Covariance Structures and Reference-Based Imputation

Why Covariance Structure Matters

In reference-based imputation, the mean restriction describes the assumed outcome trajectory after treatment discontinuation; the covariance model describes how outcomes at different visits vary together. These are distinct parts of the predictive model, but they interact: a post-dropout prediction conditioned on observed outcomes depends on the covariance between those outcomes and the missing outcomes. Two analyses can therefore use the same J2R or copy-reference mean restriction and produce different conditional predictions if they use different covariance structures.

This tutorial focuses on the conditional calculations for a multivariate normal outcome vector. It compares unstructured, compound-symmetric, and autoregressive covariance models. In each calculation, the mean vectors are those specified by the applicable reference-based restriction; the covariance structure determines how the observed residuals update the prediction. The earlier material on conditional normal distributions provides the general framework, which we apply here to understand the practical consequences of covariance choices.

Definition: Let $Y_o$ be a subject’s observed outcomes before dropout and $Y_m$ the outcomes to be imputed after dropout. Under a specified reference-based mean restriction, write their model means as $\mu_o$ and $\mu_m$. Partition the covariance matrix into observed and missing blocks, $\Sigma_{oo}$, $\Sigma_{mo}$, and $\Sigma_{mm}$. If the joint outcome vector is multivariate normal and $\Sigma_{oo}$ is invertible, then $$ Y_m\mid Y_o=y_o \sim N\left(\mu_m+\Sigma_{mo}\Sigma_{oo}^{-1}(y_o-\mu_o),\ \Sigma_{mm}-\Sigma_{mo}\Sigma_{oo}^{-1}\Sigma_{om}\right). $$ The first parameter is the conditional mean; the second is the conditional covariance.

The formula has two practical parts. The term $y_o-\mu_o$ is the observed history’s deviation from its model mean. Multiplying it by $\Sigma_{mo}\Sigma_{oo}^{-1}$ gives the adjustment to the missing-outcome mean. The conditional covariance is the uncertainty remaining after using the observed history. A covariance structure changes both parts by imposing particular relationships among visits.

Three Covariance Structures

Let there be $K$ scheduled visits, with $i$ and $j$ indexing visits. An unstructured covariance model estimates a separate variance at each visit and a separate covariance for each pair. It is flexible, but the number of covariance parameters is $K(K+1)/2$. Its estimated joint covariance matrix must be positive semidefinite, and $\Sigma_{oo}$ must be invertible for the conditional calculation to be valid; positive definiteness of the full matrix is sufficient but stronger.

Compound symmetry (CS) assumes the same variance at every visit and the same correlation between every pair of distinct visits. If $\sigma^2$ denotes the common variance and $\rho$ the common correlation, then

\operatorname{Cov}(Y_i,Y_j)= \begin{cases} \sigma^2,&i=j,\\ \sigma^2\rho,&i\ne j. \end{cases} \qquad -\frac{1}{K-1}<\rho<1.

The restriction on $\rho$ ensures a positive-definite CS matrix for $K$ visits. CS is parsimonious, but it treats near and distant visits alike: the covariance of visits 1 and 2 is the same as that of visits 1 and 6.

A first-order autoregressive model, AR(1), assumes equal visit variances and a correlation that decays geometrically with the distance between equally spaced visits:

\operatorname{Cov}(Y_i,Y_j)=\sigma^2\rho^{|i-j|}, \qquad |\rho|<1.

For positive $\rho$, adjacent visits are more strongly correlated than visits several occasions apart. AR(1) is more restrictive than an unstructured model, but it represents a pattern of declining association that CS cannot express. The equal-spacing condition matters: applying this formula to irregularly timed visits without adapting the time-distance model does not preserve the stated interpretation of $\rho$.

Worked Example: An Unstructured Conditional Prediction

Consider a hypothetical participant with observed outcomes at visits 1 and 2 and a missing outcome at visit 3. For illustration, suppose the mean vectors under the stipulated reference-based restriction are $\mu_o=(10,12)^\mathsf{T}$ and $\mu_m=13$, in outcome units, and the observed values are $y_o=(13,15)^\mathsf{T}$. Suppose the reference covariance model is unstructured with

\Sigma= \begin{pmatrix} 4&3&4\\ 3&9&6\\ 4&6&16 \end{pmatrix}.

The observed covariance block and the covariance between the missing outcome and the observed outcomes are $\Sigma_{oo}=\begin{pmatrix}4&3\\3&9\end{pmatrix}$ and $\Sigma_{mo}=(4,6)$. The determinant of $\Sigma_{oo}$ is $4(9)-3(3)=27$, so it is invertible, and

\Sigma_{oo}^{-1}=\frac{1}{27} \begin{pmatrix} 9&-3\\ -3&4 \end{pmatrix}, \qquad \Sigma_{mo}\Sigma_{oo}^{-1}=\left(\frac{2}{3},\frac{4}{9}\right).

