Why consistency is a cluster-level question
A generalized estimating equation (GEE) adds one estimating contribution from each independent cluster. The responses within a cluster may be dependent, and cluster sizes may differ. Consistency is therefore justified by averaging across independent clusters—not by treating all individual responses as independent observations. The key questions are whether the normalized sum of cluster contributions converges to a deterministic function and whether that function identifies the target parameter.
This tutorial gives sufficient conditions for a root of the GEE to converge in probability to the target. The conditions permit cluster-level contributions to have different distributions, as is natural when cluster sizes or covariates vary. Bounded cluster sizes are one useful way to control how much any one cluster can contribute. These consistency arguments do not establish asymptotic normality or justify a sandwich confidence interval.
Estimating functions and the target
Let \(n\) denote the number of independent clusters, and let cluster \(i\) contain \(m_i\) responses. Write \(Y_i\) for its response vector and \(X_i\) for its covariate information. For a parameter \(\beta\in\mathbb{R}^p\), a GEE has the form
Here \(\mu_i(\beta)\) is the vector of marginal means, \(D_i(\beta)\) is its derivative with respect to \(\beta\), and \(V_i(\beta)\) is the working covariance matrix. The consistency argument also applies to other cluster-level estimating functions; it is not tied to this particular GEE form. Assume the clusters are independent, though they need not be identically distributed. The target parameter is \(\beta_0\).
The normalization by \(n\) is by the number of clusters. If cluster sizes are uniformly bounded, the total number of responses and the number of clusters have the same order. But with very large clusters, those two normalizations can behave quite differently.
A uniform law of large numbers for cluster contributions
For \(\beta\) in a compact parameter set \(B\subset\mathbb{R}^p\), define the average expected estimating function
A useful set of sufficient conditions for uniform convergence is the following. The cluster contributions are independent for each \(n\); there is a finite constant \(C\) such that \(\sup_{i,n,\beta\in B}\operatorname{Var}(\psi_{i,r}(\beta))\leq C\) for every component \(r\); and there are nonnegative random variables \(K_{i,n}\) with \(\sup_{i,n}E[K_{i,n}]<\infty\) such that \(\|\psi_i(\beta)-\psi_i(\gamma)\|\leq K_{i,n}\|\beta-\gamma\|\) for all \(\beta,\gamma\in B\). Also assume \(g_n\) converges uniformly on \(B\) to a continuous function \(g\). Under these conditions,
The variance condition is a convenient way to obtain a pointwise law of large numbers. The random Lipschitz bounds control how much the empirical function can vary between a finite set of parameter values. Uniform convergence of \(g_n\) is a separate requirement: independence alone does not ensure that expected estimating functions from changing cluster designs approach a common limit.
Proof of uniform convergence. First consider one component of \(U_n(\beta)/n\) at a fixed \(\beta\). Independence and the uniform variance bound give
Chebyshev’s inequality implies convergence in probability to the corresponding component of \(g_n(\beta)\). Since \(p\) is finite, this holds jointly for all components at any fixed \(\beta\). For an arbitrary \(\beta\in B\), choose a point \(\gamma\) in a finite \(\delta\)-net of the compact set \(B\) with \(\|\beta-\gamma\|\leq\delta\). The Lipschitz bound yields
The expectation of the first average of \(K_{i,n}\) is uniformly bounded. Markov’s inequality therefore makes that average bounded in probability, uniformly in \(n\). For any chosen error probability, first choose a bound for this average with at least that probability, and then choose \(\delta\) small enough that both displayed differences are small on that event. The finite net has only finitely many points, so the pointwise convergence already proved holds jointly at its points. This establishes uniform convergence of \(U_n/n-g_n\) in probability. Combining it with the assumed uniform convergence of \(g_n\) to \(g\), by the triangle inequality, proves the claim. \(\square\)
Proof that the root is consistent. Fix \(\epsilon>0\). Since \(B\) is compact, \(g\) is continuous, and its only zero is \(\beta_0\), the continuous function \(\|g(\beta)\|\) has a strictly positive minimum on the closed set \(\{\beta\in B:\|\beta-\beta_0\|\geq\epsilon\}\), if that set is nonempty. Call this minimum \(c_\epsilon>0\). If \(\hat{\beta}_n\) lies in this set, then
The supremum on the right converges in probability to zero, and the norm of the approximate-root residual also converges in probability to zero. Thus the probability that \(\hat{\beta}_n\) lies at least \(\epsilon\) from \(\beta_0\) tends to zero. This is precisely \(\hat{\beta}_n\xrightarrow{p}\beta_0\). \(\square\)
Worked examples
Worked Example: A Gaussian Intercept with Alternating Cluster Sizes
Suppose \(n\) is even. Half the clusters have size one and half have size two. In cluster \(i\), let \(Y_{ij}=\beta_0+b_i+e_{ij}\), where \(b_i\) has mean zero and variance one, the \(e_{ij}\) have mean zero and variance three, and all \(b_i\) and \(e_{ij}\) are mutually independent across clusters and otherwise. Use the intercept estimating function \(\psi_i(\beta)=\sum_{j=1}^{m_i}(Y_{ij}-\beta)\).
