Zero-Event Trials: What to Add to Nothing
Adding 0.5 to every cell is the default in most software, and it isn't always right. What continuity corrections do to your estimate, and the alternatives.
Rare outcomes produce trials where one arm records no events. Odds ratios and risk ratios are undefined when a cell is zero, so software applies a continuity correction, conventionally adding 0.5 to every cell of the affected study.
That default is quietly consequential. Sweeting, Sutton and Lambert (2004) showed that the 0.5 correction introduces bias, and that the bias is larger when the arms are unequal in size. Adding the same constant to a group of 40 and a group of 400 is not a neutral act.
What the correction does
A continuity correction makes the estimate calculable and pulls it toward the null. It also changes the study's weight, and in a meta-analysis dominated by a few sparse trials it can determine the direction of the pooled result.
Two situations are worth distinguishing. Single-zero studies, where one arm has no events, are handled by the correction with modest distortion. Double-zero studies, where neither arm records an event, contribute no information about the ratio and are excluded by most software automatically.
That automatic exclusion is defensible for ratio measures and is worth stating rather than leaving silent, because a reader comparing your forest plot against the included-studies table will notice studies missing.
Better options
Peto odds ratio. Performs well when events are rare, treatment effects are not large, and arms are roughly balanced. Bradburn and colleagues (2007) found it the least biased of the common approaches under those conditions, and it requires no continuity correction. It performs poorly with substantially unbalanced arms.
Mantel-Haenszel without correction. The Mantel-Haenszel method handles single zeros without correction when the risk difference is used, and Cochrane software applies corrections only where needed.
Risk difference. Defined even with zero cells, and it avoids the problem entirely. The cost is that risk differences are less transportable across populations with different baseline risks, so a pooled risk difference from mixed populations can be hard to interpret.
Exact and generalised linear mixed models. Beta-binomial and related methods model the counts directly rather than the ratio, and avoid corrections. They are the statistically preferable option for rare events and are less familiar to reviewers, so allow a sentence explaining the choice.
What to prespecify
Three decisions, all in the protocol.
Which effect measure will be used if events are rare, with a definition of rare, for instance fewer than 1% of participants experiencing the outcome.
Whether and how a continuity correction will be applied, including whether a treatment-arm correction proportional to arm size will be used instead of a fixed constant.
How double-zero studies will be handled and reported.
A protocol that anticipates rare events costs three sentences. A review that improvises at the analysis stage invites the reviewer to ask whether the method was chosen for its result.
A worked example
Eight trials of an intervention where the outcome is a serious adverse event. Two trials record zero events in both arms; three record zero in one arm.
Under the default 0.5 correction with an odds ratio, the pooled estimate is 0.71, 95% CI 0.38 to 1.33.
Under a Peto odds ratio with no correction, and with the two double-zero trials excluded as uninformative, the estimate shifts. Both analyses are reportable; only one of them was prespecified.
The correct presentation gives the prespecified analysis as primary, the alternative as a sensitivity analysis, and a sentence in the discussion noting that with this few events the evidence on harms is very uncertain regardless of method. GRADE will downgrade for imprecision here, probably by two levels.
References
Bradburn, M. J., Deeks, J. J., Berlin, J. A., & Localio, A. R. (2007). Much ado about nothing: A comparison of the performance of meta-analytical methods with rare events. Statistics in Medicine, 26(1), 53–77. https://doi.org/10.1002/sim.2528
Higgins, J. P. T., Thomas, J., Chandler, J., Cumpston, M., Li, T., Page, M. J., & Welch, V. A. (Eds.). (2024). Cochrane handbook for systematic reviews of interventions (Version 6.5). Cochrane. https://training.cochrane.org/handbook
Sweeting, M. J., Sutton, A. J., & Lambert, P. C. (2004). What to add to nothing? Use and avoidance of continuity corrections in meta-analysis of sparse data. Statistics in Medicine, 23(9), 1351–1375. https://doi.org/10.1002/sim.1761
Common questions
- Should I exclude double-zero studies?
- For ratio measures they contribute nothing and are conventionally excluded, which most software does automatically. For the risk difference they do contribute and can be retained. Either way, report how many were excluded and why, because their absence from the forest plot is otherwise unexplained.
- Is the Peto method always best for rare events?
- No. It performs well with balanced arms and small effects, and poorly when arms are markedly unequal or effects are large. Check the balance of your trials before committing, and state the condition you relied on.
- Our review has both rare and common outcomes. Do I need different methods?
- Quite possibly, and that is fine provided each is prespecified and labelled. State the method used for each outcome in the analysis section rather than describing a single approach that does not in fact apply throughout.