Why the Outcome Model Matters
A Diggle-Kenward selection model combines two ingredients: a model for the repeated outcomes that would be defined over the study visits, and a model for the dropout process that can depend on outcomes. The outcome model is not just a technical prelude to the selection equation. Its mean and covariance determine how outcomes at different visits relate, including the unobserved outcomes after dropout that enter the joint model.
The choice is consequential. A mean model that omits a time trend can misrepresent the expected trajectory; a covariance model that imposes an implausible dependence pattern can distort the likelihood and uncertainty. Conversely, a flexible covariance model may be difficult to estimate when dropout leaves little information about later visits. The aim is to specify a scientifically reasonable outcome distribution that is explicit enough to support the selection factorization.
This is a model for the full outcome vector conditional on covariates, not merely a separate model fitted to the outcomes that happen to be observed at each visit. The selection component then models the response or dropout indicators using the relevant outcome history. In the monotone setting, define \(R_{it}=1\) when visit \(t\)'s outcome is observed. A subject who drops out after visit \(t\) has \(R_{i1}=\cdots=R_{it}=1\) and \(R_{i,t+1}=\cdots=R_{iT}=0\). The outcome distribution and the dropout model play distinct roles: specifying one does not specify the other.
Constructing the Mean Trajectory
The mean model describes the expected outcome at each scheduled visit, conditional on included covariates. Time should be represented in a way that makes the coefficients interpretable and can represent the scientific trajectory. For example, a linear time trend may be appropriate over a short interval; a categorical visit effect allows each visit to have its own mean without imposing linearity. If groups may follow different trajectories, include a time-by-group interaction rather than assuming that a group difference is constant.
For a continuous outcome and a binary baseline group indicator \(A_i\), one possible linear mean model is
Here \(t\) is the chosen numeric time scale, \(\beta_0\) is the expected outcome at \(t=0\) for the reference group, and \(\beta_3\) is the difference in linear time slopes between groups. If the trajectory is not plausibly linear, a categorical-time model can instead assign a separate mean parameter to each visit, with group-by-visit terms if needed. The time coding and reference levels should be stated: changing the origin of time changes coefficient interpretations even when it does not change fitted means.
Worked Example: Group-Specific Mean Trajectories
Consider a hypothetical four-visit study of a continuous symptom score, with larger values indicating more symptoms. Let \(t=0,1,2,3\) index visits and \(A_i=1\) denote the treatment group. Suppose the specified mean model has \(\beta_0=10\), \(\beta_1=-1.2\), \(\beta_2=0.5\), and \(\beta_3=-0.4\), all in score units per the relevant coding.
For the reference group, \(A_i=0\), the expected score is \(10-1.2t\). Its four fitted means are \(10.0,\ 8.8,\ 7.6,\) and \(6.4\) score points.
For the treatment group, \(A_i=1\), the fitted mean is \(10-1.2t+0.5-0.4t=10.5-1.6t\). The four means are \(10.5,\ 8.9,\ 7.3,\) and \(5.7\) score points. For example, at visit \(t=2\), substitution gives \(10-1.2(2)+0.5-0.4(2)=7.3\).
These are model-based conditional means, not claims that every subject follows the same trajectory. The model assumes a linear mean trend over these four coded visits and a group-specific slope. Whether those assumptions are adequate should be assessed against the measurement schedule and observed data.
Choosing a Covariance Structure
The covariance matrix describes variability at each visit and dependence between visits after accounting for the mean model. Its diagonal entries are visit-specific variances; its off-diagonal entries are covariances. A covariance structure is a set of restrictions on those entries. The restriction should reflect plausible dependence and be estimable from the available data.
