Choosing History Predictors for Dropout Models

Why Predictor Choice Is a Causal Decision

A dropout model is not merely a device for forecasting who will remain in a study. Its predictors determine which observed histories are used to explain continued follow-up. For inverse probability of censoring weighting (IPCW) to recover the target outcome distribution, the history in that model must be rich enough to make censoring conditionally independent of the future outcomes of interest.

The key question is therefore not simply, “Which variables predict dropout?” It is: Which information available before dropout is needed so that, within histories represented in the model, those who remain are informative about those who do not? A variable associated with both dropout and a future outcome is especially important. Omitting it can leave the retained sample systematically different from the target population in outcome-relevant ways.

Definition: At discrete time \(t\), let \(R_t=1\) indicate that a participant is still observed and at risk of dropout, and let \(R_t=0\) indicate that follow-up has already ended. Let \(H_t\) denote the observed history available before the censoring decision for the next interval. It may include baseline covariates, prior treatments \(A_0,\ldots,A_t\), time-varying covariates \(L_0,\ldots,L_t\), and previously measured outcomes. The retention probability is \(g_t(H_t)=P(R_{t+1}=1\mid R_t=1,H_t)\). Only information measured or known before the decision at time \(t\) can be a predictor in this model.

The history must be defined with the timing of measurement in mind. For example, an outcome measured at the end of a visit can predict dropout after that visit, but it cannot predict dropout that occurred before the measurement. Nor can a future outcome that is missing precisely because of dropout be inserted as an observed predictor. The history relevant to censoring is the history available while the participant is still at risk.

Which Variables Belong in the History?

A useful starting set consists of prior treatments, covariates, and outcome measurements that may jointly explain both continued observation and future outcomes. Their roles differ:

Prior treatments may affect subsequent outcomes and may also influence adherence, contact with the study, or the chance of withdrawal. They are especially relevant when treatment history is associated with both dropout and the future outcome.
Time-varying covariates may describe health status, access to care, or other evolving circumstances that influence both retention and prognosis.
Prior outcome measurements can be strong indicators of future outcomes and may prompt withdrawal, additional contact, or changes in follow-up.

This is a principled starting point, not a rule to include every recorded variable automatically. A variable can predict dropout but add no needed adjustment for the outcome if the remaining history already makes censoring independent of that outcome. Conversely, a variable may be only weakly associated with dropout in a marginal analysis yet still matter after conditioning on other history. Selection should focus on conditional relationships, not screening variables one at a time by their marginal association with dropout.

For a target outcome \(Y_K\), the relevant requirement can be stated in terms of the full data: among participants still observed at time \(t\), the next retention decision must be independent of the future full-data outcomes, conditional on the included history. In notation, one version is \(R_{t+1}\perp Y_K\mid R_t=1,H_t\), with the analogous condition for any other future outcomes required by the analysis. This is a substantive assumption; a well-fitting regression model does not establish it.

Worked Example: A Prior Outcome Predicts Both Dropout and Prognosis

Suppose a study measures symptom severity \(Y_0\) at baseline and plans to measure \(Y_1\) one year later. Let \(Y_0\) be binary, with \(P(Y_0=1)=0.5\). Participants with \(Y_0=1\) have a 0.8 probability of dropping out before the year-one measurement; those with \(Y_0=0\) have a 0.2 probability of dropout. Assume \(Y_1=Y_0\), so baseline severity is also fully prognostic of the future outcome.

The two baseline groups contribute the following fractions of the original cohort to the observed sample:

P(Y_0=1,R_1=1)=0.5(1-0.8)=0.10,\qquad P(Y_0=0,R_1=1)=0.5(1-0.2)=0.40.

Among those observed at year one, the fraction with \(Y_1=1\) is therefore \(0.10/(0.10+0.40)=0.20\). The target mean in the full cohort is \(E[Y_1]=0.5\). The observed mean is far too low because participants with poor baseline symptoms were disproportionately lost.

If the retention model uses \(Y_0\), the inverse retention factors are \(1/0.2=5\) for \(Y_0=1\) and \(1/0.8=1.25\) for \(Y_0=0\). The weighted contributions from the observed groups are \(0.10(5)=0.50\) and \(0.40(1.25)=0.50\). Thus the weighted fraction with \(Y_1=1\) is \(0.50/(0.50+0.50)=0.50\), recovering the target mean in this example. Here \(Y_0\) belongs in the history because it predicts both dropout and the future outcome.

