Depression Study Dropout After Symptom Changes

Why Symptom-Related Dropout Changes the Estimate

In a depression study, participants may stop attending after symptoms worsen, improve, or change in response to treatment. The remission rate among participants who complete follow-up then describes a selected group, not necessarily the full population that began the study. Weighting can correct this selection when dropout is explained by the measured history, but the weights must account for treatment assignment as well as retention.

Let \(L_t\) denote symptom severity measured at time \(t\), \(A_t\) the treatment received, and \(Y\) the indicator of remission at the end of follow-up, with \(Y=1\) for remission. Let \(R_t=1\) mean that the participant is still observed at time \(t\), with \(R_0=1\). Write \(H_t\) for the observed history available before treatment at time \(t\), including baseline covariates and prior symptoms and treatments. The history used to model retention after treatment at time \(t\) includes \(H_t\) and \(A_t\).

For example, \(g\) might specify antidepressant treatment at every visit, while a comparison strategy might specify no antidepressant treatment at every visit. The target \(E[Y^g]\) concerns the full baseline population under that strategy; it is not the remission proportion among the people who happened to follow it and remain observed.

Assumptions and the Two Sources of Weighting

The analysis requires consistency: a participant who follows \(g\) has observed outcome \(Y=Y^g\). It also requires sequential treatment exchangeability: conditional on the measured history \(H_t\), treatment at time \(t\) is independent of the potential outcomes under the strategies being compared. In addition, sequential independent censoring requires that, among those still observed, retention at the next interval be independent of \(Y^g\) conditional on the measured history and treatment received. This is an outcome-specific condition; it does not assert independence from every unobserved future quantity.

Define the treatment probability \(\pi_t(a\mid H_t)=P(A_t=a\mid R_t=1,H_t)\) and the retention probability \(q_t(H_t,A_t)=P(R_{t+1}=1\mid R_t=1,H_t,A_t)\). Both probabilities must be positive for the histories that can occur under \(g\): \(\pi_t(g_t(H_t)\mid H_t)>0\) and \(q_t(H_t,g_t(H_t))>0\). These are treatment and retention positivity conditions. They rule out histories at which the strategy is never followed or retention is impossible.

For a participant who is observed through the final outcome and whose observed treatment path follows \(g\), the combined inverse-probability weight is

Here \(I(\bar A=\bar g)\) indicates that treatment followed the strategy at every interval, and \(R_K\) indicates that the final outcome was observed. Histories and probabilities in this expression are evaluated along the participant’s observed path. The treatment factors correct for the fact that only some participants follow \(g\); the retention factors correct for the fact that only some of those participants remain observed.

Two Weighting Identities

First consider a population actually generated under intervention \(g\), rather than observational data in which participants may receive other treatments. Write \(R_K^g\) for retention under that intervention and \(q_t^g\) for the retention probability along its history. Under the stated sequential independent-censoring condition for \(Y^g\), the censoring-only identity is

To see the logic, at an interval with history fixed, the conditional expectation of the retention factor is one. In particular, when \(R_t^g=1\), independent censoring for this outcome gives

Applying this equality successively from the last interval back to the first removes each retention indicator and its probability factor, leaving \(E[Y^g]\). The argument uses independence for \(Y^g\), not for an arbitrary full-data quantity.

In observational data, this identity alone does not identify \(E[Y^g]\): a participant who received a different treatment path does not reveal \(Y^g\) by consistency. Treatment weighting is also necessary. Under consistency, sequential treatment exchangeability, sequential independent censoring for \(Y^g\), and both positivity conditions, the observed-data identification result is

A direct justification follows the sequential data-generating process. At each interval, the indicator that treatment matches \(g_t\), divided by its conditional probability, replaces the observational treatment draw with the strategy’s specified treatment. Among those retained, the next retention indicator divided by its conditional probability removes the selection due to dropout. Iterating these conditional expectation steps over \(t=0,\ldots,K-1\) leaves the distribution of histories and outcomes under \(g\), and hence \(E[Y^g]\). The treatment step uses exchangeability to equate the outcome distribution among those who received the specified treatment, conditional on history, with the corresponding potential-outcome distribution. The retention step uses the outcome-specific censoring condition. Neither step licenses replacing the strategy-specific potential outcome by an observed outcome for someone who followed another treatment path. \(\square\)

For independent participants, a simple inverse-probability weighted estimator is \(n^{-1}\sum_{i=1}^n W_i^gY_i\). A common finite-sample alternative is the Hájek ratio \(\sum_iW_i^gY_i/\sum_iW_i^g\), which normalizes by the observed sum of weights. Both target the same risk under the identification assumptions; the ratio is not exactly unbiased in finite samples. Consistency of these sample estimates additionally requires suitable sampling and moment conditions, such as independent sampling and finite expectations for the weighted terms.

Worked Examples

Diagnostics, Alternatives, and Limits

The central diagnostic question is whether the observed histories support the required probability models. Check whether the retention model includes recent symptom severity and treatment history when these predict dropout. Compare predicted retention with observed retention across symptom and treatment-history groups, and inspect whether important histories have estimated probabilities near zero. Likewise, assess whether treatment probabilities are plausible across histories. A model that omits symptom changes can leave the very selection mechanism of concern unaddressed.

Complete-case analysis is simple but generally targets the retained, selected population when retention depends on prognosis. Inverse-probability weighting instead targets the baseline population under each strategy if its assumptions hold. Likelihood-based methods and multiple imputation are alternative approaches to missing outcomes, but they also require assumptions about the missingness process and a correctly specified model; they do not remove the need to justify how dropout relates to observed and unobserved information. The introductory material on informative loss to follow-up provides broader context for these approaches.

Weighting cannot recover information absent from the measured history. If dropout depends on unrecorded worsening symptoms even after conditioning on recorded history, sequential independent censoring fails. If few participants with a particular symptom history remain observed, positivity may fail or be practically weak; estimates can then be unstable and highly dependent on a few participants. Model misspecification, measurement error in symptoms, and treatment deviations also threaten the interpretation. A weighted estimate is therefore not automatically unbiased merely because weights were calculated.

• Common mistake: Treating the retention-weighted outcome average among people who followed the strategy as an estimate for the full population, without treatment weights. The correction is to include both treatment-path and retention probabilities, under the corresponding exchangeability and positivity assumptions.
• Another pitfall: Assuming that censoring is ignorable because dropout rates are similar overall across treatment groups. Dropout can still depend strongly on recent symptoms within groups; assess retention conditional on the relevant measured history.
• Rounding check: Calculate contrasts from unrounded risks. In the example, \(1/3-2/3=-1/3\), approximately \(-0.333\); subtracting rounded values can produce a spurious final digit.

The practical conclusion is that symptom-related dropout is a feature of the causal identification problem, not just a loss of sample size. State the target population and strategy, model retention using the observed symptom and treatment history, include treatment weights when estimating observational strategy risks, and report the assumptions that connect the weighted analysis to marginal remission.