Why Available Histories Matter
Participants in a clinical trial do not always contribute the same measurements. One participant may stop attending after the second follow-up visit; another may miss the second visit but return for the third; a third may have measurements scheduled at different times. A reference-based prediction must condition on what is actually observed for that participant, not on a standard history that they do not have.
The key calculation is a conditional prediction from the joint distribution specified for the participant’s missing outcomes and available outcomes. The reference-based rule determines the relevant means and covariances; the participant’s observed history determines which entries of those quantities enter the calculation. These are distinct tasks. A correctly imposed reference rule cannot compensate for conditioning on the wrong visits or using covariance entries for the wrong times.
The Pattern-Specific Conditional Prediction
Suppose that, for a particular dropout or observation pattern, the joint model for outcomes at the relevant occasions is multivariate normal. Write its mean vector as \(\mu\) and its covariance matrix as \(\Sigma\), with entries arranged in the same order as the outcomes. The means and covariance may be restricted by a reference-based rule. In particular, they need not be the same for every outcome or every participant pattern.
Partition the mean and covariance into the observed and missing occasions. If \(\Sigma_{OO}\) is nonsingular, the conditional distribution of the missing outcomes given the available history is normal, with conditional mean and covariance
Here \(y_O\) is the participant’s observed vector, \(\mu_O\) and \(\mu_M\) are the means for the observed and target occasions, and \(\Sigma_{MO}\) is the covariance between target and observed outcomes. The expression \(y_O-\mu_O\) is the observed history’s deviation from its model mean. Thus the prediction combines the model mean for the target outcomes with an adjustment informed by the participant’s observed deviations.
For a single target occasion, the conditional variance is a scalar. For several target occasions, it is a matrix: its diagonal entries are conditional variances and its off-diagonal entries describe conditional covariances among the predicted outcomes. That joint covariance matters when predictions at multiple visits are generated together.
Different available histories call for different partitions of the same model, or for different pattern-specific joint moments if the model defines them. A visit absent from a participant’s history must not be silently inserted into \(O\). Nor should an observed value be discarded merely because another participant has a different pattern.
Worked Example: Two Participants With Different Histories
Consider a hypothetical study with outcomes at visits 1 through 4. Under the specified pattern-specific joint model, the target is the outcome at visit 4. Participant A has observed outcomes at visits 1 and 2, while participant B has observed outcomes at visits 1, 2, and 3. The following moments are supplied for the applicable pattern calculations; all outcomes are in points.
For A, suppose \(y_O=(8,10)^\mathsf{T}\), \(\mu_O=(7,9)^\mathsf{T}\), \(\mu_4=11\), \(\Sigma_{OO}=\begin{pmatrix}4&2\\2&9\end{pmatrix}\), \(\Sigma_{4O}=(1,3)\), and \(\Sigma_{44}=16\). The observed residual is \((1,1)^\mathsf{T}\). Since the determinant of \(\Sigma_{OO}\) is \(4(9)-2(2)=32>0\), it is invertible, and
Therefore A’s conditional mean is \(11+(1,3)(7,2)^\mathsf{T}/32=11+13/32=11.41\) points. The conditional variance is \(16-(1,3)(3,10)^\mathsf{T}/32=16-33/32=14.97\) points squared, so the conditional standard deviation is \(\sqrt{14.97}=3.869\) points.
For B, suppose the supplied pattern-specific moments are \(\mu_O=(7,9,10)^\mathsf{T}\), \(y_O=(8,9,11)^\mathsf{T}\), \(\mu_4=11\), \(\Sigma_{OO}=I_3\), \(\Sigma_{4O}=(0.5,1,2)\), and \(\Sigma_{44}=16\). The residual is \((1,0,1)^\mathsf{T}\), so B’s conditional mean is \(11+(0.5,1,2)(1,0,1)^\mathsf{T}=13.5\) points. The variance is \(16-(0.5^2+1^2+2^2)=10.75\) points squared, giving a standard deviation of \(3.279\) points.
The predictions differ because the observed histories and residuals differ, and because the calculation uses each pattern’s corresponding covariance entries. Neither participant’s result can be obtained by pretending that both have the same set of observed visits.