What the Longitudinal Weighting Identity Requires

The predictor choice matters because inverse retention weights can undo selection only when the modeled history supports a sequential independent-censoring condition. The following derivation states the required assumptions explicitly. Let the full-data trajectory be fixed for the purpose of conditioning, and suppose each \(H_t\) is a function of that trajectory’s history through time \(t\).

Theorem: Suppose \(R_0=1\), and for each \(t=0,\ldots,K-1\), retention is conditionally independent of the relevant future full-data trajectory given the at-risk history: \(P(R_{t+1}=1\mid R_t=1,\text{full data})=g_t(H_t)\). Assume \(g_t(H_t)>0\) almost surely for histories in the target population, and that \(Y_K\) is integrable. Then \(E\left[R_KY_K/\prod_{t=0}^{K-1}g_t(H_t)\right]=E[Y_K]\).

Proof. Conditional on the full-data trajectory, the chain rule gives the probability of remaining observed through time \(K\). Under the stated sequential condition, each factor in that probability is \(g_t(H_t)\):

P(R_K=1\mid\text{full data}) =\prod_{t=0}^{K-1}P(R_{t+1}=1\mid R_t=1,\text{full data}) =\prod_{t=0}^{K-1}g_t(H_t).

Since \(R_K\) is an indicator, conditional expectation gives

E\left[\left. \frac{R_KY_K}{\prod_{t=0}^{K-1}g_t(H_t)} \right|\text{full data}\right] = \frac{Y_K}{\prod_{t=0}^{K-1}g_t(H_t)} P(R_K=1\mid\text{full data}) =Y_K.

Taking expectations of both sides proves the identity. The integrability assumption ensures the expectations exist; positivity makes the denominators nonzero. This identity explains why the history must be sufficient for the censoring condition, but it does not guarantee that a particular fitted model correctly estimates the \(g_t\). \(\square\)

Worked Example: Why Prior Treatment Can Be Needed

Let prior treatment \(A_0\) be binary with \(P(A_0=1)=P(A_0=0)=0.5\). Suppose the future outcome is \(Y_1=2A_0\), while the probability of remaining observed is 0.5 for \(A_0=1\) and 1 for \(A_0=0\). The full-cohort mean is \(E[Y_1]=0.5(2)+0.5(0)=1\).

Among those observed, the fractions from the two treatment groups are \(0.5(0.5)=0.25\) and \(0.5(1)=0.50\), for a total observed fraction of \(0.75\). The complete-case mean is \((0.25(2)+0.50(0))/0.75=2/3\), not 1.

Using \(A_0\) in the retention model gives inverse retention factors 2 for treated participants and 1 for untreated participants. Their weighted masses are \(0.25(2)=0.50\) and \(0.50(1)=0.50\). The weighted mean is \((0.50(2)+0.50(0))/(0.50+0.50)=1\). In this example, prior treatment predicts dropout and determines the future outcome, so omitting it leaves outcome-relevant selection uncorrected.

Predictive Variables Are Not All Equally Necessary

It is important not to turn the preceding examples into the incorrect rule “include every variable associated with dropout.” A variable that predicts retention can be useful for modeling, but it is not automatically required for identification. Consider a single dropout decision and a binary variable \(Z\) with \(P(Z=0)=P(Z=1)=0.5\). Let \(Y\) be Bernoulli with probability 0.4, independent of \(Z\), and let retention probabilities be 0.2 for \(Z=0\) and 0.8 for \(Z=1\). Dropout depends strongly on \(Z\), but \(Z\) is unrelated to the outcome.

The overall retention probability is \(0.5(0.2)+0.5(0.8)=0.5\). Because \(Y\) is independent of \(Z\) and retention depends only on \(Z\), retained participants still have outcome mean 0.4. A constant inverse-retention weight of 2 therefore recovers the mean as well as the history-specific weights. In practice, adding \(Z\) may affect the variability of estimated weights or improve the model’s description of retention, but it is not needed to remove outcome selection in this stipulated example.

The principle behind this example can be verified directly: if \(Y\) is independent of \(Z\), and \(R\) is conditionally independent of \(Y\) given \(Z\), then

P(R=1,Y=y)=\sum_z P(R=1\mid Z=z)P(Y=y)P(Z=z) =P(Y=y)P(R=1).

Thus \(R\) and \(Y\) are independent. If, however, \(Z\) is associated with the outcome after accounting for the included history, and also predicts retention, omitting \(Z\) can restore precisely the selection problem the weights are meant to address. The relevant issue is not whether a variable is “significant” in a dropout regression, but whether the proposed history makes the conditional independent-censoring assumption plausible.