| Structure | Restriction | When it may be useful | Main caution |
|---|---|---|---|
| Unstructured | Each variance and covariance is estimated separately. | There are few visits and enough information to estimate many parameters. | Parameter count grows quickly; estimates may be unstable. |
| Compound symmetry | All visits share a variance and every distinct pair has the same covariance. | Dependence is plausibly similar across all visit pairs. | It ignores the possibility that more distant visits are less correlated. |
| AR(1) | With equally spaced visits, covariance decays geometrically with the visit lag. | Adjacent visits are expected to be more alike than distant visits. | The equally spaced, stationary pattern may not fit irregular schedules or changing variability. |
| Random intercept and slope | Shared subject-specific effects induce dependence, with a specified residual variance structure. | Subjects plausibly vary in their baseline levels and trajectories. | The induced covariance pattern may be too restrictive for the data. |
An unstructured covariance matrix is flexible, but with \(T\) visits it requires \(T(T+1)/2\) covariance parameters. This can be costly when later visits have few observed outcomes. More structured alternatives reduce the number of parameters by imposing assumptions. For example, a homogeneous AR(1) model for equally spaced visits uses a common variance \(\sigma^2\) and a correlation parameter \(\rho\), with covariance \(\sigma^2\rho^{|t-s|}\). A compound-symmetry model instead uses one common variance and one common correlation for all distinct visit pairs.
Worked Example: Covariance From Random Intercepts and Slopes
Suppose a hypothetical continuous outcome is measured at times \(t=0,1,2\), and the model includes a subject-specific random intercept \(b_{0i}\), a random slope \(b_{1i}\), and independent visit-level errors \(\epsilon_{it}\): \(Y_{it}=\mu_{it}+b_{0i}+b_{1i}t+\epsilon_{it}\). Assume the random effects have mean zero, \(\operatorname{Var}(b_{0i})=4\), \(\operatorname{Var}(b_{1i})=0.25\), \(\operatorname{Cov}(b_{0i},b_{1i})=-0.5\), and \(\operatorname{Var}(\epsilon_{it})=1.5\). The errors are independent across visits and independent of the random effects.
The covariance between visits \(t\) and \(s\) is
For example, the variance at \(t=1\) is \(4-0.5(2)+0.25(1)(1)+1.5=4.75\). The covariance between \(t=0\) and \(t=2\) is \(4-0.5(2)+0.25(0)(2)=3.0\). Applying the same calculation to all entries gives
The random-effects covariance matrix is valid here because its determinant is \(4(0.25)-(-0.5)^2=0.75>0\), and its first diagonal entry is positive; hence it is positive definite. Adding independent errors with positive variance adds \(1.5\) to each outcome variance and preserves positive definiteness. This model implies decreasing variability across these visits, so that feature should be plausible for the outcome and time coding before the structure is adopted.
What the Covariance Assumption Changes
Covariance assumptions affect the distribution of observed outcomes and how the model links observed history to unobserved future outcomes. The following calculation illustrates why two structures that have the same marginal variance can imply different uncertainty for a change.
Worked Example: A Change Has Different Variance Under Two Structures
Suppose three equally spaced visits have marginal variance \(4\) score-squared units. Under an AR(1) model with adjacent-visit correlation \(\rho=0.6\), the covariance between visits 1 and 3 is \(4(0.6)^2=1.44\) score-squared units. Therefore,
Under compound symmetry with the same variance and a common correlation of \(0.6\), the covariance between visits 1 and 3 is \(4(0.6)=2.4\) score-squared units. The corresponding calculation is \(4+4-2(2.4)=3.2\) score-squared units. Thus, although both models use the same marginal variance and adjacent-visit correlation, they make different assumptions about the association between visits 1 and 3 and yield different variability for the change. Neither result is universally preferable; the structure must be justified for the particular outcome and visit schedule.
For a Gaussian outcome model, a useful marginalization fact follows directly from the multivariate normal distribution: if only visits in a set \(O\) are considered, their marginal distribution is normal with mean vector \(\mu_{i,O}\) and covariance submatrix \(\Sigma_{i,OO}\). This fact supplies the outcome marginal for observed visits. It does not mean that the entire Diggle-Kenward observed-data likelihood is just that marginal density. The response-process factors also enter, and outcomes after dropout may need to be integrated out in the joint likelihood.
Worked Example: Extracting the Observed-Visit Marginal
Suppose a hypothetical four-visit continuous outcome follows a Gaussian AR(1) model with visit-specific mean vector \((12,10,8,6)^\mathsf{T}\), common variance \(9\) score-squared units, and \(\rho=0.5\). A subject remains observed through visit 3 and then drops out. The outcome marginal for the observed visits uses the first three means and the covariance submatrix for visits 1, 2, and 3.
