Decomposing Marginal Covariance by Pattern

Why Marginal Covariance Depends on Dropout Patterns

A population covariance summarizes how outcomes vary together across everyone in the target population. In a dropout setting, however, individuals belong to different patterns, and their outcome distributions can differ. The covariance in the full population therefore reflects two sources of variation: variation among individuals within the same pattern, and variation between patterns whose mean outcome vectors differ.

This distinction matters because averaging the covariance matrices within patterns does not generally give the population covariance matrix. That average omits the variation attributable to different pattern means. Conversely, a large population covariance need not mean that individuals have strong within-pattern association: it can arise partly, or even mainly, because patterns have different means.

Let \(G\) denote the dropout pattern, taking values \(1,\ldots,K\), and let \(\mathbf{Y}\) be a common \(q\)-dimensional outcome vector, such as measurements at two visits. The vector is defined for every individual in the target population, including outcomes that are unobserved after dropout. For pattern \(g\), define its weight \(\pi_g=P(G=g)\), mean vector \(\boldsymbol{\mu}_g=E(\mathbf{Y}\mid G=g)\), and covariance matrix \(\boldsymbol{\Sigma}_g=\operatorname{Cov}(\mathbf{Y}\mid G=g)\). The population mean is \(\boldsymbol{\mu}=E(\mathbf{Y})\), and the population covariance matrix is \(\boldsymbol{\Sigma}=\operatorname{Cov}(\mathbf{Y})\).

The weights are nonnegative and sum to one, and the patterns form an exhaustive, mutually exclusive partition of the population. The outcome vector must have finite second moments for its covariance to exist. No normality assumption is needed for this decomposition. In a dropout application, the essential additional requirement is that each pattern-specific distribution be defined for the same full outcome vector; this includes assumptions or sensitivity specifications for outcomes that are unobserved in a pattern.

Deriving the Decomposition

The population mean is the weighted average of the pattern-specific means:

To decompose covariance, write the centered outcome as the sum of a within-pattern deviation and a pattern-mean deviation:

Here \(\boldsymbol{\mu}_G\) means the mean vector for the pattern an individual belongs to. Take the outer product of this equality and then its expectation. The two cross-product terms have expectation zero: conditional on \(G=g\), \(E(\mathbf{Y}-\boldsymbol{\mu}_g\mid G=g)=\mathbf{0}\), while \(\boldsymbol{\mu}_g-\boldsymbol{\mu}\) is fixed. The remaining terms yield the decomposition.

This is the multivariate law of total covariance. Each diagonal entry decomposes a population variance into within-pattern and between-pattern variance. Each off-diagonal entry decomposes a population covariance in the same way. The between-pattern matrix is positive semidefinite because it is a weighted sum of outer products; its off-diagonal entries, however, can be positive, negative, or zero.

What the Components Mean

The within-pattern term answers a conditional question: after fixing dropout pattern, how do outcomes vary and covary? The between-pattern term answers a different question: how much population variation is associated with the pattern groups having different mean vectors? These are components of a population covariance, not necessarily separate causal effects of dropout. Pattern membership can be associated with observed or unobserved characteristics, and this decomposition alone does not explain why the means differ.

For any pair of visits \(j\) and \(k\), the marginal covariance is

If patterns with above-average means at one visit also have above-average means at the other, the between-pattern covariance tends to be positive. If one visit's pattern means are high where the other's are low, it can be negative. Consequently, the marginal covariance can differ substantially from every pattern-specific covariance.

Assumptions, Diagnostics, and Limitations

The decomposition itself is an identity under the stated partition and finite-moment conditions. The challenging assumptions in dropout applications concern the inputs, not the identity. Pattern weights can often be estimated from observed pattern membership. By contrast, pattern-specific means and covariances for the full outcome vector can involve unobserved outcomes. The observed-data distribution alone may not determine those quantities; assumptions, restrictions, or sensitivity choices are needed.

Useful checks include verifying that weights are nonnegative and sum to one; that each covariance matrix is symmetric with nonnegative variances; and that the calculated between-pattern and total covariance matrices are symmetric and positive semidefinite. A negative diagonal variance or a markedly asymmetric result signals an arithmetic or implementation error. An off-diagonal covariance can be negative without being an error. Also check that every pattern uses the same outcome ordering and units, and that the full vector's definition is consistent across patterns.

When pattern-specific means are nearly equal, the between-pattern component should be small. If the means are exactly equal, it is the zero matrix, although within-pattern covariance matrices can still differ. When there is only one pattern, the between-pattern component is zero and the population covariance equals that pattern's covariance.

A common mistake is to report \(\sum_g\pi_g\boldsymbol{\Sigma}_g\) as the marginal covariance. This is only the within-pattern component; it equals the marginal covariance only when all pattern means are equal. Another mistake is to calculate a covariance for a dropout pattern using only individuals with both visits observed and treat it as \(\boldsymbol{\Sigma}_g\). That calculation describes an observed subset, not necessarily the full pattern-specific distribution. The correction is to state which distribution the covariance represents and how unobserved outcomes are handled.

The decomposition is useful for interpreting a marginal covariance, but it does not select a pattern-mixture model, identify missing outcomes, or establish a causal explanation for differences between patterns. It also does not replace comparisons with other modeling approaches. A selection model represents the joint distribution through a different factorization; a pattern-mixture model makes the pattern-specific outcome distributions explicit. Regardless of representation, a marginal covariance derived for the same full population must account for both within-group variation and differences in group means.