Why Compare Dropout Across Outcome Histories?
A dropout rate averaged over everyone at risk can hide important differences. Patients with a poor recorded outcome at one visit may leave follow-up more often before the next visit than patients with a good recorded outcome. Comparing these rates describes how dropout is related to information that was observed before dropout. It does not, by itself, show why people left or what their unrecorded outcomes would have been.
The denominator matters. At a given visit, the relevant probability is calculated among people still under follow-up and eligible to drop out at that visit—not among everyone initially enrolled. Each person’s outcome history used in the comparison must have been recorded before the dropout event being studied.
This definition fixes the timing: the history is known before the dropout decision or event represented by \(D_t\). If outcomes are recorded at a withdrawal visit, or if a study defines dropout at a different point in the visit sequence, the event and its risk set must be defined accordingly. The calculation is only meaningful when dropout status and prior history are correctly classified.
Reading Conditional and Overall Probabilities
A history-specific dropout probability is a conditional probability. For example, \(q_t(h)=0.30\) means that, among participants at risk at visit \(t\) whose recorded history is \(h\), 30% drop out before their visit-\(t\) outcome is recorded. It is not the percentage of all enrolled participants who drop out, and it does not say that the history caused dropout.
The overall visit-specific probability averages across histories in the risk set. If the possible histories are \(h_1,\ldots,h_K\), then the law of total probability gives
Thus, the overall rate is a weighted average of the history-specific rates, with weights equal to the proportions of each history in that visit’s risk set. It can change between visits even when the history-specific probabilities do not, because the composition of the risk set changes.
Worked Example: Comparing Dropout After Poor and Good Outcomes
Suppose 100 hypothetical participants are at risk before visit 3. Their visit-2 outcomes have been recorded and classified as poor or good. Among the 30 with a poor outcome, 12 drop out before visit 3. Among the 70 with a good outcome, 14 drop out.
For the poor-history group, the estimated visit-specific probability is
For the good-history group, it is
The observed dropout probability is 20 percentage points higher after a poor rather than a good recorded outcome. The overall rate is \( (12+14)/100=0.26 \), or 26%. The weighted-average check agrees:
These calculations support the descriptive statement that visit-3 dropout was more common among participants with a poor recorded visit-2 outcome in this hypothetical risk set. They do not establish that the prior outcome caused dropout, nor do they reveal the visit-3 outcomes that dropouts would have had.
Why a History Association Does Not Settle MAR or MNAR
The outcome-history comparison concerns whether dropout is associated with observed information. In the notation used here, it describes how \(P(D_t=1\mid H_{t-1},\text{at risk})\) varies with \(H_{t-1}\). A difference across histories makes dropout predictable from recorded history, but it does not determine whether dropout also depends on outcomes that were not observed.
Under a sequential MAR condition, dropout at visit \(t\), conditional on the recorded history and other recorded covariates included in the condition, does not depend on the unobserved outcomes. This condition permits dropout probabilities to vary with observed history. By contrast, MNAR involves dependence on unobserved outcomes even after conditioning on the observed information. A history-specific rate comparison alone cannot test that additional dependence: the outcomes after dropout are unavailable to make the needed comparison.
Likewise, MCAR means dropout is independent of the data relevant to the missingness process, rather than merely having an overall rate that can be calculated. If history-specific rates differ in the population, that pattern is inconsistent with MCAR relative to that recorded history. In a finite sample, however, apparent differences can arise by chance. Neither a difference nor a lack of a visible difference by itself establishes a missingness mechanism.
Worked Example: A History-Specific Rate Is Not a Test of MNAR
In a hypothetical study, 80 participants at risk before visit 4 have a mild recorded visit-3 outcome, and 80 have a severe recorded visit-3 outcome. Before visit 4, 8 participants in the mild group and 16 in the severe group drop out.
The estimated probabilities are
The severe-history group has a 10-percentage-point higher observed dropout rate. This shows that the recorded visit-3 category predicts dropout in these data. It does not identify the relationship between dropout and the unobserved visit-4 outcome.
To see why, suppose the observed dropout statuses and all recorded outcomes remain exactly as stated. In one possible completion of the data, the missing visit-4 outcomes among dropouts could have a distribution similar to the outcomes among participants who remain, after conditioning on recorded history. In another possible completion, dropouts could have systematically worse missing outcomes within each recorded-history group. Both completions agree with the observed dropout counts and histories, but they imply different relationships between dropout and the missing visit-4 outcomes. The observed rate comparison cannot distinguish them.
Therefore, the defensible conclusion is about observed-history prediction, not about unobserved outcomes. Additional assumptions or sensitivity analyses are needed to reason about dependence on those unobserved outcomes.
Visit-Specific Interpretation and a Short Derivation
The weighted-average identity follows by partitioning the visit-\(t\) risk set according to recorded history. The events \(H_{t-1}=h_k\) are mutually exclusive and cover the risk set. Therefore,
The second equality is the conditional-probability product rule. This derivation also explains why an overall dropout rate should not be compared across visits without considering both the history-specific probabilities and the distribution of histories among those still at risk.
