Why the Mean Structure Matters
In a longitudinal study with monotone dropout, participants contribute different numbers of repeated outcomes. A pattern-mixture model describes the outcome distribution separately for each dropout pattern. Its mean structure determines how much the expected outcome trajectory is allowed to differ between patterns. A saturated structure permits each pattern-by-visit mean to vary freely; a parsimonious structure links some of those means.
That choice is consequential. The data directly inform means at visits observed within a pattern. A mean after that pattern has dropped out is not observed in that pattern. A model may nevertheless assign it a value by imposing restrictions that link it to observed means. Such a value is an implication of the restrictions, not a direct estimate from outcomes collected after dropout.
This tutorial concerns mean structures, not the covariance structure within a pattern. As in the earlier material on pattern-specific outcome distributions, a full pattern-specific outcome model can describe outcomes at all scheduled visits, including those not observed for a particular pattern. The observed-data likelihood, however, contains outcome information only for visits actually observed.
Saturated and Parsimonious Structures
A saturated mean structure assigns a separate parameter to every pattern-by-visit cell, \(\mu_{dt}\), for \(d,t=1,\ldots,K\). It has \(K^2\) mean parameters if the full \(K\)-visit vectors are specified for every pattern. This places no equality or shape restrictions on the trajectories. In particular, the post-dropout means are free parameters too, but the observed outcomes within that pattern do not directly estimate them.
There are only \(K(K+1)/2\) observed pattern-by-visit cells under monotone dropout, because pattern \(d\) contributes observed outcomes at visits \(1,\ldots,d\). A saturated model for those observed cells therefore has \(K(K+1)/2\) directly observed means. The other \(K(K-1)/2\) cells in the full \(K\)-by-\(K\) matrix are post-dropout means. Without restrictions or additional information, they are not identified by the observed outcomes.
A common parsimonious alternative is an additive pattern-and-visit structure:
Here \(\alpha_t\) describes the visit-specific mean in the complete-visit pattern, and \(\gamma_d\) is the pattern offset relative to that pattern. The reference constraint \(\gamma_K=0\) is needed because adding a constant to every \(\alpha_t\) and subtracting it from every \(\gamma_d\) would otherwise leave all \(\mu_{dt}\) unchanged. This additive model has \(K\) visit parameters and \(K-1\) pattern parameters, or \(2K-1\) mean parameters.
The additive model says more than “there are fewer parameters.” It assumes a pattern’s mean offset is the same at every visit. For example, if pattern \(d\) is lower than the complete-visit pattern by 2 outcome units at visit 1, the model requires it to be lower by 2 units at every later visit as well. Its post-dropout predictions follow from that same restriction. A still more restrictive structure, \(\mu_{dt}=\alpha_t\), assumes no pattern differences at all.
Worked Example: Counting Parameters and Checking Additivity
Consider a hypothetical study with three visits and three dropout patterns. Suppose the following matrix gives a set of hypothetical pattern-specific means, in outcome units:
| Last observed visit \(d\) | Visit 1 | Visit 2 | Visit 3 |
|---|---|---|---|
| 1 | 10 | 9 | 8.5 |
| 2 | 11 | 10 | 9.5 |
| 3 | 12 | 11 | 10.5 |
The saturated full mean matrix has \(3^2=9\) parameters. Only \(1+2+3=6\) cells correspond to observed visits: the pattern-1 mean at visit 1, the two pattern-2 means at visits 1 and 2, and all three pattern-3 means. The remaining three cells are post-dropout means.
For the additive structure, take \((\alpha_1,\alpha_2,\alpha_3)=(12,11,10.5)\), \(\gamma_3=0\), \(\gamma_2=-1\), and \(\gamma_1=-2\). Then, for example, \(\mu_{22}=\alpha_2+\gamma_2=11+(-1)=10\), and \(\mu_{13}=\alpha_3+\gamma_1=10.5+(-2)=8.5\). The same calculation gives every entry in the table. Thus this particular matrix is compatible with additivity, even though most entries are not observed for their respective patterns.
With three visits, the additive model uses \(2(3)-1=5\) parameters rather than 9 for the full matrix. Among observed cells it imposes one restriction: the pattern-2 versus pattern-3 difference must be the same at visits 1 and 2. Here both differences are \(11-12=-1\) and \(10-11=-1\), as required. The model also fixes the post-dropout means through its shared parameters; the data do not verify those unobserved entries directly.
What the Additive Restriction Implies
The main restriction can be derived directly. For any two patterns \(d\) and \(e\), at any visit \(t\) where both patterns have observed outcomes, the additive model gives:
The visit term cancels. Therefore, pattern differences must be constant across visits where that pair of patterns is observed. For patterns \(d\) and \(e\), the shared observed visits are \(t\le \min(d,e)\). If those differences change over the shared visits, the observed means cannot all satisfy the additive restriction.
This is a useful diagnostic, but it is not a test of whether an unobserved post-dropout mean is correct. The diagnostic examines equalities among observed means. Even perfect agreement with those equalities cannot validate a restriction for a visit at which a pattern has no observations.
Worked Example: A Shared-Visit Contrast as a Diagnostic
Suppose hypothetical observed means, in outcome units, are 11 and 9 for the pattern with last observed visit 2 at visits 1 and 2, and 12 and 11 for the complete-visit pattern at those same visits. Both patterns are observed at both visits, so their differences can be compared.
