Covariance Structures Within Dropout Patterns

Why Model Covariance Within Dropout Patterns?

A pattern-mixture model describes outcome distributions separately for groups defined by their dropout patterns. Within a pattern, the covariance matrix describes how repeated outcomes vary together. Its structure matters: it determines how many parameters must be estimated and which similarities across visits the model assumes. A parsimonious structure can be useful when pattern-specific samples are small, but it can also impose restrictions that do not fit the data.

Here, let \(G_i=g\) indicate that participant \(i\) belongs to dropout pattern \(g\), and let \(q_g\) be the number of scheduled visits with observed outcomes in that pattern. Under monotone dropout, these are the first \(q_g\) visits. Write the observed outcome vector as \(Y_{i,g}^{\mathrm{obs}}=(Y_{i1},\ldots,Y_{iq_g})^\mathsf{T}\). The mean vector is \(\mu_g\), and the covariance matrix is \(\Sigma_g^{\mathrm{obs}}\). The covariance models below describe dependence among outcomes within a pattern; they do not, by themselves, specify the pattern-specific means or the probability of belonging to a pattern.

For a Gaussian pattern-specific model, the observed outcomes in pattern \(g\) are modeled as \(Y_{i,g}^{\mathrm{obs}}\mid G_i=g\sim N_{q_g}(\mu_g,\Sigma_g^{\mathrm{obs}})\). The covariance structures can also describe second moments without assuming normality, but Gaussian likelihood calculations additionally rely on the normal distributional assumption. In either case, the covariance matrix must be positive definite for a non-degenerate multivariate normal model.

Three Covariance Structures

An unstructured covariance makes no equality restrictions among variances or covariances. For \(q_g\) visits, it estimates each diagonal variance and each distinct off-diagonal covariance separately. It is the most flexible of the three structures considered here.

Because a covariance matrix is symmetric, the number of distinct covariance parameters is \(q_g(q_g+1)/2\): there are \(q_g\) variances and \(q_g(q_g-1)/2\) distinct covariances.

A compound-symmetry (CS) structure assumes the same variance at every visit and the same covariance between every distinct pair of visits. In correlation terms, every pair of visits has the same correlation \(\rho_g\).

where \(I_{q_g}\) is the identity matrix and \(J_{q_g}\) is the matrix of ones. When \(q_g>1\), positive definiteness requires \(\sigma_g^2>0\) and \(-1/(q_g-1)<\rho_g<1\). CS has two covariance parameters when at least two visits are observed: a common variance and a common correlation. For a single visit, only the variance is estimable.

A first-order autoregressive, or AR(1), structure assumes a common variance and correlations that decay geometrically with the number of visit intervals between outcomes. For equally spaced visits, its entries are

Here \(\sigma_g^2>0\), and positive definiteness requires \(|\rho_g|<1\). The correlation between adjacent visits is \(\rho_g\), between visits two intervals apart it is \(\rho_g^2\), and so on. Thus AR(1) uses two covariance parameters when at least two visits are observed. It is most naturally interpreted for regularly spaced visits; irregular visit times require a model that defines dependence using actual time gaps rather than simply visit-number differences.

Parameter Counts and Model Implications

The count is for covariance parameters only; it does not include mean parameters or other model components. If each pattern has its own covariance parameters, the total covariance count is the sum of the counts for the patterns. For example, an unstructured model with pattern-specific matrices requires \(\sum_g q_g(q_g+1)/2\) covariance parameters, while CS or AR(1) requires two per pattern with at least two observed visits and one for a single-visit pattern. Sharing covariance parameters across patterns reduces the count but adds a cross-pattern equality assumption.

