Bound a Binary Risk Difference Under Missing Outcomes

Define the full eligible denominator

For each group, record observed events, the number with an observed binary outcome, and the number whose outcome is missing. Confirm that observed and missing counts partition the intended population at the same follow-up time. Exclude structural non-applicability only under the prespecified outcome definition, not to make the bounds narrower.

Cochrane's missing-outcome bias guidance emphasizes that missingness can affect interpretation depending on its relationship to outcomes. A simple percentage missing cannot establish that an analysis is unbiased.

Construct the extreme risks

If E events are observed among O observed outcomes and M outcomes are missing, the smallest full-population event proportion is E/(O+M), obtained by assigning every missing outcome as a non-event. The largest is (E+M)/(O+M), obtained by assigning all missing outcomes as events.

For a risk difference A minus B, the lowest possible difference combines A's lowest risk with B's highest; the highest difference combines A's highest risk with B's lowest. Keep the event direction explicit, since the meaning of a favorable extreme depends on whether the event is desirable.

Recalculate a fictional example

Suppose group A has 20 events among 80 observed outcomes and 20 missing outcomes. Its full denominator is 100, with risk bounded between 20% and 40%. Group B has 30 events among 90 observed outcomes and ten missing, giving bounds from 30% to 40%.

The full-population risk difference A minus B can therefore range from −20 to +10 percentage points under these extreme assignments. The observed-case difference is 20/80 minus 30/90, approximately −8.33 percentage points, which answers a different descriptive calculation. Use the missing-outcome bounds tool to reproduce these invented counts.

Interpret the range correctly

The bounds represent possible completed-data proportions under the entered counts, not a 95% confidence interval. They do not incorporate sampling uncertainty, establish which missingness scenario is credible, or supply a recommended imputation procedure. Extreme assignments can be implausible and still serve as a transparent initial sensitivity check.

Use the missing-outcome worksheet to retain denominator definitions, reasons for missingness, and the assumptions behind any narrower scenarios. If no outcomes are missing, the logical range collapses to the observed risk difference, but statistical uncertainty about population effects still remains.

Review patterns by group and the timing of losses. Consider whether additional information supports a more informative sensitivity analysis, while documenting the assumptions that information introduces. Do not treat a range crossing zero as proof of no effect or a range staying on one side as proof of causality.

Finish with the observed counts, extreme bounds, and a clear distinction between arithmetic possibilities and the inferential analysis. This makes the impact of unobserved outcomes visible without pretending the calculator has recovered them.

Source checked on 4 November 2026; fictional bounds independently recalculated.