Worked Example: A Covariate That Predicts Dropout but Not the Outcome

Use the preceding setup: \(P(Z=0)=P(Z=1)=0.5\), \(Y\sim\operatorname{Bernoulli}(0.4)\) independently of \(Z\), and retention probabilities \(g(0)=0.2\), \(g(1)=0.8\). The retained fractions are 0.10 in group \(Z=0\) and 0.40 in group \(Z=1\), so total retention is 0.50. Since \(Y\) remains independent of \(Z\) among those retained, the retained outcome mean is 0.4.

With a constant retention probability of 0.5, each retained observation receives weight 2. Its expected weighted outcome contribution is \(E[2R Y]=2(0.5)(0.4)=0.4\), and its expected total weight is \(E[2R]=2(0.5)=1\). Their ratio is 0.4.

With history-specific weights, the expected weighted outcome contribution is \(0.5(0.2)(0.4)(1/0.2)+0.5(0.8)(0.4)(1/0.8)=0.2+0.2=0.4\). The expected total weight is \(0.5(0.2)(1/0.2)+0.5(0.8)(1/0.8)=0.5+0.5=1\). The mean is again 0.4. The variable strongly predicts retention, yet its inclusion is not necessary for this particular outcome mean because it carries no outcome information.

Practical Selection, Diagnostics, and Limitations

A defensible selection process starts with the time ordering and the scientific roles of the variables, then checks whether the fitted retention models are usable. The following sequence is a practical guide:

Write the at-risk history.
For each time point, list only treatment, covariate, and outcome information known before the retention decision.

Identify outcome-relevant selection.
Ask which prior variables may be associated with future outcomes and with retention, conditional on the other included history.

Fit a plausible conditional model.
Allow appropriate time effects and, when scientifically warranted, nonlinearities or interactions. A single linear predictor may not capture different retention patterns across histories.

Check support and fit.
Look for histories with very small estimated retention probabilities, sparse treatment-history combinations, and systematic discrepancies between predicted and observed retention.

Very small retention probabilities signal a positivity problem: some histories have little chance of remaining observed. Correctly adding predictors cannot create information where follow-up is nearly absent. A highly flexible model can also become unstable when histories are sparse, while an overly simple model can leave important conditional dependence unaccounted for. Diagnostics can reveal warning signs, but cannot prove sequential independent censoring or positivity in the target population.

The main alternative is outcome regression, which models the future outcome from observed history rather than reweighting retained participants. It relies on its own modeling assumptions and can be sensitive to outcome-model misspecification; it does not make the censoring mechanism irrelevant. Complete-case analysis is simpler, but is justified for a target mean only under conditions that make the observed outcomes representative of the target population. These approaches answer related problems in different ways; predictor choice for a censoring model is not a substitute for specifying the causal estimand or the assumptions behind it.

A common mistake is to select predictors solely by a statistical test of association with dropout, or to include every variable measured anywhere in the study. The correction is to respect time ordering and ask whether the included history supports conditional independence of censoring and the future outcomes of interest. A future outcome cannot be used to predict an earlier dropout decision, and a predictor of dropout alone need not be essential for a particular outcome. Conversely, omitting a prior outcome, treatment, or covariate that jointly predicts dropout and prognosis can make the weighted analysis systematically wrong even if the regression software reports a converged fit.

Key takeaway: Choose dropout-model predictors to make the censoring decision conditionally independent of the relevant future outcomes, using only information available before that decision. Prior treatments, covariates, and outcomes that relate to both dropout and prognosis are central candidates; prediction alone is not the criterion.

Check Your Understanding

Answer each question using the timing and conditional-independence principles above.

A symptom score measured at visit 2 predicts both dropout after visit 2 and the outcome at visit 4. Should it be considered for the visit-2 retention model? Solution: Yes. It is available before the dropout decision and is related to both retention and a future outcome, so omitting it may leave outcome-relevant selection.
A laboratory value is measured only at visit 4, after some participants have already dropped out following visit 2. Can it be a predictor in the visit-2 retention model? Solution: No. It is not available before that decision, and it may be missing because of dropout. A predictor must belong to the history observed at the time of the decision.
A baseline variable predicts retention but is independent of the future outcome and of other prognostic history. Is it always required for unbiased estimation of the future mean? Solution: No. As the binary example shows, when censoring depends only on an outcome-independent variable, marginal retention can still be independent of the outcome. Including the variable may affect precision or model behavior, but prediction alone does not establish necessity.
Why can a model with excellent apparent prediction of dropout still fail to support valid IPCW? Solution: Prediction does not establish that censoring is independent of future outcomes given the modeled history. The model can omit an important prognostic variable, use information measured after dropout, or encounter histories with near-zero retention probabilities.