Comparing Dropout Models With Different Outcome Histories

Why the Outcome History Matters

A dropout model describes how the chance of leaving a longitudinal study changes with information available at a visit. Even when candidate models use the same link and include the same current outcome dependence, they can give different dropout probabilities because they represent past outcomes differently. One model may use only the most recent observed outcome; another may allow each past outcome to contribute separately; a third may use a summary such as the past mean.

These choices are consequential: a model that retains only the latest value cannot distinguish two participants with the same latest value but very different earlier trajectories. A full-history model can represent that distinction, but estimates more parameters and may be unstable when histories are long or sparse. A summary is more compact, but it treats histories as equivalent whenever they have the same summary.

For clarity, consider logistic dropout models with a common visit-specific intercept and baseline covariate component. In all three specifications below, the coefficient \(\delta\) on \(Y_{it}\) is included as the same form of direct outcome dependence. The models differ in how they use the prior observed history:

Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\), so a linear predictor \(\eta\) gives dropout probability \(1/\{1+\exp(-\eta)\}\). The dependence on \(Y_{it}\) makes the selection model MNAR when that outcome is missing at dropout: the dropout probability then depends on an unobserved value. Dependence only on observed past outcomes does not, by itself, create that feature. This distinction matters even though all three candidates are being compared as MNAR models: the common current-outcome term supplies their direct MNAR dependence, while their history terms encode different observed-history relationships.

What the Three History Specifications Assume

The latest-outcome specification assumes that, once \(Y_{i,t-1}\), \(X_i\), and the current outcome term are accounted for, earlier outcomes do not add further information about dropout. This can be a useful working approximation when recent status is expected to dominate, but it is not a consequence of having longitudinal data.

The full-history specification allows each prior visit to have its own coefficient. It can distinguish timing and trajectory shape, but requires more parameters as the number of visits increases. The summary specification imposes a particular compression. For example, if \(S(H_{i,t-1})\) is the unweighted mean of the prior outcomes, every prior visit has the same coefficient per unit of outcome in the linear predictor. Choosing a summary is therefore a substantive modeling restriction, not merely a computational shortcut.

These specifications can be nested. A latest-outcome model is a restriction of the full-history model when all coefficients on earlier outcomes are zero. A summary model is a restriction of the full-history model when the full-history coefficient vector has the specified weights of that summary. This gives a precise way to understand what information a compressed history retains.

Thus, for the weighted summary \(S(H)=\sum_jw_jY_{ij}\), the summary and full-history linear predictors agree exactly under those coefficient restrictions. If the restrictions fail, the summary cannot reproduce the full-history predictor for every possible history. This algebraic fact does not establish that either restriction is scientifically appropriate.

When Summaries Hide Trajectory Differences

A direct way to see what is lost by compression is to compare histories with the same summary. Suppose the two hypothetical histories are \((2,4,5)\) and \((4,2,5)\). Both have mean \(11/3\) and latest value 5. Consequently, the latest-outcome and mean-summary predictors in the preceding example assign the same dropout probabilities to both histories, provided the current outcome and other covariates are also the same.

The full-history predictor can distinguish them because the coefficients vary by visit. For \((2,4,5)\), the history contribution is \(0.1(2)+0.2(4)+0.3(5)=2.5\). For \((4,2,5)\), it is \(0.1(4)+0.2(2)+0.3(5)=2.3\). With the shared contribution \(-1+0.5(6)=2\), the respective linear predictors are \(4.5\) and \(4.3\). The corresponding probabilities are approximately \(0.9890\) and \(1/\{1+\exp(-4.3)\}\approx0.9866\). The latest and mean models cannot express this difference, because their inputs are identical for these two histories.

Assumptions, Comparison, and Failure Modes

A meaningful comparison requires aligned models. Use the same definition of dropout, outcome scale, link, baseline covariates, current-outcome term, visit structure, and analysis population. Otherwise, a change in predictions cannot be attributed specifically to the history specification. The history must also be available as defined for every participant at risk; when prior visits themselves are missing, the rule for forming or modeling the history must be explicit. A model using a mean of observed prior values, for example, is not automatically the same as one using a mean over all scheduled prior visits.

The logistic form assumes a linear relationship between each included predictor and the log odds of dropout, conditional on the other terms. It also assumes the chosen history representation is adequate for the intended comparison. Inference further relies on a correctly specified likelihood and appropriate treatment of repeated observations within participants. Predictions should not be extrapolated to covariate or outcome histories unsupported by the data; estimates of current-outcome dependence are especially consequential because that outcome is missing for dropouts.

Model comparisons should therefore examine what the predictor structure can and cannot express, not just compare a single predicted probability. The following table summarizes the central trade-offs.

Warning signs include implausibly large or unstable history coefficients, fitted probabilities near 0 or 1 for many participants, and substantial changes in coefficients when a small number of history terms are added or removed. These symptoms can reflect sparse histories, strong correlation among repeated outcomes, or an overly ambitious full-history model. A summary model can also fail in a less conspicuous way: its fit may appear stable while its imposed equivalence of distinct trajectories is scientifically implausible.

Observed-data fit alone cannot settle how dropout depends on the missing current outcome. Different assumptions about that unobserved value can be compatible with the same observed information. The history comparison is useful for exposing one part of the modeling choice, but it does not validate the unobserved-outcome dependence. Interpret conclusions as conditional on the specified selection model, and distinguish observed-history flexibility from assumptions about outcomes after dropout.

Common Mistake

A second pitfall is treating the latest-outcome model as simply a smaller version that must be adequate. It is a restriction: earlier outcomes have zero additional contribution conditional on the latest outcome and other terms. Likewise, a summary is not neutral compression; it makes histories sharing that summary equivalent in the dropout predictor. State these restrictions before interpreting differences among candidate models.