Why Tabulate Dropout by Observed History?
An overall dropout percentage can hide an important pattern. If participants with poor outcomes at one visit are more likely to withdraw before the next, the overall percentage combines groups with different observed histories and may conceal that relationship. A table organized by visit, treatment arm, and prior outcome category makes such patterns visible and gives each reported rate a clear denominator.
These tables describe the observed follow-up process; they do not, by themselves, identify why withdrawal occurred or establish a missingness mechanism. In particular, a difference across observed-history groups is useful diagnostic evidence, not proof that a particular mechanism label applies.
Define the Risk Set Before Calculating a Rate
Index treatment arms by \(a\), scheduled visits by \(t\), and categories of the prior observed outcome by \(c\). Consider the interval after the prior scheduled visit and before the outcome is collected at visit \(t\). The risk set for arm \(a\) and visit \(t\) consists of participants who are still under follow-up and eligible for that visit when the interval begins. A participant is counted as a dropout in that interval if they permanently withdraw before the visit-\(t\) outcome is collected. A withdrawal after that outcome belongs to a later interval.
For a straightforward table, prior outcome categories should be defined in advance, for example as low, medium, and high using clinically meaningful cut points. When each participant has an observed outcome at the immediately preceding visit, the participant belongs to the category containing that outcome. If the immediately preceding outcome is unavailable but the participant remains under follow-up, do not assign a guessed value or silently remove them from the risk set. Show a separate “prior outcome unavailable” history row, and clarify what earlier recorded outcome, if any, was used to define that row. Such participants may have a different time gap since measurement, so their rates should not be interpreted as directly comparable without that qualification.
A valid table requires more than arithmetic. The categories should be mutually exclusive, and the displayed categories—including any separate unavailable-history row—should account for the intended risk set. Withdrawal status through the interval must be ascertained well enough to distinguish permanent withdrawal from a missed visit by someone who remains in follow-up. Participants whose planned follow-up ends before the interval should not be treated as dropouts. Report the counts as well as the percentages: a rate from a small denominator is less stable and less informative than the same-looking rate from a large one.
Worked Tables and Calculations
Worked Example: Comparing Prior Outcome Categories at One Visit
Suppose a hypothetical study has 100 participants entering the interval before visit 2 in one treatment arm. All are eligible for visit 2, and each had an observed visit-1 outcome. Categories were set before examining dropout: low, medium, and high. The counts are:
| Visit 1 outcome category | At risk, \(n_{a2c}\) | Withdraw before visit 2, \(d_{a2c}\) | Dropout rate |
|---|---|---|---|
| Low | 20 | 8 | 8/20 = 40% |
| Medium | 40 | 8 | 8/40 = 20% |
| High | 40 | 4 | 4/40 = 10% |
| Total | 100 | 20 | 20/100 = 20% |
For the low category, \(\hat r_{a,2,\mathrm{low}}=8/20=0.40\); for the high category, \(\hat r_{a,2,\mathrm{high}}=4/40=0.10\). In this hypothetical arm, the observed visit-2 dropout proportion is 30 percentage points higher among participants in the low prior-outcome category than among those in the high category. This is a descriptive comparison for this interval and these observed groups; it does not establish that a low outcome caused withdrawal.
Worked Example: Keeping Visit and Treatment Arm Visible
In a second hypothetical study, 100 participants in each of two arms enter the interval before visit 2. The visit-2 counts by prior outcome category are as follows. By visit 3, 84 participants remain eligible in each arm; the visit-3 categories are defined from their observed visit-2 outcomes.
| Arm | Visit | Prior category | At risk | Withdrawals | Rate |
|---|---|---|---|---|---|
| A | 2 | Low | 20 | 8 | 8/20 = 40% |
| A | 2 | High | 80 | 8 | 8/80 = 10% |
| B | 2 | Low | 20 | 4 | 4/20 = 20% |
| B | 2 | High | 80 | 12 | 12/80 = 15% |
| A | 3 | Low | 30 | 9 | 9/30 = 30% |
| A | 3 | High | 54 | 5 | 5/54 = 9.26% |
| B | 3 | Low | 42 | 8 | 8/42 = 19.0% |
| B | 3 | High | 42 | 4 | 4/42 = 9.52% |
For example, in arm A before visit 3, the total is \(9+5=14\) withdrawals among \(30+54=84\) at-risk participants, giving \(14/84=16.7\%\). In arm B the corresponding calculation is \((8+4)/(42+42)=12/84=14.3\%\). At visit 2, both arms have 16 withdrawals among 100 participants, or 16%. These summaries show why arm and visit should remain explicit: the overall arm-specific percentages do not display the different category patterns at visit 2, and they do not replace the visit-3 rates. The category counts entering visit 3 are consistent with 84 participants remaining after the visit-2 intervals in each arm.