The observed residuals are $(13-10,15-12)^\mathsf{T}=(3,3)^\mathsf{T}$. Therefore, the conditional mean is $13+\frac{2}{3}(3)+\frac{4}{9}(3)=13+2+1.333=16.33$ outcome units, to four significant figures. For the variance, $16-(\frac{2}{3})(4)-(\frac{4}{9})(6)=16-\frac{16}{3}=10.67$ outcome-units squared, to four significant figures. The conditional standard deviation is $\sqrt{10.67}=3.266$ outcome units, to four significant figures.

These calculations assume the stated mean restriction and a positive-definite joint covariance matrix. Here the leading principal minors of $\Sigma$ are $4$, $27$, and $288$, all positive, so the symmetric matrix is positive definite. This check supports the use of the multivariate normal conditional formula.

How Structure Changes the Update

The next example holds the specified mean vectors and observed values fixed, changing only the covariance model. This isolates the role of covariance: it determines how much each observed residual contributes to the prediction. It does not itself decide whether J2R or copy-reference is the appropriate post-dropout restriction.

Worked Example: Comparing CS and AR(1) Weights

Suppose there are three equally spaced visits, visits 1 and 2 are observed, and visit 3 is missing. For this hypothetical calculation, let $\mu_o=(10,12)^\mathsf{T}$, $\mu_m=13$, and $y_o=(13,15)^\mathsf{T}$. Both observed residuals are 3 outcome units. Assume a common visit variance of 9 outcome-units squared and correlation parameter $\rho=0.5$.

Under CS, $\Sigma_{oo}=\begin{pmatrix}9&4.5\\4.5&9\end{pmatrix}$, $\Sigma_{mo}=(4.5,4.5)$, and $\Sigma_{mm}=9$. Direct multiplication gives $\Sigma_{mo}\Sigma_{oo}^{-1}=(1/3,1/3)$. Thus the conditional mean is $13+\frac{1}{3}(3)+\frac{1}{3}(3)=15.00$ outcome units. The conditional variance is $9-\frac{1}{3}(4.5)-\frac{1}{3}(4.5)=6.000$ outcome-units squared.

Under AR(1), visits 1 and 2 have covariance $9(0.5)=4.5$, while visits 1 and 3 have covariance $9(0.5)^2=2.25$. Hence $\Sigma_{oo}$ is unchanged, but $\Sigma_{mo}=(2.25,4.5)$. The conditional weights are $(0,0.5)$, giving conditional mean $13+0(3)+0.5(3)=14.50$ outcome units. The conditional variance is $9-0(2.25)-0.5(4.5)=6.750$ outcome-units squared.

The CS prediction uses both observed residuals equally because all distinct visits have the same correlation. The AR(1) prediction, for this three-visit pattern, uses the most recent residual and gives the earlier residual weight zero after conditioning on visit 2. Its conditional variance is also larger here. These are consequences of the two covariance assumptions, not evidence that either mean restriction is automatically preferable.

Covariance Choice Within J2R and Copy-Reference

J2R and copy-reference specify reference-based behavior after dropout; they do not by themselves require that the reference covariance be unstructured, CS, or AR(1). Once the applicable mean restrictions and covariance assumptions have been stated, the same conditional Gaussian calculation applies. In particular, conditional prediction uses the mean vector appropriate to the restriction being analyzed and the covariance blocks appropriate to the chosen model.

A useful way to avoid mixing up these roles is to write the prediction in two stages. First specify $\mu_o$ and $\mu_m$ under the selected reference-based rule. Then choose and estimate $\Sigma$, partition it by observed and missing visits, and calculate the conditional mean and covariance. If comparing J2R with copy-reference, keep the covariance assumption fixed to isolate differences due to the restrictions. If comparing covariance structures, hold the mean restriction fixed.

Worked Example: Separating Mean Restrictions From Covariance Assumptions

Suppose a hypothetical analysis has one observed outcome with residual $y_o-\mu_o=2$ outcome units and one missing post-dropout outcome. Under a specified reference covariance model, let both marginal variances be 9 outcome-units squared and their covariance be 4.5 outcome-units squared. The conditional weight on the observed residual is $4.5/9=0.5$, and the conditional variance is $9-(4.5/9)(4.5)=6.750$ outcome-units squared.