Checking the Restrictions and Their Consequences

The positive-definiteness restrictions are not optional technical details. A covariance matrix that is not positive definite cannot represent a non-degenerate joint distribution with the specified variances and covariances. One justification of the AR(1) restriction is its stationary representation \(Y_{ij}-\mu_{gj}=\rho_g(Y_{i,j-1}-\mu_{g,j-1})+\epsilon_{ij}\), with innovation variance \(\sigma_g^2(1-\rho_g^2)>0\). This requires \(|\rho_g|<1\) and yields lag-\(h\) covariance \(\sigma_g^2\rho_g^h\). For CS, vectors orthogonal to the all-ones vector have eigenvalue \(\sigma_g^2(1-\rho_g)\); the all-ones direction has eigenvalue \(\sigma_g^2[1+(q_g-1)\rho_g]\). Both eigenvalues must be positive, giving \(\rho_g<1\) and \(\rho_g>-1/(q_g-1)\).

What Dropout Changes—and What It Does Not

Patterns with different lengths have covariance matrices of different observed dimensions. A pattern with one observed visit supplies information about its variance but cannot identify a within-pattern correlation. A pattern with two visits supplies only one pairwise relationship. Consequently, a fully unstructured covariance may be feasible in a large, well-populated pattern but unstable in a small pattern with many visits. A simpler structure reduces the number of estimated quantities by imposing assumptions; it does not create information absent from the data.

There is also an important distinction between the covariance of observed outcomes and a full covariance model for all scheduled outcomes. If a pattern ends after visit \(q_g\), the observed-data likelihood informs the covariance among visits \(1,\ldots,q_g\). It does not directly reveal covariances involving outcomes after dropout. A model that specifies the full outcome vector for every pattern must therefore make additional assumptions about those unobserved components. The observed covariance block alone cannot establish those assumptions.

Covariance restrictions are also separate from assumptions about dropout. Choosing CS, AR(1), or an unstructured matrix does not make dropout ignorable and does not determine the pattern weights. Pattern-specific covariance modeling addresses dependence among outcomes conditional on pattern membership; it is not, by itself, a solution to missing-data identification.

Choosing a Structure: Assumptions, Alternatives, and Limits

Unstructured covariance is appropriate when flexibility is a priority and there is enough pattern-specific information to estimate all distinct entries. It avoids imposing common variances or a particular lag pattern, but costs parameters and can be unstable. CS is a compact choice when equal variances and a common correlation across visits are substantively reasonable. AR(1) is compact when equal variances and decreasing correlation with visit separation are plausible. With just two observed visits, CS and AR(1) both impose equal marginal variances and a single correlation, so their covariance forms coincide; their difference becomes visible with three or more visits.

These are modeling assumptions, not facts guaranteed by the dropout-pattern definition. Useful diagnostics include plotting or tabulating empirical variances and pairwise correlations by visit within each adequately sized pattern, checking whether correlations appear roughly constant or decrease with lag, and examining residual dependence after fitting the mean structure. Such summaries can be noisy in small patterns, so they should be interpreted alongside sample sizes and uncertainty rather than as definitive tests.

• Warning sign for CS: pairwise correlations show a clear pattern by visit lag, or visit variances differ substantially.
• Warning sign for AR(1): correlations do not decline geometrically with lag, or the visit spacing makes a unit-lag interpretation inappropriate.
• Warning sign for an unstructured model: estimates are highly variable, correlations are extreme, or the fitted matrix is not positive definite.
• General limitation: no observed-data diagnostic can determine the covariance of outcomes that are never observed within a pattern without additional assumptions.

A common mistake is to say, “AR(1) has two parameters, so it assumes the same correlation at every visit.” That confuses parameter count with the restriction itself. AR(1) has a common variance and a lag-one correlation parameter, but the correlation at lag \(h\) is \(\rho_g^h\); for example, if \(\rho_g=0.60\), the lag-two correlation is \(0.36\). CS, in contrast, assigns the same correlation to every distinct pair. A second mistake is to treat a low parameter count as evidence that a structure fits. Parsimony is useful only when the imposed covariance pattern is plausible for the outcomes and visits under study.