The lag-one covariance is \(9(0.5)=4.5\), and the lag-two covariance is \(9(0.5)^2=2.25\). Thus the observed-visit marginal has mean \((12,10,8)^\mathsf{T}\) and covariance
If the observed scores are \(11,9,7\), their outcome-density contribution is the trivariate normal density evaluated at \((11,9,7)^\mathsf{T}\) with that mean and covariance. In a selection model, this is not by itself the full likelihood contribution for the subject: the dropout factor must also be included, with the unobserved visit-4 outcome handled as required by the joint factorization. This example checks the covariance submatrix and mean used for the observed outcomes; it does not assume that dropout is independent of the outcomes.
Justification, Diagnostics, and Limits
The marginalization statement used above can be justified by partitioning a jointly Gaussian vector into observed and unobserved components. Write \(Y_i=(Y_{i,O}^\mathsf{T},Y_{i,M}^\mathsf{T})^\mathsf{T}\), where \(O\) and \(M\) index the observed and missing visits. Any linear combination of \(Y_{i,O}\) is also a linear combination of the full Gaussian vector, so it is normally distributed. Its expectation and covariance are the corresponding subvector and submatrix, \(\mu_{i,O}\) and \(\Sigma_{i,OO}\). Therefore \(Y_{i,O}\mid X_i\sim N(\mu_{i,O},\Sigma_{i,OO})\). This establishes the observed-outcome marginal, but not the response-process contribution to the selection-model likelihood.
Before using a covariance structure, check that its parameters produce a symmetric positive-definite covariance matrix for the scheduled visits. Also check whether the visit spacing, variability, and scientific process make its restrictions plausible. Useful diagnostics include plots of observed outcomes and residuals against time, comparisons of fitted means with visit-specific observed summaries, and inspection of fitted variances and correlations. These checks can reveal a poor fit among observed data, but they cannot directly verify the model for outcomes after dropout: those values are not observed.
A practical comparison is to fit a small number of scientifically defensible alternatives, such as an unstructured covariance when the number of visits and information permit it, and a simpler structure motivated by the trajectory or measurement process. Assess stability, convergence, estimated correlations, and whether substantive conclusions depend heavily on the choice. In a selection model, a covariance structure can also affect how information from observed outcomes relates to unobserved outcomes. Agreement among observed-data diagnostics is therefore not proof that the extrapolation beyond dropout is correct.
State the time coding, covariates, and any group-by-time terms needed to represent the expected trajectory.
Match its restrictions to the number and spacing of visits, and explain why its dependence pattern is plausible.
Confirm symmetry, positive definiteness, sensible variances and correlations, and estimability with the available observed data.
Use the outcome model for the joint distribution of repeated outcomes and the selection model for dropout; do not infer one from the other.
A frequent mistake is to estimate a separate mean and variance from the observed subjects at each visit and treat those summaries as the full outcome model. That reasoning ignores that the observed group can change across visits because of dropout. The correction is to define a coherent longitudinal outcome distribution conditional on covariates, then combine it with the selection model and the appropriate integration over unobserved outcomes. Another pitfall is choosing an elaborate covariance merely because it is flexible: if late-visit information is sparse, the extra parameters may be weakly estimated or the fit may fail. Simplicity is useful when justified, not when it silently imposes an implausible pattern.
No covariance structure makes the selection assumptions empirically certain. Different full-data models can agree closely on observed outcomes while implying different distributions for outcomes after dropout. The mean and covariance should therefore be reported as components of the joint model, along with their restrictions and diagnostic evidence. Their adequacy is a modeling judgment rather than a conclusion guaranteed by observed-data fit alone.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In a mean model with numeric time \(t\), why might a time-by-group interaction be needed?
- An unstructured covariance has \(T=5\) visits. How many distinct covariance parameters does it require?
- Under an AR(1) model with variance \(9\) and \(\rho=0.5\), what is the covariance at lag 3?
- Why is the Gaussian marginal density for observed visits not, by itself, the complete observed-data likelihood in a selection model?
Solutions. (1) It allows the expected time trend to differ between groups; without it, the model imposes a common slope when time is numeric. (2) The count is \(T(T+1)/2=5(6)/2=15\) parameters: five variances and ten distinct covariances. (3) The covariance is \(9(0.5)^3=9(0.125)=1.125\) outcome-squared units. (4) The selection-model likelihood also includes response-process factors, and outcomes after dropout may need to be integrated out. The observed-outcome marginal accounts only for the outcome density at the observed visits.