Why Overall Rates Are Weighted Averages
When the categories partition the risk set, the pooled rate is determined by the category counts and their denominators. The following identity also explains why a simple average of category percentages is generally wrong.
This identity applies when the displayed categories collectively cover the risk set and its total size is positive. The sum on the right includes only categories with \(n_{atc}>0\), since a zero-denominator category has no defined rate. To verify the identity, substitute \(d_{atc}=n_{atc}\hat r_{atc}\) for each positive-denominator category. A zero-count category contributes zero to the numerator on the left and need not be assigned a rate.
Worked Example: An Empty Category and the Correct Pooled Rate
Suppose a hypothetical arm-visit table has 0 participants in the low category, 10 in the medium category, and 30 in the high category. There are 1 withdrawal in the medium category and 9 in the high category. The low-category rate is undefined because its denominator is zero; it is not zero percent.
The pooled rate is \((1+9)/(0+10+30)=10/40=25\%\). The positive-denominator category rates are \(1/10=10\%\) and \(9/30=30\%\). Their risk-set weights are \(10/40=0.25\) and \(30/40=0.75\), so the weighted calculation is \(0.25(10\%)+0.75(30\%)=2.5\%+22.5\%=25\%\). The unweighted average, \((10\%+30\%)/2=20\%\), is not the overall dropout rate because the two categories contain different numbers of at-risk participants.
Reading the Patterns, and Knowing Their Limits
A useful diagnostic table makes several comparisons possible without collapsing them into a single number: across outcome categories within a visit and arm; across visits within an arm; and across arms at a comparable visit and prior-outcome category. Interpret these comparisons with the counts visible. For instance, a large percentage difference based on only a few participants may be highly unstable. If rates are presented as percentages, state the numerator and denominator or include them in adjacent columns.
Check the construction before interpreting the pattern. Confirm that categories are defined consistently, each person is counted once in the relevant risk set, withdrawals are assigned to the correct interval, and administrative end of follow-up is not counted as withdrawal. Check whether “prior outcome unavailable” is included and whether its definition is consistent across arms and visits. A sudden change in denominators may reflect a change in eligibility, follow-up schedule, or data capture as well as attrition. If a participant can miss a visit and later return, a missing outcome at one visit is not automatically a permanent dropout; the table must keep those statuses distinct.
Stratified tables are often more informative than a single pooled rate because pooling can conceal heterogeneity across observed histories. Conversely, a table with many categories, visits, and arms may become sparse and hard to interpret. A pooled table is compact but hides those differences. A plot of the same rates can make trends easier to see, but should retain denominators or counts. Model-based summaries can adjust or smooth comparisons and handle multiple predictors, but require additional modeling choices and assumptions; they do not make the raw definitions of the risk set less important.
Most importantly, these are descriptions of observed follow-up. A relationship between prior observed outcome and subsequent dropout is compatible with dropout depending on recorded history, but does not show that all relevant history was recorded or rule out dependence on unobserved outcomes. Likewise, similar rates across observed-history categories do not establish independence from unobserved outcomes. The earlier material on MCAR, MAR given the recorded outcome history, and MNAR distinguishes these claims by their conditioning information; a diagnostic table alone cannot resolve them.
Specify the scheduled visit, who is eligible at interval start, and which event counts as permanent withdrawal before the visit outcome.
Set prior-outcome categories consistently and represent unavailable prior outcomes separately rather than excluding eligible participants without explanation.
For each arm, visit, and category, display \(d_{atc}\), \(n_{atc}\), and \(d_{atc}/n_{atc}\) when \(n_{atc}>0\). Mark empty cells as undefined.
Verify that the categories cover the intended risk set, investigate unusual denominators, and describe patterns without claiming that the table proves a missingness mechanism.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a hypothetical arm and visit, 12 of 30 participants in the low prior-outcome category withdraw, while 4 of 20 in the high category withdraw. Calculate both rates and the pooled rate.
- A prior-outcome category contains no eligible participants. What rate should be reported for it, and should it contribute to the weighted-average calculation?
- At a visit, one participant misses the outcome measurement but remains enrolled and in follow-up. Should that person automatically count as a permanent dropout?
- What can a table showing higher dropout among participants with low prior outcomes support, and what can it not establish by itself?
Solutions. 1. The low-category rate is \(12/30=40\%\); the high-category rate is \(4/20=20\%\). The pooled rate is \((12+4)/(30+20)=16/50=32\%\). 2. Report the category rate as undefined because its denominator is zero. Exclude it from the weighted-rate sum, while retaining it in the displayed table if it is part of the planned categories. 3. No. A missed measurement is not automatically permanent withdrawal; classify the participant using their follow-up status, and distinguish temporary absence from dropout. 4. The table supports a descriptive statement that the observed dropout proportion differs across the displayed prior-outcome groups in that arm and interval. It cannot, by itself, establish causation, prove MAR, or rule out dependence on unobserved outcomes.