For illustration, if the mean restriction for a chosen analysis sets the missing-outcome mean to 13 outcome units, its conditional prediction is $13+0.5(2)=14.00$ outcome units. If a different stipulated mean restriction instead sets that missing-outcome mean to 12, while leaving the same covariance and observed residual unchanged, the prediction is $12+0.5(2)=13.00$ outcome units. In each calculation the covariance contributes the same 1.000-unit conditional adjustment and the same conditional variance; the different starting mean contributes the one-unit difference between predictions.

The example is an arithmetic illustration, not a claim that these particular means define J2R or copy-reference in every model. The required means must follow the chosen restriction. This separation makes it possible to explain exactly which assumption produced a change in a result.

Calculation, Diagnostics, and Limitations

For a chosen covariance model, calculate a conditional prediction in a fixed order. This reduces the risk of using an inappropriate mean vector or covariance block.

Specify the restriction and mean vectors.
Identify the J2R or copy-reference model being used, then state $\mu_o$ and $\mu_m$ for the observed and missing visits under that model.

Specify the covariance structure.
State whether the covariance is unstructured, CS, AR(1), or another model, and give the assumptions relevant to its validity.

Partition and check the covariance matrix.
Construct $\Sigma_{oo}$, $\Sigma_{mo}$, and $\Sigma_{mm}$. Check that $\Sigma_{oo}$ is invertible and that the fitted covariance matrix is positive definite.

Compute both conditional quantities.
Use the conditional mean formula for the prediction and the Schur complement for the conditional covariance. Report units for the mean and squared units for the variance.

Several diagnostics can reveal problems. An unstructured estimate that is nearly singular can produce unstable inverse matrices, extreme conditional weights, or a very small estimated conditional variance. A fitted CS model is questionable if empirical correlations clearly decline with visit separation or differ substantially by visit pair. AR(1) is questionable if correlations do not approximately follow a distance-decay pattern, if variances change materially over time, or if visits are not equally spaced under the stated model. In all cases, inspect the fitted covariance matrix, conditional variances, and predictions across observed dropout patterns; a conditional variance should not be negative.

These models make different compromises. Unstructured covariance offers the fewest structural restrictions but can require many parameters and may be unstable with limited data. CS is parsimonious but assumes a common correlation for all distinct visit pairs. AR(1) imposes a distance-based pattern and can be efficient when that pattern is reasonable, but it may misrepresent changing variances or non-decaying correlations. A more parsimonious model is not automatically better: its assumptions should be scientifically and empirically defensible.

The conditional variance above treats the model parameters as fixed. In practice, the mean and covariance parameters are estimated, and uncertainty in those estimates adds uncertainty beyond the conditional outcome variation shown here. The displayed formula is therefore a model-based conditional distribution given the fitted parameters, not by itself a complete account of uncertainty in an entire imputation-and-inference procedure.

Common Pitfall

Common mistake: “J2R means the covariance structure is irrelevant, because post-dropout outcomes follow the reference mean.” This is incorrect. Even when a rule specifies the post-dropout marginal mean, the conditional mean given observed outcomes includes $\Sigma_{mo}\Sigma_{oo}^{-1}(y_o-\mu_o)$, and the conditional variance also depends on $\Sigma$. A reference-based mean restriction does not eliminate the need to specify and assess covariance assumptions.

A second frequent error is to interpret a covariance model as if it were a dropout rule. CS or AR(1) describes dependence across visits; it does not determine what mean trajectory should apply after treatment discontinuation. State the mean restriction and covariance assumption separately before interpreting a prediction.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

Under CS with three visits, common variance 4, and correlation $0.25$, what is the covariance between visits 1 and 3?
For an AR(1) model with variance 9 and $\rho=0.5$, what are the covariances between visits 1 and 2 and between visits 1 and 3?
Why can two analyses using the same reference-based mean restriction give different conditional post-dropout means?
In the CS calculation in the second worked example, what would the conditional mean be if both observed residuals were zero?
Name one limitation of an unstructured covariance model and one assumption required by the AR(1) model used here.

Solutions. 1. CS gives covariance $4(0.25)=1.000$ outcome-units squared for every distinct pair. 2. The covariances are $9(0.5)=4.500$ and $9(0.5)^2=2.250$ outcome-units squared. 3. The covariance structure changes the conditional weights $\Sigma_{mo}\Sigma_{oo}^{-1}$, which determine how observed residuals update the prediction. 4. With both residuals zero, the adjustment is zero, so the conditional mean is the stipulated $\mu_m=13$ outcome units. 5. An unstructured model can require many parameters and yield unstable or nearly singular estimates; the stated AR(1) model assumes equally spaced visits, equal variances, and correlations that decay geometrically with visit separation.