A trial can lose participants’ outcome measurements without losing their names from the dataset. A report may still describe the analysis as intention to treat. Neither circumstance tells you what the missing outcomes would have been.
A practical reading question is: how different would the unobserved outcomes need to be for the conclusion to change? Answering it requires explicit assumptions, rather than treating every missing outcome as a success, failure or copy of the last observation by habit.
This article uses original fictional numbers to teach that reasoning. The programmes and outcome threshold are invented; the exercise is not evidence about an intervention.
First, keep the randomized and observed denominators separate
A fictional Canadian trial assigns 100 people to programme A and 100 to programme B. At twelve weeks, it records whether each person reaches a prespecified task-performance threshold. Reaching it is the favourable outcome.
| Group | Randomized | Follow-up outcome observed | Observed successes | Outcome missing |
|---|---|---|---|---|
| A | 100 | 80 | 56 | 20 |
| B | 100 | 60 | 45 | 40 |
Among participants with observed outcomes, success is 56/80 = 70% in A and 45/60 = 75% in B. The complete-case difference, A minus B, is −5 percentage points.
That is an accurate description of the observed subset. It is not automatically the result for everyone randomized. The observed groups may exclude different kinds of participants, and the missing proportions differ.
An intention-to-treat label does not resolve this issue. Cochrane’s risk-of-bias guidance asks whether missingness could depend on the outcome’s true value and whether sensitivity analyses support the result; a single safe percentage of missing data cannot settle that assessment. Cochrane Handbook chapter 8, section 8.5.
Write the unknowns before guessing them
Let x be successes among the 20 missing A participants, and y successes among the 40 missing B participants. Then:
A success proportion = (56 + x) / 100
B success proportion = (45 + y) / 100
A − B difference, in percentage points = 11 + x − y
Here x ranges from 0 to 20 and y from 0 to 40. Each additional success changes its group’s percentage by one point because the group denominator is 100. That convenient correspondence would not hold with every sample size.
The formula exposes the unresolved information. We know the observed successes; we do not know which values of x and y occurred.
Calculate the extreme bounds
The least favourable completion for A assigns no successes to its missing participants and successes to all of B’s missing participants:
56/100 − 85/100 = −29 percentage points
The most favourable completion for A does the reverse:
76/100 − 45/100 = +31 percentage points
Thus the possible completed-data differences range from −29 to +31 percentage points, holding the observed data fixed. This is not a confidence interval. It describes the range produced by all allowable completions of these missing binary outcomes. It does not include a separate calculation of sampling uncertainty or assign probabilities to the extreme scenarios.
The bounds show that the observed records alone do not determine the direction of the completed-data difference. They do not establish that the extremes are equally plausible—or plausible at all.
Move from extreme bounds to a reasoned scenario
Suppose a teaching scenario assumes half of the missing participants in each group succeeded. Then x = 10 and y = 20:
- A: 66/100 = 66%.
- B: 65/100 = 65%.
- Difference: +1 percentage point.
The sign reverses from the complete-case result. Equal assumed success proportions among missing participants did not mean equal numbers of added successes, because there were twice as many missing outcomes in B.
Alternatively, suppose missing participants have their own group’s observed success proportion. Then x = 14 and y = 30, giving 70% versus 75% again. This completion reproduces the complete-case difference; that agreement comes from the assumption, not independent confirmation.
Neither scenario uncovers what actually happened. It asks what follows if a specified pattern is true. A sound strategy pairs a primary analysis under defensible assumptions with sensitivity analyses exploring departures from them. White et al.: Strategy for intention to treat analysis in randomised trials with missing outcome data.
Locate a tipping point—and name what is tipping
Hold B’s missing successes at y = 20. The difference becomes:
11 + x − 20 = x − 9 percentage points
At x = 9, both groups have 65 successes and the difference is zero. At x = 10, A leads by one percentage point. At x = 8, A trails by one.
This is a tipping point for the sign of the point estimate, conditional on the specified value for y. It is not a tipping point for statistical significance, an important-benefit threshold or a treatment recommendation. Those questions require additional definitions and analysis. A real tipping-point display can vary assumptions in both groups rather than fixing one.
The next appraisal question is whether nine or ten successes among A’s missing participants is plausible given recorded reasons for missingness and other observed information. The arithmetic cannot answer that on its own.
Look for mechanisms, not just method names
Imagine that A participants missed follow-up because a scheduling system failed, while B participants with worsening function were more likely to withdraw. Those are different possible missingness processes. Conversely, if highly improved participants stopped answering messages, missing outcomes might be better than observed outcomes.
The report should help you distinguish such possibilities. Reasons, timing and predictors of missingness are useful, although they cannot fully verify assumptions about unobserved outcomes. Multiple imputation and other models can incorporate observed information, but their names do not establish that assumptions are satisfied. Sensitivity analyses remain relevant. Cochrane Handbook chapter 8.
Do not confuse stopping the assigned programme with having no outcome measurement. Someone can discontinue treatment and still provide follow-up information. Keeping those events separate makes the analysis question clearer.
Transfer exercise
A second fictional trial randomizes 50 participants per group. A has 30 observed successes and five missing outcomes; B has 25 observed successes and ten missing outcomes. All other outcomes are observed failures.
Find the extreme bounds for A minus B. Then suppose five of B’s missing participants succeeded. How many successes among A’s missing participants produce a tie, and how many are needed for A to lead?
Reasoned solution: A’s completed success proportion ranges from 30/50 to 35/50, or 60% to 70%. B’s ranges from 25/50 to 35/50, or 50% to 70%. The difference therefore ranges from −10 to +20 percentage points.
With five added successes, B has 30/50 = 60%. A ties with no added successes and leads with at least one, because one participant now represents two percentage points. These remain completed-data scenarios, not estimated confidence intervals or proof of a benefit.
Takeaway for a paper discussion
Record the observed result, missing numbers by group, primary assumptions, and the sensitivity scenario that changes the interpretation. Finish with what remains uncertain. “The conclusion survives this plausible departure” is a more informative claim than “the authors used intention to treat.”
Sources and further reading
- Cochrane Handbook, chapter 8 — missing-outcome risk of bias; relevant full-text sections reviewed.
- White et al.: Strategy for intention to treat analysis in randomised trials with missing outcome data, 2011 — explicit assumptions and sensitivity analyses; indexed publisher excerpts reviewed.