At visit 1 the pattern-2 minus complete-pattern difference is \(11-12=-1\) unit. At visit 2 it is \(9-11=-2\) units. The contrast between these differences is
The contrast is not zero, so these four means do not satisfy an additive pattern-and-visit structure. A saturated structure can represent them without that restriction. If these were estimates from a finite sample, sampling variability would matter when assessing the evidence against additivity; the arithmetic discrepancy alone is a diagnostic, not a significance test.
Worked Example: What a Parsimonious Model Extrapolates
Worked Example: A Two-Visit Dropout Pattern
Imagine a hypothetical two-visit study. The observed mean at visit 1 among participants whose last observed visit is 1 is 8 outcome units. Among participants observed at both visits, the means are 10 units at visit 1 and 8 units at visit 2. These values are chosen for illustration.
Under the additive structure, \(\alpha_1=10\) and \(\alpha_2=8\), because \(\gamma_2=0\) for the complete-visit pattern. The first pattern’s visit-1 mean implies \(\gamma_1=8-10=-2\) units. The model consequently predicts its visit-2 mean as
That visit-2 outcome is not observed for pattern 1. The prediction of 6 units follows from the assumption that this pattern remains 2 units below the complete pattern at visit 2, just as it was at visit 1. A saturated full mean structure would leave \(\mu_{12}\) unrestricted; the observed data alone do not choose between 6 units and other values.
There is an important edge case here. With two visits, the three observed cells are \((1,1)\), \((2,1)\), and \((2,2)\). The additive model has \(2(2)-1=3\) parameters, so it can match any three observed means exactly while still determining an unobserved fourth mean. Thus an exact fit to observed cell means need not mean the post-dropout prediction is empirically checked.
Assumptions, Estimation, and Diagnostics
The choice between structures is a modeling decision. A saturated structure assumes no equality among pattern-specific means, but it still requires enough observed information to estimate the means it includes. Parsimonious structures assume specific equalities or links. For the additive structure, those include constant pattern differences across shared visits. To identify its parameters from observed data in the usual design, each pattern must have observations and the complete-visit pattern must provide outcomes at all \(K\) visits; the observed-cell design must also have adequate information to estimate the parameters.
In a normal pattern-specific model, the mean parameters are estimated from the observed outcome vectors for each pattern, with the likelihood accounting for the distribution of those vectors. A saturated full matrix cannot obtain information about post-dropout means merely by adding them as parameters: those means do not enter the observed-outcome contribution for the affected pattern. Restrictions such as additivity can connect unobserved cells to estimable parameters, but the resulting values depend on those restrictions.
Practical checks include examining observed pattern-by-visit means and their uncertainty, plotting trajectories only over visits shared by the patterns being compared, and calculating the shared-visit contrasts implied by the model. Sparse pattern sizes make cell means unstable, while empty patterns or missing visits within a pattern can compromise identifiability. A numerical fitting routine’s convergence does not establish that a mean structure is plausible.
The principal alternatives can be summarized as follows:
| Mean structure | What it permits | Main consequence |
|---|---|---|
| Saturated pattern-by-visit means | Every cell mean is free | Flexible for observed means; post-dropout means are not identified from observed outcomes alone |
| Additive pattern and visit effects | Pattern offsets are constant across visits | Fewer parameters; post-dropout means are implied by the additive restriction |
| Common visit means | No pattern differences at any visit | Most restrictive; rules out mean differences associated with dropout pattern |
The saturated structure is a useful reference for flexibility, but its large number of parameters can be difficult to estimate in small pattern strata, and its post-dropout means remain unlearned from observed outcomes. The additive structure reduces dimension and supplies extrapolations, but its simplicity is not evidence that constant offsets are scientifically credible. A common-mean structure is simpler still, but may erase meaningful differences among dropout patterns. No choice of mean parameterization alone resolves the assumptions needed to learn about outcomes that were not observed.
Common Pitfall
- Wrong reasoning: “The additive model fits all observed pattern means, so its post-dropout predictions are confirmed by the data.”
- Correction: A model can fit observed cells exactly and still impose untestable restrictions on unobserved cells. In the two-visit example, there are as many additive parameters as observed means; the predicted visit-2 mean for the early-dropout pattern is nevertheless an extrapolation.
- Wrong reasoning: “Saturated means are always safer because they make no assumptions.”
- Correction: Saturation avoids equality restrictions, but it does not create information about post-dropout outcomes. Those means remain unidentified from the observed data without additional assumptions or information.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- With four visits and four monotone dropout patterns, how many mean parameters are in the saturated full matrix, and how many are directly observed pattern-by-visit cells?
- For the additive model \(\mu_{dt}=\alpha_t+\gamma_d\), what does it require of the difference between patterns 1 and 3 at visits 1 and 2, if both patterns are observed at both visits?
- A pattern’s observed visit-1 mean is 14 units, while the complete-pattern visit-1 and visit-2 means are 16 and 13 units. What visit-2 mean does the additive model imply for that pattern?
- Can an additive model fit all observed means exactly and still make an unverified post-dropout prediction? Explain using the two-visit case.
Solutions. (1) The saturated full matrix has \(4^2=16\) means. The observed cells number \(1+2+3+4=10\); the remaining 6 are post-dropout cells. (2) The two pattern differences must be equal across the shared visits, because each equals \(\gamma_1-\gamma_3\); with \(\gamma_4=0\), this is still a constant pattern contrast. (3) The pattern offset is \(14-16=-2\) units, so the implied visit-2 mean is \(13+(-2)=11\) units. (4) Yes. With two visits, the additive model has three parameters and there are three observed cells. It can match those observed means while its restriction determines the fourth, unobserved cell; that prediction is not directly verified by the observed outcomes.