Confidence Interval Propagation¶
Uncertainty propagation method — instantiates Effect Size Standardization
Carries a raw estimate's uncertainty through the standardizing transformation so the reported effect keeps a valid interval instead of collapsing to a point.
Standardizing an effect is a transformation, and a transformation applied to a point estimate must also be applied to that estimate's uncertainty — or the uncertainty is silently thrown away. Confidence Interval Propagation is the mechanism that stops that loss. It takes the standard error or interval on the raw effect and pushes it through the same conversion that produced the standardized magnitude, so the output arrives as a range, not a bare number. Its defining commitment is that it operates on the spread, not the center: it never changes the point estimate, and it never judges whether the effect is meaningful — its one job is to make sure the interval that leaves the pipeline is the correct image of the interval that entered it.
Example¶
A sports scientist standardizes a small training study's effect on VO₂max. The raw mean difference is 3.5 mL/kg/min with a 95% interval of [1.1, 5.9]; converting to a standardized mean difference gives d = 0.55. The naive move — report "d = 0.55" and stop — discards everything the interval told us. Instead the analyst propagates: the standardized effect's standard error is derived from the group sizes and the point estimate itself (larger d's carry a bit more sampling variance), yielding an SE of about 0.21 and a 95% interval of roughly [0.14, 0.96]. That range is decision-changing: the effect might be trivial or might be large, and the study simply cannot tell. The record keeps the raw interval, the propagation formula used, and the resulting standardized interval side by side, so a reviewer can confirm the width was transformed, not invented — and can see the estimate was never precise to begin with.
How it works¶
The distinguishing question is how uncertainty rides through a given transformation, since each metric has its own variance formula:
- Use the metric's variance expression, not the raw one. A standardized mean difference has a known large-sample variance in terms of group sizes and the effect; a log risk ratio's variance comes from the event counts. The right formula depends on which sibling produced the effect.
- Work on the scale where the interval is symmetric. For ratios, build the interval in log space and exponentiate the endpoints — never add and subtract a margin on the raw ratio.
- Apply the delta method for derived quantities. When a transform has no closed-form variance, approximate it from the transform's derivative — or resample when even that is unsafe.
- Log the propagation. Record raw interval → formula/scale → standardized interval so the width is auditable.
It carries uncertainty; it does not correct bias in the point estimate and does not display the interval graphically.
Tuning parameters¶
- Propagation method — closed-form variance vs. delta-method approximation vs. bootstrap. Closed forms are fast and exact where they exist; the bootstrap is safest for messy transforms but costs computation.
- Confidence level and shape — 90% vs. 95%, and whether the interval is symmetric or (for ratios/bounded metrics) asymmetric on the natural scale.
- Scale of construction — log vs. raw for ratios, Fisher-z vs. raw for correlations; building on the wrong scale produces intervals that cross impossible values.
- Correlation between inputs — whether covariances among terms are included; ignoring them understates the propagated width.
When it helps, and when it misleads¶
Its strength is that it prevents the archetype's "uncertainty detachment" failure directly: a standardized magnitude without an interval invites overconfident ranking, and this mechanism guarantees the interval survives the rescaling. The delta method is the general-purpose tool that makes propagation possible for almost any transform.[n1]
Its danger is false precision by construction: the propagated interval is only as honest as the variance formula and the assumptions behind it, and applying a large-sample formula to a tiny study, or building a symmetric interval on a ratio, produces a tidy range that is simply wrong — sometimes spanning impossible values like a negative risk. The classic misuse is reporting a crisp standardized interval whose width was computed under normality the data violate. The guarding discipline is to match the propagation method to the metric and sample size, construct on the natural scale, and treat any interval that reaches an impossible value as a signal the method, not the effect, has failed.
How it implements the components¶
uncertainty_attachment— its core act: it attaches a correctly transformed interval or standard error to the standardized magnitude.raw_effect_estimate_record— preserves the original standard error/interval and its scale as the input the propagation transforms.conversion_traceability_table— logs the raw interval, the propagation formula and scale, and the resulting standardized interval so the width is auditable.
It does not compute the point estimate it wraps — the base magnitude and its standardization_rule come from Standardized Mean Difference Calculation; it never removes point-estimate bias, which is Hedges Correction Application via small_sample_bias_correction_flag; and it does not draw the interval — that display is Forest Plot or Effect Table Display.
Related¶
- Instantiates: Effect Size Standardization — supplies the uncertainty that must travel with every standardized magnitude.
- Consumes: Standardized Mean Difference Calculation — takes its point estimate and raw variance as the inputs to propagate.
- Sibling mechanisms: Standardized Mean Difference Calculation · Hedges Correction Application · Correlation or Regression Coefficient Transformation · Risk Ratio or Odds Ratio Standardization · Absolute Risk Difference Translation · Meta-Analytic Effect Harmonization · Minimal Important Difference Anchoring · Forest Plot or Effect Table Display
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Carries a raw estimate's uncertainty through the standardizing transformation so the reported effect keeps a valid interval instead of collapsing to a point, making its operative form a computation, comparison, model, or analytic representation used to infer, estimate, or choose.
Independent corroboration: The frozen evidence defines Confidence Interval Propagation as 'Carries a raw estimate's uncertainty through the standardizing transformation so the reported effect keeps a valid interval instead of collapsing to a point', so its operative form is Analysis, Modeling & Optimization.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Statistics & Experimental Design
Origin pattern: Single lineage
Present-day reach: Specialized
Rationale: Statistical inference established delta-method, exact-transform, and resampling procedures for carrying interval uncertainty through transformations.
Review resolution: Statistical inference established delta-method, exact-transformation, and resampling procedures for carrying interval uncertainty through transformations. Calculus and probability are formal prerequisites rather than a separate origin domain.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] The delta method approximates the variance of a transformed estimate from the variance of the original and the derivative of the transformation, giving a first-order standard error where no exact formula exists. It is the standard bridge for propagating uncertainty through nonlinear effect-size conversions, with resampling as the fallback when the linear approximation is unreliable. ↩