Why the Factorization Needs an Assumption
A selection model for longitudinal data separates two ingredients: how the outcomes are distributed, and how response depends on outcomes. The Diggle–Kenward construction is useful because its response component can depend on the outcome at the current visit as well as earlier outcomes. That allows dropout to depend on an outcome that is itself missing when dropout occurs.
The separation is not, by itself, a justification for any particular response model. The joint distribution can always be factored by the probability chain rule, but the response transitions in that unrestricted factorization may depend on the entire outcome vector, including future outcomes. The Diggle–Kenward restriction rules out that future dependence, conditional on the outcomes through the current visit. The distinction is central: without it, the proposed factorization using only current and past outcomes need not represent the joint distribution.
The response probability \(q_t\) is defined only for a participant still at risk of dropout at visit \(t\). Because of monotonicity, being at risk means all earlier response indicators equal one. It is important that \(q_t\) conditions on \(Y_t\), not just outcomes through \(t-1\). This permits dependence on the current outcome. If \(Y_t\) is not observed when a participant drops at visit \(t\), dependence on \(Y_t\) is dependence on a latent, unobserved value for that participant.
Let \(f_Y(y_1,\ldots,y_T)\) denote the joint outcome density (or probability mass function). The defining restriction for the Diggle–Kenward response component is, for each visit \(t\geq2\),
Thus, among those still observed, the transition at visit \(t\) is conditionally independent of future outcomes \(Y_{t+1},\ldots,Y_T\), given the outcomes through \(t\). This is a model restriction, not a consequence of defining \(q_t\). The outcome component \(f_Y\) and the functions \(q_t\) together determine the joint distribution of outcomes and response histories under this restriction.
Deriving the Joint Factorization
For a fixed full outcome vector \(y\), consider a response path that drops out at visit \(d\), where \(2\leq d\leq T\). Its response indicators are \(R_1=\cdots=R_{d-1}=1\) and \(R_d=\cdots=R_T=0\). Apply the probability chain rule to the response transitions conditional on the full outcome vector. At each visit before \(d\), the participant remains observed; at \(d\), the participant drops out. By the stated restriction, the corresponding transition probabilities are \(q_t(y_{1:t})\) and \(1-q_d(y_{1:d})\). Later indicators are zero deterministically by monotonicity. Multiplying by the outcome distribution gives the joint factorization below. If there is no dropout, every at-risk transition is a response.
For dropout at visit 2, the product before \(1-q_2\) is empty and equals 1. No factors are included for transitions after dropout: those indicators are already determined to be zero, and the participant is no longer at risk. Including further dropout factors would incorrectly treat a participant who has already dropped out as making repeated dropout decisions.
Worked Example: Two-Visit Joint Probabilities
Suppose a hypothetical study records a baseline score \(Y_1=10\) for everyone and a follow-up score \(Y_2\), measured in points. The outcome model assigns probability 0.60 to \(Y_2=0\) and 0.40 to \(Y_2=1\). The response model gives \(q_2(10,0)=0.80\) and \(q_2(10,1)=0.50\). For this two-visit example, the response model is defined after observing the visit-2 outcome, and there are no future outcomes whose values could affect the transition.
For \(Y_2=0\) and a complete response path, the joint probability is \(0.60\times0.80=0.48\). For \(Y_2=0\) and dropout at visit 2, it is \(0.60\times(1-0.80)=0.60\times0.20=0.12\).
For \(Y_2=1\), the complete-path joint probability is \(0.40\times0.50=0.20\), and the dropout-path joint probability is \(0.40\times(1-0.50)=0.40\times0.50=0.20\).
The four joint probabilities sum to \(0.48+0.12+0.20+0.20=1.00\). The example illustrates the factorization, not evidence that the response model is appropriate in an actual study. With only two visits, there are no post-visit-2 outcomes; in longer follow-up, the restriction about future outcomes must also be considered.
Why the Restriction Matters
The justification can also be checked by summing the response-path probabilities conditional on a fixed outcome vector. At visit 2, either dropout occurs, with probability \(1-q_2\), or the participant remains observed, with probability \(q_2\). If the participant remains observed, the same alternatives apply at visit 3, and so on. The conditional probabilities of the mutually exclusive dropout paths therefore add to one:
Here each \(q_t\) is evaluated at the relevant fixed outcome history. The equality follows by cancellation: each negative product term is paired with the same product as the positive term for the next path. This confirms that the factorized response probabilities form a valid distribution over monotone response paths, conditional on \(Y=y\). It does not show that the restriction is true for a particular study.
Worked Example: A Three-Visit Path
Consider a hypothetical three-visit study and a fixed outcome vector \(y=(4,6,5)\), with scores measured in points. Suppose the outcome model assigns \(f_Y(4,6,5)=0.20\), and the response model gives \(q_2(4,6)=0.75\) and \(q_3(4,6,5)=0.60\). Assume the response transitions satisfy the future-outcome restriction. These values concern this particular outcome history; they are not estimated from a claimed real study.
For the path \((R_1,R_2,R_3)=(1,1,1)\), the joint probability is \(0.20\times0.75\times0.60=0.090\).
For dropout at visit 3, the path is \((1,1,0)\). Its joint probability is \(0.20\times0.75\times(1-0.60)=0.20\times0.75\times0.40=0.060\).
