Unstructured Correlation for Fixed Visit Schedules

What “Unstructured” Means

For repeated measurements, the working correlation describes how responses from the same cluster are related after accounting for their means and marginal variances. With a fixed schedule of \(m\) visits, an unstructured working correlation assigns a separate correlation parameter to every distinct pair of visits. It imposes no requirement that correlations be equal at different lags or follow a particular time-decay rule.

Let the scheduled visits be indexed \(1,\ldots,m\). Write \(\rho_{jk}\) for the working correlation between visits \(j\) and \(k\), where \(j<k\). The working correlation matrix \(R\) is symmetric, has ones on its diagonal, and has \(\rho_{jk}\) in the two positions corresponding to visits \(j\) and \(k\). In a standard GEE, this matrix contributes to the working covariance; the marginal variances are represented separately.

“Unstructured” does not mean that entries can be chosen without constraints. The symmetry and unit diagonal are part of the specification, and the entire matrix must satisfy a joint positive-definiteness condition. The entries are also not separate variance parameters: they describe correlations, while the marginal variances are handled elsewhere in the working covariance.

Counting the Correlation Parameters

There is one free parameter for each unordered pair of distinct visits. The pair \((j,k)\) is the same correlation as \((k,j)\), so symmetry prevents counting it twice. There are \(m\) choices for the first visit and \(m-1\) remaining choices for the second; dividing by two removes the duplicate orderings.

For example, two visits require one correlation parameter, three visits require three, and four visits require six. The number grows quadratically with the number of visits. This count describes the entries in the model’s common correlation matrix; it does not guarantee that the available data contain enough information to estimate every entry.

Positive Definiteness Is a Joint Requirement

A correlation matrix is positive definite if \(x^\mathsf{T}Rx>0\) for every nonzero vector \(x\). This condition ensures that the matrix is invertible and is the relevant requirement when standard GEE calculations use its inverse. Each pairwise correlation must lie strictly between \(-1\) and \(1\) when the matrix is positive definite, but those separate bounds do not ensure that a larger matrix is positive definite.

For three visits, write \(a=\rho_{12}\), \(b=\rho_{13}\), and \(c=\rho_{23}\). The matrix and its determinant are

For this three-visit matrix, positive definiteness holds exactly when \(|a|<1\) and \(\det(R)>0\). In particular, the determinant condition adds a joint restriction beyond the individual pairwise bounds.

To verify the criterion, partition the matrix into its first two rows and columns and its final row and column. The leading \(2\times2\) block \(B=\begin{pmatrix}1&a\\a&1\end{pmatrix}\) is positive definite exactly when \(|a|<1\). For this condition, completing the square in the quadratic form for \(R\) shows that its remaining one-dimensional part has coefficient \(1-(b^2-2abc+c^2)/(1-a^2)\). The full quadratic form is positive for every nonzero vector exactly when that coefficient is positive. Multiplying it by \(1-a^2>0\) gives \(1-a^2-b^2-c^2+2abc>0\), which is \(\det(R)>0\). This proves the stated condition. \(\square\)

For matrices with more than three visits, checking each pair separately is still insufficient. The full matrix must be positive definite. In practice, a matrix factorization or an eigenvalue check can assess this condition; for a symmetric matrix, positive definiteness is equivalent to all eigenvalues being positive. A matrix that is merely positive semidefinite may represent a degenerate correlation structure, but it is singular and does not have the inverse required by standard GEE calculations.

Cluster Sizes and the Visit Schedule

A common unstructured matrix is most straightforward when every cluster has a complete set of the same \(m\) scheduled visits. Then each cluster’s working correlation has the same dimension and the same visit labels, and the full \(m\times m\) matrix applies. There must be at least two scheduled visits for the matrix to contain any pairwise correlation parameters; for one visit, the matrix is simply \((1)\).

Clusters with different numbers of observed visits can still be associated with a common full-schedule matrix when each observed response can be matched to its correct scheduled visit. If cluster \(i\) is observed at the subset \(S_i\) of scheduled visits, its working correlation is the submatrix formed by retaining the rows and columns indexed by \(S_i\). This uses the same underlying schedule-specific parameters, rather than treating a cluster’s first observed measurement as the same visit regardless of its actual scheduled position.

To prove this result, take a nonempty subset \(S\) of the visit indices and a nonzero vector \(y\) with one component for each index in \(S\). Form a vector \(x\) of length \(m\) by placing the components of \(y\) at the indices in \(S\) and zeros elsewhere. Then \(x\) is nonzero and \(y^\mathsf{T}R_{S,S}y=x^\mathsf{T}Rx>0\), because \(R\) is positive definite. This is precisely the definition of positive definiteness for the principal submatrix \(R_{S,S}\). \(\square\)

This matrix fact concerns validity, not whether every parameter can be estimated from the observed data. If a pair of scheduled visits is never jointly observed, the corresponding pairwise relationship is not directly represented by an observed pair in that sample. The parameter count for a full unstructured matrix remains \(\binom{m}{2}\), but the count alone does not establish that all those parameters are estimable.

When the Unstructured Choice Is Useful—and Its Limits

An unstructured matrix is flexible when the study has a modest, fixed set of visits and there is no reason to impose a common-correlation or time-decay pattern. It lets each visit pair have its own working correlation. That flexibility comes with a rapidly increasing number of parameters: moving from four to eight visits increases the count from \(6\) to \(28\). This makes the choice increasingly demanding as the schedule grows.

The fixed-schedule qualification matters. The entries \(\rho_{jk}\) refer to correlations for particular scheduled visits. If measurement times vary across clusters, the same visit indices must still identify comparable occasions for a common matrix to have that interpretation. A visit-indexed unstructured matrix does not automatically encode how correlation changes with elapsed time.

Common mistake: “Every proposed pairwise correlation is between \(-1\) and \(1\), so the full unstructured matrix is valid.” The correction is that these are necessary pairwise bounds, not a sufficient joint condition. The whole matrix must be positive definite for standard inverse-based GEE calculations. For three visits, the determinant check makes the additional restriction explicit; for larger matrices, check positive definiteness of the full matrix.

A second practical distinction is between the number of entries and the amount of information in the data. A schedule with \(m\) visits defines \(m(m-1)/2\) parameters, but unequal cluster sizes or unobserved visit pairs can leave some relationships weakly represented or unavailable in the observed sample. Positive definiteness guarantees a usable inverse for the specified matrix; it does not, by itself, guarantee reliable estimation of every correlation.