Adding Previous Outcomes to the Dropout Hazard

Why Include the Previous Outcome?

A dropout decision may relate both to a participant’s current, possibly unobserved outcome and to the most recent outcome already recorded. For example, a high symptom score at the last completed visit could predict later withdrawal even after accounting for the current symptom level in the selection model. Including both predictors makes these conditional relationships explicit. It does not, by itself, make the current outcome observable when dropout occurs or establish that either relationship is causal.

The model below extends a complementary log-log current-outcome hazard by adding the last observed outcome. It is a selection equation: it describes dropout conditional on the underlying outcome history. If dropout happens before the current outcome is recorded, that current value is latent in the dropout case, even though it appears in the model.

What the Two Coefficients Mean

The equation models the hazard among those still at risk at visit \(t\), not the probability of ever dropping out. Its left-hand side is the complementary log-log transform of the hazard. Thus, for a one-unit increase in the current outcome while the lagged outcome is held fixed, the transformed hazard changes by \(\gamma_c\). For a one-unit increase in the lagged outcome while the current outcome is held fixed, it changes by \(\gamma_l\). Equivalently, the corresponding \(-\log(1-h_{it})\) quantity is multiplied by \(\exp(\gamma_c)\) or \(\exp(\gamma_l)\), respectively.

These are conditional interpretations. The coefficient for the previous outcome does not describe its unadjusted relationship with dropout: it describes the relationship at the same current outcome. Likewise, the current-outcome coefficient compares individuals at the same lagged outcome. Because the link is not the logit link, exponentiating a coefficient does not give a dropout odds ratio. A coefficient’s effect on the probability scale also depends on the starting hazard.

The model requires an outcome scale that makes the coefficients interpretable. If outcomes are centered or rescaled, the corresponding coefficients refer to those transformed units. At the first visit there is no \(Y_{i,t-1}\) from the study’s outcome history. The first-visit hazard therefore needs its own specification without a lagged study outcome; one must not insert an invented or unobserved prior value.

Conditional Comparisons and Visit-to-Visit Risk

A useful way to understand the addition of the lagged outcome is to compare hazards while explicitly holding the other predictor fixed. On the transformed scale, changing one predictor by one unit adds its coefficient to the linear predictor. On the probability scale, the resulting change is nonlinear. The following calculations use the same hypothetical coefficients and outcome scaling as above.

At successive visits, hazards are conditional on still being under follow-up. Consequently, a hazard at a later visit is not added directly to an earlier hazard to obtain cumulative dropout risk. If the visit-specific hazards along a specified history are \(h_{i1},\ldots,h_{iT}\), the probability of remaining under follow-up through visit \(T\), conditional on that history, is the product of the corresponding conditional survival probabilities, \(\prod_{t=1}^{T}(1-h_{it})\). The cumulative dropout probability is one minus that product.

Justification and Assumptions

The inverse link follows directly from the hazard definition. Write the linear predictor as \(\eta_{it}=\alpha_t+\gamma_cY_{it}+\gamma_lY_{i,t-1}\). If the model states that \(\log\{-\log(1-h_{it})\}=\eta_{it}\), exponentiating both sides gives \(-\log(1-h_{it})=\exp(\eta_{it})\). Multiplying by \(-1\), exponentiating again, and rearranging yields \(1-h_{it}=\exp\{-\exp(\eta_{it})\}\), hence \(h_{it}=1-\exp\{-\exp(\eta_{it})\}\). This also shows why a coefficient shift multiplies \(-\log(1-h_{it})\) rather than the odds \(h_{it}/(1-h_{it})\).

The interpretation and use of this equation depend on several conditions:

• The dropout event and the risk set must be defined consistently at each visit. The hazard is conditional on remaining under follow-up to that visit.
• The previous outcome used at visit \(t\) must be the last outcome recorded before the visit-\(t\) dropout decision. The model’s timing must match how measurements and dropout decisions occur.
• The complementary log-log link and additive linear predictor must be an adequate specification over the outcome ranges and visits being analyzed. The displayed equation does not include covariates or interactions.
• The joint model must account for the distribution of outcomes, including current outcomes that are unobserved for participants who drop out before measurement. Including a latent current outcome in a selection equation does not make it observed.
• There must be sufficient information in the data and the specified outcome and dropout models to estimate the parameters. In particular, a coefficient’s conditional interpretation does not guarantee that the coefficient is well identified.

The derivation concerns the link and coefficient interpretation; it does not establish that the model is correct for a given application. The data and study design must support the assumed timing, outcome scale, and functional form.

Diagnostics, Alternatives, and Limitations

Useful checks begin with the risk sets and timing. Confirm that each participant contributes to a visit-specific hazard only while still under follow-up, and that the lagged outcome is genuinely observed before the dropout decision being modeled. Check the coding, units, missingness, and ranges of both outcome predictors. If the same participant can re-enter follow-up, or if dropout is not an absorbing event, the simple hazard setup needs to be adapted.

Assess whether the linear predictor is plausible across the observed ranges of current and lagged outcomes. Sparse combinations of the two outcomes can make conditional comparisons unstable; a few observations may determine the fitted coefficient in a region with little overlap. Extreme fitted hazards, strong sensitivity to outcome scaling, or marked departures between observed and model-implied dropout patterns are warnings. These checks can reveal a poor fit or weak information, but cannot verify the model’s dependence on unobserved current outcomes: those values are missing precisely for some dropout cases.

Compared with a current-outcome-only selection equation, the extended model allows the last observed outcome to carry additional conditional information about dropout. A lagged-outcome-only equation instead omits the current outcome from the selection predictor. These models answer different conditional questions and make different assumptions; neither becomes adequate merely by adding a predictor. A selection equation depending only on observed history has a different missingness interpretation from one that depends on the latent current outcome. The latter retains a dependence on an unobserved value and therefore requires assumptions about the outcome distribution as well as the selection equation.

A common mistake is to say, “\(\exp(\gamma_l)\) is the odds ratio for dropout per one-unit increase in the previous outcome.” That is incorrect for this complementary log-log model. The correction is that \(\exp(\gamma_l)\) multiplies \(-\log(1-h_{it})\), conditional on the current outcome and the other model terms. Another mistake is to call \(\gamma_l\) an unconditional or causal effect. It is a model coefficient describing a conditional association; it does not alone establish causation or resolve uncertainty about unobserved outcomes.

Adding the lagged outcome can improve the description of observed-history dependence, but it does not guarantee that the selection model is identified, correctly specified, or robust to unmeasured predictors of dropout. The numerical estimates and inferences can depend on assumptions about the unobserved outcomes. Those assumptions should be stated and their consequences considered rather than treated as confirmed by the observed data.