pmatrix} 3&-\sqrt6/4\\ -\sqrt6/4&2 \end{pmatrix}.

Then $U=D^TV^{-1}(r_1,r_2)^T$, or explicitly,

U= \begin{pmatrix} \frac{8}{45}\{6r_1+(6- \sqrt6)r_2\}\\[2pt] \frac{8}{45}\{-3\sqrt6\,r_1/4+6r_2\} \end{pmatrix}.

Both residuals have expectation zero: $E[r_1]=E[Y_1]-2=0$ and $E[r_2]=E[Y_2]-3=0$. Substituting those expectations into each displayed component gives $E[U]=(0,0)^T$. The nonzero off-diagonal weight changes the linear combinations of residuals, but not their zero means.

Worked Example: Random Covariates and Conditional Centering

Suppose a cluster has two conditionally independent Bernoulli responses with a shared binary covariate $X$, where $P(X=0)=P(X=1)=1/2$. Their common conditional probability is $p(X)=\operatorname{logit}^{-1}(\beta_0+\beta_1X)$, with $\beta_0=0$ and $\beta_1=\log 3$. Use a working correlation of $0.5$, although the true conditional correlation is zero.

When $X=0$, $p=1/2$ and $v=p(1-p)=1/4$. When $X=1$, $p=3/4$ and $v=3/16$. For either value of $X$, the working covariance is $V=v\begin{pmatrix}1&1/2\\1/2&1\end{pmatrix}$, which is positive definite because its correlation matrix has determinant $3/4>0$. The derivative matrix has two identical rows, each equal to $v(1,X)$, since $\partial p/\partial(\beta_0,\beta_1)=v(1,X)$.

Write $r_j=Y_j-p(X)$. Conditional on $X$, $E[r_1\mid X]=E[r_2\mid X]=0$. The working inverse is $V^{-1}=(4/(3v))\begin{pmatrix}1&-1/2\\-1/2&1\end{pmatrix}$, so

U(\beta_0,\beta_1) = \begin{pmatrix}v\\vX\end{pmatrix} \frac{4}{3v} \begin{pmatrix}r_1-\tfrac12r_2\\-\tfrac12r_1+r_2\end{pmatrix}.

For $X=0$, the leading vector is $(1/4,0)^T$ and the inverse multiplier is $16/3$; for $X=1$, the leading vector is $(3/16,3/16)^T$ and the inverse multiplier is $64/9$. In both cases the conditional expectation of the displayed residual combinations is zero. Hence $E[U\mid X=0]=E[U\mid X=1]=0$, and averaging over $X$ gives $E[U]=\tfrac12(0)+\tfrac12(0)=0$. Conditioning is what makes the argument valid when covariates are random.

What the Result Does and Does Not Say

A working correlation changes the weighting of residuals. When it is wrong, the resulting contribution can have a different variance and the estimating equation can weight patterns of residuals less effectively. But if the weight is fixed given the covariates, it cannot turn a conditionally mean-zero residual vector into one with a nonzero conditional expectation. That is the precise reason correlation misspecification does not destroy this centering property.

Independence within a cluster is not assumed in the theorem. The first example deliberately used dependent responses. Independence between clusters is a separate sampling assumption: it is not needed for the expectation of a finite sum to be zero, but it supports the standard independent-cluster analysis of sums of contributions.

Common pitfall: “The working covariance is wrong, so the expected GEE must be nonzero.” This reasoning confuses a covariance model with a mean model. The correction is to calculate the conditional expectation: if $V_i$ and $D_i$ are determined by $X_i$, then $E[U_i\mid X_i]=D_i^TV_i^{-1}E[Y_i-\mu_i\mid X_i]=0$ under a correct conditional marginal mean. Conversely, this argument does not justify using a weight matrix computed from the same cluster’s outcomes: then one cannot generally pull that matrix outside the conditional expectation.

Finally, “unbiased GEE” is shorthand for an estimating function centered at $\beta_0$. It is not a guarantee that a finite-sample root has expectation $\beta_0$. Nor does it say that any choice of working correlation is equally useful: the correlation choice can affect precision even when the population equation remains centered.

Check Your Understanding

Use the conditional-mean argument, rather than assuming that the working covariance is correct.

Which condition allows $D_i^TV_i^{-1}$ to be taken outside the conditional expectation given $X_i$?
Why is independence between clusters not needed to show that the expectation of a finite sum of cluster contributions is zero?
In the binary example, calculate the true correlation from the stated joint probabilities.
Does a centered population estimating function imply that a finite-sample solution $\hat\beta$ is unbiased? Explain.
What may fail in the centering argument if the weight matrix is computed from the same cluster’s observed outcomes?