Why Means Do Not Determine Covariances
Reference-based imputation specifies how outcomes after dropout are related to outcomes in a reference arm. A crucial distinction is that a restriction on means does not, by itself, specify how outcomes vary together. Two imputation distributions can have the same post-dropout marginal mean and different variances, covariances, and conditional predictions.
This distinction matters when comparing jump-to-reference (J2R) with copy-reference (CR). J2R is commonly described through a post-dropout mean restriction. To obtain a complete predictive distribution, an implementation must also state which covariance matrix it uses. Copy-reference, by contrast, uses the reference-arm conditional distribution for the missing outcomes. The distinction is clearest when the treatment and reference arms have different covariance matrices.
The formulas below use a multivariate normal model, as in the earlier material on conditional normal distributions for missing visits. They describe a covariance convention, not a claim that the convention is determined by the J2R mean restriction itself.
A common, explicitly stated convention pairs the J2R mean restriction with the treatment-arm covariance matrix. Call this the treatment-covariance J2R convention. Under it, the imputation model uses the J2R mean vector \(\mu_J\), whose observed-period components follow the treatment-arm means and whose post-dropout components follow the reference-arm means, together with \(\Sigma_T\). This is one way to complete the J2R mean restriction into a joint distribution; it is not the only possible covariance convention.
Under copy-reference, the missing outcomes are predicted using the reference-arm distribution, including its covariance structure. In particular, the reference-arm conditional relationship is applied to the participant’s observed history. Thus, if arm covariance matrices differ, CR and treatment-covariance J2R can differ in both the conditional mean and conditional covariance.
Conditional Distributions and the Covariance Choice
For any jointly normal outcome vector partitioned into observed and missing components, write its mean and covariance in matching blocks:
Provided \(\Sigma_{OO}\) is nonsingular, the conditional distribution of the missing outcomes given the observed history \(Y_O=y_O\) is normal. Its mean adjusts the marginal missing-outcome mean according to the observed residuals, and its covariance is the Schur complement of \(\Sigma_{OO}\):
The second argument of \(N(\cdot,\cdot)\) is a covariance matrix (a variance in the one-dimensional case). Under treatment-covariance J2R, use \(\mu_J\) and \(\Sigma_T\) in this formula. Under CR, use \(\mu_R\) and \(\Sigma_R\). In both cases the conditioning values are the participant’s observed values; what changes is the model used to relate that history to the missing outcomes.
Why the Conditional Covariance Has This Form
A short derivation makes the role of covariance explicit. For a jointly normal vector with the block covariance above, define the residual
Its covariance with \(Y_O\) is zero:
Joint normality means that zero covariance here implies independence. Therefore, conditional on \(Y_O=y_O\), the residual \(Z_M\) retains its unconditional covariance. Expanding that covariance gives
The remaining nonrandom part of \(Y_M\) conditional on \(Y_O=y_O\) is the stated conditional mean. This derivation also shows why changing covariance assumptions changes more than a reported uncertainty: the regression adjustment \(\Sigma_{MO}\Sigma_{OO}^{-1}\) in the conditional mean can change too.
Worked Example: Different Covariances Give Different Predictions
Consider hypothetical two-visit outcome scores, measured in points, for a treatment participant who drops out after visit 1. The visit 1 score is 60. The treatment-arm means at visits 1 and 2 are 58 and 61; the reference-arm means are 54 and 55. Suppose the treatment covariance matrix is
and the reference covariance matrix is
Both matrices are valid covariance matrices: their variances are positive and their determinants are positive (\(16(25)-12^2=256\), and \(9(16)-6^2=108\)).
Under treatment-covariance J2R, the missing visit 2 marginal mean is the reference mean, 55 points, while the visit 1 mean remains the treatment mean, 58 points. The conditional mean and variance for visit 2 are
Under CR, the reference-arm conditional distribution gives
Both rules set the missing-outcome marginal mean to the reference mean of 55 points. Their conditional means differ because the observed score is compared with different visit 1 means and because the covariance-based regression slopes differ. Their conditional variances also differ: 16 versus 12 points squared. The calculations therefore do not support the claim that the two methods merely use different labels for the same predictive distribution.
What the Covariance Convention Implies
Under treatment-covariance J2R, the covariance matrix is retained from the treatment arm even though the post-dropout marginal means are set to reference values. Consequently, the model can combine reference-level post-dropout means with treatment-arm variability and cross-visit association. Under CR, both the conditional prediction and its uncertainty are based on the reference-arm distribution. The two approaches may therefore encode different assumptions about how much missing outcomes vary and how they co-move with the observed history.
The choice should be documented precisely. A statement such as “we used J2R” identifies a mean restriction, but may not fully identify the predictive distribution. A clear report says whether the J2R mean vector was paired with treatment, reference, or common covariance parameters, and states how the missing outcomes were conditioned on observed values.
