From a Response Pattern to a Likelihood Contribution
A selection model describes both the longitudinal outcomes and how the observed response pattern depends on those outcomes. To use it with incomplete data, the likelihood must reflect what was actually observed: the outcome density is evaluated at observed values, while outcomes that were not observed are marginalized. The response pattern also contributes selection probabilities. The crucial bookkeeping is to include the selection factors associated with the visits at which dropout could occur, without treating outcomes after dropout as if they had been measured.
Use the following convention. There are \(T\) scheduled visits, and \(Y_t\) is the outcome at visit \(t\). Assume the baseline outcome \(Y_1\) is observed for everyone. Let \(R_t=1\) if the outcome at visit \(t\) is observed and \(R_t=0\) otherwise. Dropout is monotone: once a participant misses a visit, later outcomes are also unobserved. Define \(D\) as the first missed visit, with \(D=T+1\) denoting complete follow-up. Thus a participant with \(D=d\), for \(2\le d\le T\), has observed outcomes \(Y_1,\ldots,Y_{d-1}\), but not \(Y_d,\ldots,Y_T\).
At visit \(t\), conditional on still being followed, let \(h_t(y_{1:t},x)\) be the probability of dropout at that visit, given the latent outcome history \(y_{1:t}=(y_1,\ldots,y_t)\) and observed covariates \(x\). The convention here is that the dropout decision may depend on the current outcome \(Y_t\), even when that outcome is unobserved because dropout occurs at visit \(t\). This is why \(Y_t\) must be treated as latent in the contribution for dropout at \(t\). The outcome model has joint density or mass function \(f_\theta(y_{1:T}\mid x)\).
The Contributions by Dropout Time
For complete follow-up, all outcomes are observed and the participant stayed in follow-up at each dropout decision. For dropout at visit \(d\), the participant stayed through visits \(2,\ldots,d-1\) and dropped at \(d\). There are no subsequent at-risk dropout decisions for this participant, because dropout is absorbing.
The second expression is a density or mass for the observed outcomes and the dropout pattern. Its integral ranges over every unobserved outcome, from \(Y_d\) through \(Y_T\). The survival factors before dropout involve observed outcome histories, because \(Y_1,\ldots,Y_{d-1}\) are available. The dropout factor \(h_d\) can involve the unobserved current outcome \(Y_d\), so it remains inside the integral. In a discrete outcome model, replace integration over a missing outcome by summation over its possible values.
If there are several participants with conditionally independent response and outcome histories, the full likelihood is the product of their individual contributions. Covariates \(x\) are conditioned on here because they are assumed observed; they are not integrated out as missing outcomes.
Why the Formula Has This Form
The derivation follows directly from the factorization of the joint model and marginalization. For a complete latent trajectory \(y_{1:T}\), the outcome model contributes \(f_\theta(y_{1:T}\mid x)\). The response path contributes a factor \(1-h_t\) at each visit when the participant remains under follow-up, and a factor \(h_d\) at the first missed visit. Multiplying these terms gives the joint density of the full trajectory and the specified response path.
For complete follow-up, the full trajectory is observed, so that joint density is already the observed-data contribution. For dropout at \(d\), only \(y_{1:d-1}\) is observed. The observed-data contribution is therefore the joint density integrated over the remaining coordinates. In particular, the response path for dropout at \(d\) contributes no hazard factors after \(d\): the participant is no longer at risk of a further dropout event. Integrating over the unobserved outcomes accounts for their uncertainty without adding hypothetical post-dropout response decisions.
Worked Contributions
Worked Example: Complete Follow-Up Through Three Visits
Suppose a hypothetical participant has observed outcomes \(y_1=2\), \(y_2=3\), and \(y_3=1\). At these values, the outcome model gives \(f_\theta(2,3,1\mid x)=0.0120\) per outcome-unit cubed. The dropout hazards are \(h_2(2,3,x)=0.10\) and \(h_3(2,3,1,x)=0.25\). Assume the model and the stated timing convention apply.
The participant completed follow-up, so the contribution is the joint outcome density multiplied by the probabilities of staying at visits 2 and 3:
The contribution is \(0.00810\) per outcome-unit cubed. No outcomes are integrated out because all three were observed. Using \(h_2\) or \(h_3\) instead of their complements would assign dropout, not continued follow-up, at those visits.