Worked Example: Changing Risk-Set Composition Across Visits
Consider a hypothetical follow-up with 100 participants at risk before visit 2: 40 have a poor visit-1 outcome and 60 have a good one. Before visit 2, 12 of the 40 with poor history and 6 of the 60 with good history drop out. The respective probabilities are \(12/40=0.30\) and \(6/60=0.10\), and the overall visit-2 dropout probability is \(18/100=0.18\).
Among the 82 who remain, suppose visit-2 outcomes are recorded. Before visit 3, 20 participants are at risk with a poor visit-2 outcome and 62 with a good visit-2 outcome. If 8 of the 20 and 6 of the 62 drop out, then
The visit-3 overall rate is \(14/82\approx0.171\). Its weighted calculation is
The visit-2 and visit-3 overall rates are 18.0% and about 17.1%, respectively, but those summaries do not describe the same risk set or the same prior-outcome classification. In addition, only participants who remained through visit 2 could contribute visit-2 outcomes to the visit-3 history comparison. The visit-3 result is conditional on having survived follow-up to that risk set; it is not a comparison for the original 100 participants.
Assumptions, Diagnostics, and Alternatives
The basic proportions are descriptive estimates. To interpret them as estimates of the corresponding visit-specific probabilities, the risk set, dropout event, and prior history must be defined consistently, and dropout status and recorded outcomes must be measured accurately. Each person must contribute to the denominator only when eligible and at risk for the event. If the study has sampling or clustering features, uncertainty calculations must account for them; raw proportions alone do not quantify uncertainty.
Inspect whether history groups are sufficiently populated. A rate based on one dropout among two people is highly unstable, while a history group with no participants has no estimable rate. When many detailed histories are considered, some cells may be sparse even in a large study. Combining histories can improve stability but changes the question and can conceal meaningful differences. Any grouping should be justified by the scientific meaning of the outcomes, not chosen only to produce a desired rate pattern.
A useful conceptual comparison is between three summaries:
| Summary | What it answers | What it does not answer |
|---|---|---|
| Overall visit-specific dropout probability | What fraction of this visit’s risk set drops out? | Whether dropout differs by recorded history |
| History-specific probabilities | How does observed dropout vary across specified recorded histories? | Whether unobserved outcomes influence dropout |
| Mechanism assumption such as sequential MAR | What conditional independence restriction is being assumed? | It is not established merely by observing similar or different rates across histories |
A model that adjusts for baseline covariates or additional recorded history can answer a more conditional predictive question than a simple grouping. But its interpretation depends on the covariates and functional form used, and it still cannot recover the unobserved outcomes without assumptions. The introductory distinction between MCAR, MAR, and MNAR is useful here: history-specific rates describe observable patterns, while the mechanism labels concern conditional relationships that include unobserved data.
Common warning signs include an incorrect denominator that includes people who already dropped out, using an outcome recorded after the dropout event as if it were prior history, and treating a small difference in noisy rates as decisive evidence. A further pitfall is to say, “dropout is predictable from prior outcomes, therefore it is MAR.” The correction is that predictability from observed history is compatible with MAR, but it does not rule out additional dependence on unobserved outcomes. Conversely, finding similar rates across observed histories does not prove MCAR or MAR.
The main alternative to history-specific comparisons is an overall rate, which is concise but can mask variation and shifts in risk-set composition. A regression model can summarize multiple covariates or histories, but it adds modeling assumptions and should not be interpreted as solving the unobserved-outcome problem. The choice should match the question: use conditional comparisons to describe observed-history patterns, and use explicit missingness assumptions when making claims that involve outcomes not recorded.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- At a visit, 18 of 60 participants with a poor prior outcome drop out, compared with 12 of 120 with a good prior outcome. Calculate both probabilities and their difference in percentage points.
- Why is it incorrect to use the full enrollment count as the denominator for a visit-specific dropout probability if some participants already dropped out?
- If dropout rates differ across recorded outcome histories, what conclusion is warranted, and what conclusion about unobserved outcomes is not warranted?
- At a later visit, history-specific dropout probabilities remain the same, but the overall rate changes. Give one possible explanation.
Solutions. (1) The poor-history probability is \(18/60=0.30\); the good-history probability is \(12/120=0.10\). The difference is \(0.30-0.10=0.20\), or 20 percentage points. (2) Participants who already dropped out are not eligible to experience the visit-specific event in the current risk set. Including them changes the denominator and does not estimate the conditional probability defined for that visit. (3) The rates support an observed association between recorded history and dropout in the risk set. They do not establish whether dropout depends on outcomes that were unobserved after dropout. (4) The proportions of the different histories among those at risk may have changed, changing the weights in the overall weighted average even if each history-specific probability stayed constant.