Worked Example: Same J2R Means, Different Missing-Outcome Variance
Suppose the visit 1 and visit 2 means are 58 and 55 points under the J2R mean restriction. Consider two possible covariance conventions for this same mean vector. One uses the treatment covariance entries from the preceding example; the other uses the reference covariance entries. The marginal variance assigned to the missing visit 2 outcome is 25 points squared under the first convention and 16 points squared under the second.
These are not conditional variances. Given the observed visit 1 score, the treatment-covariance convention gives \(25-12^2/16=16\) points squared, while the reference-covariance convention gives \(16-6^2/9=12\) points squared. Thus the same J2R marginal mean restriction permits distinct predictive uncertainties, and the conditioning step preserves that difference.
This comparison isolates the point of the example: the post-dropout mean vector alone cannot select either variance. The covariance assumption has to be supplied separately. Changing covariance can also change the conditional mean through its regression slope, so implementations that differ in covariance need not agree even on individual imputations.
Worked Example: When the Covariance Distinction Disappears
Now suppose, hypothetically, that the analysis uses a common covariance matrix for both arms:
This is a valid covariance matrix because its determinant is \(16(25)-8^2=336>0\). For the same visit 1 value of 60, treatment-covariance J2R uses the visit 1 treatment mean of 58 and the visit 2 reference mean of 55:
CR uses the reference visit 1 mean of 54 and reference visit 2 mean of 55, but the same covariance matrix:
The conditional variances coincide because the covariance matrices are equal. The conditional means do not coincide because the mean vectors used for the observed history differ. Equal covariance assumptions remove this particular covariance contrast; they do not make J2R and CR identical distributions in general.
Checks, Comparisons, and Limitations
The conditional formulas require a coherent covariance model. In practice, check that the relevant observed-outcome covariance block is invertible and that the full covariance matrix is positive definite (or otherwise valid for the specified model). Implausible conditional variances, extreme regression slopes, or large changes in predictions under modest covariance changes can signal unstable estimation or a poorly supported covariance assumption. These checks do not establish that the post-dropout assumption is true; the identifying restriction is an assumption about unobserved outcomes.
The main distinctions can be summarized as follows:
| Feature | Treatment-covariance J2R convention | Copy-reference |
|---|---|---|
| Post-dropout marginal mean | Set to the reference-arm mean | Reference-arm distribution |
| Covariance used for conditioning | Treatment-arm covariance, under this stated convention | Reference-arm covariance |
| Conditional mean | Uses the J2R mean vector and treatment covariance | Uses reference mean and reference covariance |
| Conditional covariance | Schur complement from treatment covariance | Schur complement from reference covariance |
A common-covariance model makes the covariance columns agree, but the mean restrictions can still yield different predictions. Conversely, using reference covariance with a J2R mean vector is a possible documented implementation, but it should not be called the same thing as treatment-covariance J2R without explaining that choice. These methods specify predictive distributions under assumptions; they do not reveal the actual unobserved outcomes or prove that a reference behavior is correct.
Common Pitfall
- Wrong reasoning: “J2R sets the missing mean to the reference mean, so its missing outcomes must have the reference variance and covariance.”
- Correction: A mean restriction fixes mean components only. The covariance must be specified separately. Under the treatment-covariance J2R convention, the missing mean can be reference-based while the covariance is treatment-based; CR uses the reference distribution for its conditional prediction.
- Another trap: “The conditional mean of a missing outcome is always its marginal reference mean.” This ignores the observed-history adjustment. Use the conditional-normal formula and check both the mean vector and covariance matrix that the chosen rule specifies.
Check Your Understanding
Answer each question, then compare with the short solutions.
- Does setting a missing visit’s marginal mean to the reference mean determine its conditional variance? Why?
- With treatment-covariance J2R, what covariance matrix is used in the conditional-normal formula?
- In the first worked example, why did the two conditional means differ even though both rules used 55 points as the missing visit’s marginal mean?
- If treatment and reference covariance matrices are equal, which covariance-based feature of the two rules must agree, and which feature may still differ?
- Name two checks that can identify a problematic covariance model, and explain why passing them does not verify the post-dropout assumption.
Solutions. (1) No. The mean restriction does not specify covariance; the conditional variance requires the covariance blocks through the Schur complement. (2) The treatment-arm covariance matrix, under the stated treatment-covariance J2R convention. (3) The rules used different observed-history means and covariance-based regression slopes, so their conditional adjustments differed. (4) The conditional covariance from the shared covariance matrix agrees, while the conditional means may differ because the mean vectors differ. (5) Check positive definiteness and invertibility of the observed block, and inspect implausible conditional variances or unstable slopes. These are mathematical or model-stability checks; they cannot establish behavior for outcomes that were not observed.