Worked Example: Dropout at the First Follow-Up Visit
Consider a hypothetical two-visit study with binary outcomes. A participant has observed \(Y_1=0\) and drops at visit 2, so \(Y_2\) is unobserved. The joint outcome model assigns probability \(0.30\) to \((Y_1,Y_2)=(0,0)\) and probability \(0.20\) to \((Y_1,Y_2)=(0,1)\). The dropout hazards at visit 2 are \(0.10\) when \(Y_2=0\) and \(0.40\) when \(Y_2=1\). Because the outcome is discrete, marginalizing over \(Y_2\) means summing over its two possible values.
The observed-data contribution is \(0.110\). Both possible values of \(Y_2\) contribute: each joint outcome probability is multiplied by the dropout probability corresponding to that latent current outcome. The probabilities \(0.30\) and \(0.20\) sum to \(0.50\), the marginal probability of \(Y_1=0\); multiplying that marginal by one hazard would generally be wrong because the hazard differs across the unobserved values of \(Y_2\).
Worked Example: Dropout at the Last Visit of a Three-Visit Study
In a hypothetical three-visit study, a participant has observed \(Y_1=1\) and \(Y_2=0\), then drops at visit 3. The joint outcome model assigns probabilities \(0.18\) and \(0.12\) to the trajectories \((1,0,0)\) and \((1,0,1)\), respectively. At visit 2 the dropout hazard is \(0.20\). At visit 3, the hazards are \(0.30\) when \(Y_3=0\) and \(0.60\) when \(Y_3=1\). The participant stayed through visit 2, so the factor \(1-0.20=0.80\) applies before summing over the unobserved \(Y_3\).
The observed-data contribution is \(0.1008\). The outcome \(Y_3\) is integrated out by summing over its two values, and its dropout hazard stays inside that sum because it depends on \(Y_3\). No factor for continued follow-up at visit 3 is included: this participant dropped at visit 3 rather than staying through it.
Checks, Alternatives, and Limitations
A useful diagnostic is to check each response pattern against its timeline. List observed outcomes first, mark the first missed visit, and then verify that the contribution has exactly the factors implied by that path. For dropout at \(d\), there should be \(d-2\) pre-dropout survival factors, one dropout hazard at \(d\), and marginalization over \(T-d+1\) outcomes. For complete follow-up, there should be \(T-1\) survival factors and no marginalization. These counts can reveal an off-by-one error before any computation.
Other warning signs include evaluating a current-outcome hazard at an outcome treated as observed when it is actually missing, omitting a pre-dropout survival factor, or including a hazard after dropout. A contribution should also be nonnegative and should have the appropriate probability or density scale. In a discrete model, summing a dropout contribution over all possible observed histories and dropout times should be compatible with a joint probability model; probabilities exceeding one can indicate duplicated or incorrectly placed response factors. These checks do not establish that the model is substantively appropriate, but they help detect bookkeeping errors.
| Approach | How it handles the response pattern | What to keep in mind |
|---|---|---|
| Selection-model observed-data likelihood | Models the outcome distribution and dropout probabilities, then marginalizes unobserved outcomes. | Current-outcome dependence can make the hazard at dropout depend on a latent outcome. |
| Complete-case analysis | Uses participants with complete outcomes and omits the incomplete-pattern contributions. | It does not generally use all available observed outcomes and can target a distorted result if complete cases are not representative. |
| Pattern-mixture model | Models outcome distributions conditional on response pattern. | It organizes the model by response pattern rather than by factoring outcome and selection components. |
The selection-model contribution is a construction, not a guarantee of identification or a test of the assumptions. In particular, a hazard that depends on an unobserved current outcome involves information that the observed data alone may not determine. The assumed outcome and selection models, and any restrictions they impose, therefore matter to the resulting inference. Missing-data concepts such as missing at random are relevant for comparing assumptions, but do not remove the need to specify which outcomes are observed in each contribution.
Common Mistake
Check Your Understanding
Answer each question, then compare with the solutions below.
- With \(T=4\), which outcomes are observed for a participant with \(D=3\), and which are marginalized?
- For dropout at visit 3, how many pre-dropout survival factors and how many dropout hazards appear?
- A participant completes a two-visit study. Should the contribution include \(h_2\) or \(1-h_2\)?
- Why must a dropout hazard that depends on the current outcome remain inside the integral or sum when that outcome is missing?
Solutions. (1) \(Y_1,Y_2\) are observed; \(Y_3,Y_4\) are marginalized. (2) There is one pre-dropout survival factor, \(1-h_2\), and one dropout hazard, \(h_3\). (3) It includes \(1-h_2\), because the participant stayed through visit 2. (4) The hazard can take different values for different latent current outcomes, so the observed-data contribution must average the joint outcome-and-selection term over those possible values rather than apply a single hazard outside the marginalization.