The equation \(U_n(\beta)=0\) has root \(\hat{\beta}_n=\sum_i\sum_jY_{ij}/\sum_i m_i\). There are \(n/2\) clusters of each size, so the total number of responses is \(n/2+n=3n/2\). Since every response has mean \(\beta_0\), \(E[\hat{\beta}_n]=\beta_0\).
For a size-one cluster, \(\operatorname{Var}(\sum_jY_{ij})=\operatorname{Var}(b_i+e_{i1})=1+3=4\). For a size-two cluster, \(\sum_jY_{ij}=2\beta_0+2b_i+e_{i1}+e_{i2}\), so its variance is \(4(1)+3+3=10\). Independence across clusters gives the variance of the pooled mean:
Chebyshev’s inequality now gives \(\hat{\beta}_n\xrightarrow{p}\beta_0\). The within-cluster dependence is present, but the cluster sizes are bounded and the independent clusters provide an increasing number of contributions.
Worked Example: A Binary Intercept with Unequal Cluster Sizes
Again let \(n\) be even, with half of the clusters of size one and half of size two. Suppose each response is Bernoulli with success probability \(p_0=0.4\), independent across responses and clusters. For a logistic intercept, use \(\psi_i(\beta)=\sum_{j=1}^{m_i}\{Y_{ij}-p(\beta)\}\), where \(p(\beta)=1/(1+\exp(-\beta))\).
The expected normalized estimating function is \(g_n(\beta)=(3/2)\{0.4-p(\beta)\}\), because the average cluster size is \(3/2\). Its unique zero is \(\beta_0=\log(0.4/0.6)=\log(2/3)\): the logistic function is strictly increasing, and \(p(\beta_0)=0.4\). Thus the limiting function identifies the target.
When the pooled sample proportion \(\bar Y\) is strictly between zero and one, the GEE root is \(\hat{\beta}_n=\log\{\bar Y/(1-\bar Y)\}\). There are \(3n/2\) responses, and their independence gives
Hence \(\bar Y\xrightarrow{p}0.4\). Since the logit function is continuous at \(0.4\), \(\hat{\beta}_n\xrightarrow{p}\log(2/3)\) whenever the finite root exists. The probability of a boundary sample proportion tends to zero, because \(\bar Y\) converges in probability to the interior value \(0.4\). Defining the estimator arbitrarily on that event does not alter consistency.
Worked Example: One Unbounded Cluster Dominates the Estimate
Consider \(n\) independent clusters. Cluster 1 has size \(n\), and each of its responses equals the same random variable \(Z\), where \(E[Z]=0\) and \(\operatorname{Var}(Z)=1\). Each of the other \(n-1\) clusters has size one and response zero. The marginal mean is zero for every response. Apply the intercept equation \(U_n(\beta)=\sum_i\sum_j(Y_{ij}-\beta)=0\).
There are \(n\) responses in the first cluster and \(n-1\) in the other clusters, so the root is \(\hat{\beta}_n=nZ/(2n-1)\). Therefore \(\hat{\beta}_n\xrightarrow{p}Z/2\), not zero: for example, \(P(|Z/2|>1/4)>0\) for a standard normal \(Z\). The clusters are independent and the estimating equation is unbiased at zero, but the single large cluster retains a nonvanishing share of all responses. This example shows why independence and unbiasedness alone do not guarantee consistency.
What bounded cluster sizes do—and do not—guarantee
A common sufficient setup is that there is a fixed \(M<\infty\) with \(m_i\leq M\) for all clusters and sample sizes. Together with bounded covariates, well-behaved working covariance matrices, and suitable response moments, this can help ensure that cluster contributions have controlled variances and parameter dependence. It also ensures that the total number of responses is at most \(Mn\), so no one cluster can dominate merely by containing arbitrarily many observations.
Bounded cluster sizes alone do not prove consistency. The expected estimating functions must still approach a limit, and that limit must have a unique zero at the target. For a GEE with a correctly specified marginal mean, working-correlation misspecification can leave the expected contribution zero at \(\beta_0\). But a zero at the target is not enough: the limiting function must distinguish \(\beta_0\) from other parameter values. The design must provide information for the parameter being estimated.
The central consistency logic is therefore: independent clusters support averaging; size and moment conditions control the contributions; convergence of expected functions supplies a stable limit; and identification forces an approximate root toward \(\beta_0\). These ingredients establish convergence in probability, but not a limiting distribution or a valid variance estimator. Those require additional arguments.
Check Your Understanding
Use the definitions and conditions in this tutorial to answer the following.
- Why is the natural normalization of the GEE in this argument by the number of clusters rather than automatically by the total number of responses?
- What role does the unique-zero condition on the limiting function \(g\) play in the consistency proof?
- Which part of the uniform convergence argument uses a finite net of the compact parameter set?
- In the Gaussian intercept example, why does the within-cluster dependence not prevent consistency?
- What feature of the final example makes unbiased estimating contributions insufficient for consistency?