Use Measurement Times, Not Just Visit Labels
Visit labels are convenient only when they identify the same measurement occasions across participants. If one schedule measures at weeks 2, 6, and 12 while another measures at weeks 3, 8, and 12, “visit 2” does not identify a common time. The mean and covariance must correspond to actual measurement times, or to a model that explicitly defines the relationship between those times. A covariance for measurements at weeks 3 and 8 cannot be substituted for one at weeks 2 and 6 merely because both are called visits 1 and 2.
A continuous-time model may provide means \(\mu(t)\) and covariances \(C(s,t)\) directly at scheduled times. A discrete-visit model may instead define moments only for specified occasions. In either case, use only moments supported by the model. Do not create an outcome at an unscheduled time by treating it as a missing visit unless the estimand and model explicitly call for that prediction.
Worked Example: Conditioning at Unequally Spaced Times
For a hypothetical participant, suppose the observed measurements were scheduled at weeks 2 and 5, and the missing scheduled measurement is at week 6. A fitted reference-based joint model supplies the relevant means in millimetres: \(\mu_O=(5,7)^\mathsf{T}\), \(\mu_6=10\); and covariances in square millimetres: \(\Sigma_{OO}=\begin{pmatrix}4&1\\1&4\end{pmatrix}\), \(\Sigma_{6O}=(1,2)\), and \(\Sigma_{66}=9\). The observed values are \(y_O=(6,8)^\mathsf{T}\). Assume these supplied moments are from a valid joint Gaussian model and that \(\Sigma_{OO}\) is invertible; its determinant is \(15>0\).
The residual vector is \((1,1)^\mathsf{T}\). Since \(\Sigma_{OO}^{-1}=\frac{1}{15}\begin{pmatrix}4&-1\\-1&4\end{pmatrix}\), multiplying this inverse by the residual gives \((3/15,3/15)^\mathsf{T}\). The conditional mean is therefore
For the conditional variance, \(\Sigma_{OO}^{-1}\Sigma_{O6}=(2/15,7/15)^\mathsf{T}\), so \(\Sigma_{6O}\Sigma_{OO}^{-1}\Sigma_{O6}=(2+14)/15=16/15\). The conditional variance is \(9-16/15=7.933\) square millimetres, and the conditional standard deviation is \(2.817\) millimetres. These calculations use the covariance for weeks 2, 5, and 6—not a covariance chosen by matching visit numbers from another schedule.
Scheduled Missing Outcomes and Unscheduled Times
A prediction is appropriate only for an outcome included in the analysis target. If a participant was scheduled at weeks 0, 3, and 6, has observed outcomes at weeks 0 and 3, and missed week 6, then week 6 belongs to \(M\). A week-4 measurement that was never scheduled does not. It should not be added to the missing-outcome vector simply to make the participant’s record resemble another participant’s schedule.
Worked Example: A Missing Scheduled Visit, Not an Extra Visit
Suppose a hypothetical participant has an observed outcome of 23 units at week 3 and a scheduled but missing outcome at week 6. Under the specified pattern-specific model, let \(\mu_3=20\), \(\mu_6=24\) units, \(\Sigma_{33}=9\), \(\Sigma_{63}=4\), and \(\Sigma_{66}=16\), with the variances and covariance in square units. The observed residual is \(23-20=3\) units, and the observed variance is positive.
The conditional mean at week 6 is \(24+(4/9)(3)=25.33\) units. Its conditional variance is \(16-4(1/9)4=14.22\) square units, so its conditional standard deviation is \(3.771\) units. Week 4 is not part of the calculation: it was not a scheduled outcome, and no prediction at week 4 is being requested.
Justification, Diagnostics, and Limitations
The conditional formula follows by subtracting the linear prediction from the missing outcomes. Define \(R=Y_M-\mu_M-\Sigma_{MO}\Sigma_{OO}^{-1}(Y_O-\mu_O)\). Its covariance with \(Y_O\) is zero, because
For jointly normal variables, zero covariance implies independence. Thus \(R\) is independent of \(Y_O\), has mean zero, and has covariance \(\Sigma_{MM}-\Sigma_{MO}\Sigma_{OO}^{-1}\Sigma_{OM}\). Adding back the linear prediction gives the stated conditional mean and covariance. The argument requires a valid joint Gaussian model and an invertible observed-history covariance; it does not justify the reference-based restrictions themselves.