For dropout at visit 2, the path is \((1,0,0)\). Its joint probability is \(0.20\times(1-0.75)=0.050\). There is no \(q_3\) factor for this path, because after dropout at visit 2 the participant is not at risk at visit 3. The three path probabilities sum to \(0.090+0.060+0.050=0.200\), the outcome-vector probability. Equivalently, conditional on this outcome vector, the response-path probabilities sum to one.
Worked Example: When an Observed-History Transition Is Not Enough
Take fixed \(Y_1\) and \(Y_2\), and let \(Y_3\) be Bernoulli with \(P(Y_3=0)=P(Y_3=1)=0.50\). Suppose the response process is \(R_2=R_3=Y_3\), with \(R_1=1\). This is monotone, but the visit-2 response decision depends on the future outcome \(Y_3\).
Conditioning only on the history available through visit 2 gives \(P(R_2=1\mid R_1=1,Y_1,Y_2)=0.50\). If this marginal probability is incorrectly used as \(q_2(Y_{1:2})\), the proposed factorization assigns probability \(P(Y_3=0)\{1-q_2\}=0.50\times0.50=0.25\) to the joint event \(Y_3=0,(R_1,R_2,R_3)=(1,0,0)\).
But under the specified data-generating process, when \(Y_3=0\), both \(R_2\) and \(R_3\) equal zero with certainty. The true joint probability of that event is \(P(Y_3=0)=0.50\), not 0.25. The chain rule itself remains valid if the response transition is conditioned on the full outcome vector; what fails is replacing that full-vector conditional probability by a transition depending only on outcomes through the current visit. This process violates the Diggle–Kenward restriction.
Assumptions, Diagnostics, and Alternatives
The factorization has both structural and statistical requirements. Monotone dropout must describe the response process, or the model must be extended to account for intermittent missingness. The visit-1 convention \(R_1=1\) must fit the study design; if initial nonresponse is possible, it requires its own response component. The future-outcome restriction must be scientifically defensible as a modeling assumption. In addition, the specified outcome distribution and response model must be adequate for the outcomes, histories, and response patterns being analyzed. A probability model must assign valid probabilities, and fitted values should be checked for numerical stability and implausible predictions.
Diagnostics can identify some problems, but cannot generally verify the key restriction. Check whether observed response rates vary across recorded outcome histories in ways the chosen response model can represent. Examine fitted probabilities for values near zero or one, sparse histories, and poor overlap between histories represented among those retained and those who drop out. These checks can reveal an inadequate model or unstable estimates. They cannot reveal the unobserved outcomes after dropout, so they cannot establish that dropout is conditionally independent of those future outcomes. This is an identification limitation, not something resolved by a good fit to observed data.
The factorization clarifies how the Diggle–Kenward selection representation differs from common alternatives. It is not the same as missing at random: its response transition may depend on the current outcome, which can be missing for a participant who drops out. A pattern-mixture representation instead describes outcome distributions conditional on missingness patterns. Under suitable specifications, selection and pattern-mixture representations can express the same joint distribution, but their assumptions and modeling coordinates differ. Neither representation makes the unobserved part of the distribution empirically determined without additional assumptions or information. The earlier material on identifiability and on equivalent selection and pattern-mixture representations is relevant to interpreting that limitation.
| Representation | Factor being modeled | Key distinction |
|---|---|---|
| Diggle–Kenward selection factorization | Outcome distribution and response transitions conditional on current and past outcomes | Requires the stated restriction excluding future-outcome dependence from at-risk transitions |
| Missing at random response model | Response conditional on observed data | Does not allow response dependence on an unobserved current outcome after conditioning on observed data |
| Pattern-mixture representation | Outcome distributions within response patterns | Organizes the joint distribution by missingness pattern rather than by response transitions |
A common mistake is to write \(f_Y(y)\) times response probabilities conditional on \(Y_{1:t}\), then call the result a chain-rule identity. The chain rule initially conditions each response transition on the full outcome vector and the preceding response history. The replacement by a probability conditional only on current and past outcomes is valid only under the explicit future-outcome restriction. Another mistake is to interpret a fitted \(q_t\) as a directly observed dropout probability for every outcome history: values of \(Y_t\) missing at dropout are not observed for those who drop out, so the response dependence on such values cannot be checked directly from their data.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- State the condition that permits an at-risk response transition to depend on \(Y_{1:t}\) rather than the full vector \(Y_{1:T}\).
- In a three-visit study, write the response factor for dropout at visit 2.
- Suppose a fixed outcome vector has \(q_2=0.70\) and \(q_3=0.50\). What are the conditional probabilities of dropout at visit 2, dropout at visit 3, and complete response?
- Why can observed response rates not, in general, verify the future-outcome restriction?
Solutions. (1) Conditional on response through visit \(t-1\) and \(Y_{1:t}\), \(R_t\) must be independent of \(Y_{t+1:T}\). (2) The response factor is \(1-q_2(y_{1:2})\); later indicators are zero by monotonicity and add no transition factors. (3) The probabilities are \(1-0.70=0.30\) for dropout at visit 2, \(0.70(1-0.50)=0.35\) for dropout at visit 3, and \(0.70(0.50)=0.35\) for complete response; they sum to 1.00. (4) Outcomes after dropout are unobserved for those participants, so the dependence of their dropout decisions on those future values cannot generally be assessed from observed response rates alone.