Before using the result, check that the selected observed values, target outcomes, and time ordering match the model inputs. Check that \(\Sigma_{OO}\) is invertible and that the conditional covariance is positive semidefinite, up to small numerical rounding error. A negative conditional variance is not a meaningful prediction; it signals incompatible moments, an indexing error, or numerical instability. Near-singularity can also make predictions highly sensitive to small changes in the inputs.
The model’s assumptions deserve separate scrutiny. The joint distribution must be adequate for the outcome and time scale, and the reference-based restrictions must represent the intended post-deviation behavior. A conditional calculation does not establish that unobserved outcomes follow those restrictions. Predictions at times with little or no model support may rely heavily on extrapolation. Likewise, a history with unusual values may produce a conditional mean outside a familiar range; that is a prompt to inspect model fit and support, not an automatic reason to truncate the prediction.
| Approach | What it uses | Main caution |
|---|---|---|
| Pattern-specific conditional prediction | Each participant’s observed set, target set, and corresponding joint moments | Requires correct indexing and adequate pattern-specific moments |
| One common history for everyone | A fixed set of visits, often created by padding or dropping measurements | Can condition on outcomes a participant never had or discard observed information |
| Sequential visit-by-visit prediction | Earlier outcomes, with predictions potentially used as later conditioning values | Is not automatically the same as conditioning jointly on the original observed history |
The pattern-specific calculation is preferable when observed histories differ because it makes the conditioning set explicit. A common-history shortcut is valid only if it preserves each person’s actual information and the model’s intended joint distribution. Sequential prediction can be useful when the model and imputation procedure are defined sequentially, but it should not be substituted for the joint conditional distribution without justification. The introductory material on reference-based imputation establishes the role of the reference restriction; here, the central practical limitation is that a correct restriction still requires a coherent schedule, history, and covariance mapping.
Common Mistakes
- Using the wrong covariance entries. Matching “visit 2” across schedules with different times can use a covariance for the wrong pair of occasions. Select entries by their actual modeled occasions or times.
- Padding an absent history with an invented value. A participant with no measurement at a scheduled occasion has no observed value there to condition on. Do not insert a mean or zero as if it were observed.
- Imputing a visit that was never scheduled. An unscheduled time is not automatically a missing outcome. Define the target occasions before calculating.
- Using the reference mean but ignoring the cross-covariance. The conditional mean is generally adjusted by \(\Sigma_{MO}\Sigma_{OO}^{-1}(y_O-\mu_O)\); it equals the marginal target mean only in special cases, such as zero cross-covariance or zero observed residuals.
- Treating a numerical failure as a harmless detail. A singular observed covariance or negative conditional variance requires investigation of the model and indexing, not an arbitrary replacement value.
Check Your Understanding
Use the conditional Gaussian formula and keep the measurement occasions explicit.
- A participant has one observed outcome \(y_1=7\), with \(\mu_1=6\), target mean \(\mu_2=10\), variance \(\Sigma_{11}=4\), covariance \(\Sigma_{21}=1\), and target variance \(\Sigma_{22}=9\). Find the conditional mean and variance for outcome 2.
- A participant has an observed outcome at week 5 and a missing scheduled outcome at week 10. A week-7 measure was never scheduled. Should week 7 be included in the missing vector? Explain.
- Two participants have different observed visit sets. What must be checked before applying the conditional formula to each participant?
- Suppose the observed-history covariance matrix is singular. Is the usual inverse-based formula available as written? State an appropriate response.
Solutions. (1) The observed residual is \(7-6=1\), so the conditional mean is \(10+(1/4)(1)=10.25\) units. The conditional variance is \(9-1(1/4)1=8.75\) square units. (2) No. Week 7 was not scheduled, so it is not a missing scheduled outcome to impute; the target is the missing week-10 outcome. (3) Check each participant’s observed and target sets, actual measurement times and ordering, and the matching pattern-specific means and covariance blocks. Also verify that the observed covariance is invertible and the resulting conditional covariance is valid. (4) No. The inverse-based expression is not available as written. Investigate redundant or incorrectly indexed measurements and the model specification; use an appropriate revised model or a justified method for the singular case rather than silently forcing an inverse.