Sensitivity Analysis With E-values

Define the sensitivity analysis

Load a lecture result or enter a custom adjusted risk ratio and plausible hidden-confounder strengths.

Lecture presets

Adjusted association

Adjusted risk ratio

0.25 protective1 null4 harmful

95% CI lower

95% CI upper

The confidence interval is used for uncertainty display. The lecture's primary E-value is calculated from the point estimate.

Possible hidden confounder U

Exposure–U strength, RR_EU

1 none8 very strong

U–outcome strength, RR_UY

1 none8 very strong

These are sensitivity parameters on the risk-ratio scale. They are not estimated from the observed dataset.

Interpretation options

Show the optional E-value for the confidence-limit closest to the null.

Show the three lecture methods on the E-value curve.

Candidate hidden confounder

Actions

Interpretation of the current analysis

Integrates the adjusted association, sampling uncertainty, point-estimate E-value, confidence-limit E-value, method agreement, and the selected hidden-confounder scenario.

Report-ready interpretation

Interpretive principle. The E-value is a continuous, context-dependent robustness benchmark. It should be interpreted above and beyond the measured adjustment set, alongside the confidence interval, plausible strengths of omitted confounders, the associations of measured covariates where available, and other sources of bias. There is no universal “large enough” cutoff.

Mathematical formulation

Point-estimate E-value, risk-ratio bias factor, equal-strength derivation, confidence-limit extension, and interpretation conditions.

GRADUATE-LEVEL NOTATION

Lecture definition

For {RR}_{obs} \geq 1, E = {RR}_{obs} + \sqrt{{RR}_{obs} ({RR}_{obs} - 1)}

Direct lecture formula

{RR}_{obs} > 1 : E = {RR}_{obs} + \sqrt{{RR}_{obs} ({RR}_{obs} - 1)}

This is the formula used in the lecture note and in the classroom examples.

Protective association extension

{RR}_{obs} < 1 : R = \frac{1}{{RR}_{obs}}, E = R + \sqrt{R (R - 1)}

If the observed risk ratio is below 1, first invert it and then apply the same lecture formula.

Risk-ratio bias factor

B ({RR}_{EU}, {RR}_{UY}) = \frac{{RR}_{EU} {RR}_{UY}}{{RR}_{EU} + {RR}_{UY} - 1}

Sensitivity-parameter meaning: RR_EU bounds the strongest residual exposure–U association, and RR_UY bounds the strongest residual U–outcome association on the risk-ratio scale. These are sensitivity parameters defined after adjustment for the measured covariates X; in the standard bound, RR_UY is interpreted within levels of exposure and X. They are not ordinary regression coefficients.

Explain-away condition

\frac{R}{B} \leq 1 \Leftrightarrow B \geq R

Here R is the observed association oriented away from the null. The factor is a worst-case bound, not a literal bias correction.

Unequal-link boundary

{RR}_{UY, required} = \frac{R ({RR}_{EU} - 1)}{{RR}_{EU} - R}, {RR}_{EU} > R

If one link is relatively weak, the other must be stronger. No finite second link suffices when the fixed first link is at or below R.

Equal-strength benchmark

\begin{matrix} {RR}_{EU} = {RR}_{UY} = r \Rightarrow B = \frac{r^{2}}{2 r - 1} \\ \frac{r^{2}}{2 r - 1} = R \Rightarrow r^{2} - 2 R r + R = 0 \\ r = R \pm \sqrt{R (R - 1)} \end{matrix}

Why the larger root is used

E = R + \sqrt{R (R - 1)} \geq 1

The smaller root is below 1, whereas the sensitivity strengths are defined away from the null and are at least 1.

Optional confidence-limit E-value

R_{CI} = \{\begin{cases} L, & L > 1 \\ \frac{1}{U}, & U < 1 \\ 1, & L \leq 1 \leq U \end{cases} E_{CI} = R_{CI} + \sqrt{R_{CI} (R_{CI} - 1)}

CI excludes the null

Use the confidence limit closest to 1, orient it away from the null, and calculate its E-value.

CI includes the null

The confidence-limit E-value is 1 because no unmeasured confounding is needed to move the interval to include the null.

Lecture convention: the uploaded lecture reports point-estimate E-values only. This confidence-limit calculation is included as an optional advanced extension and is labeled separately throughout.

What the E-value addresses

One sensitivity question: how strong residual unmeasured confounding would need to be, on the risk-ratio scale, to move an adjusted association to the null.

What it does not prove

A large E-value does not establish causation, and a small E-value does not establish no effect. The E-value is a robustness benchmark, not a posterior probability.

Other threats remain

Measurement error, reverse causality, selection bias, poor overlap, model misspecification, interference, and wrong time ordering are not repaired by an E-value.

Scale matters

The standard formula is on the risk-ratio scale. A risk difference cannot be inserted directly into the E-value formula.

Precise reading: an unmeasured confounder with both sensitivity links equal to E could explain away the point estimate. If both links are weaker than E, they cannot fully explain it under the standard bias-factor bound. One link may be below E only if the other is correspondingly stronger.

1. Hidden-confounder graph and current sensitivity parameters

Measured covariates X are adjusted for. The two red U arrows represent residual associations above and beyond X.

causal path T → Ymeasured adjustment pathsunmeasured-confounding links

Arrow thickness is a pedagogical representation of the selected sensitivity parameters. It is not a formal structural coefficient or a fitted DAG edge weight.

2. Selected-confounder stress test

Applies the user-specified hidden-confounder strengths to the point estimate and the confidence interval.

3. E-value Function

This panel follows the lecture note: for risk ratios above 1, E = RR + √{RR(RR − 1)}.

E-value formulacurrent / selected resultIPWOutcome regressionDoubly robust

The lecture curve is drawn on the RR ≥ 1 scale. If the observed adjusted risk ratio is below 1, the current point is plotted at 1/RR before applying the lecture formula.

4. Bias-factor surface: can the selected U explain away the result?

Green cells satisfy B ≥ R; blue cells leave residual association away from the null.

strong enough under the boundnot strong enoughcurrent U

The dashed boundary is B(RR_EU,RR_UY) = R. The equal-strength point on this boundary is the E-value.

5. Unequal-strength trade-off

Fix one hidden-confounder link and calculate the minimum strength required for the other.

explain-away boundaryequal E-value pointcurrent U

6. Point estimate and confidence-limit robustness

The lecture's primary E-value uses the point estimate; the CI E-value is displayed as an optional extension.

7. Three adjusted methods from the lecture

Click a row or point to load that method into the complete sensitivity dashboard.

Method	Adjusted RR	95% CI	p-value	Point E-value	CI E-value

8. Statistical significance and causal sensitivity answer different questions

Neither number replaces the other.

p-value / confidence interval

Sampling question: Is the adjusted association statistically distinguishable from RR = 1 under the fitted analysis and its uncertainty procedure?

E-value

Sensitivity question: How strong would residual unmeasured confounding need to be to move the adjusted association to the null?

A small p-value does not establish exchangeability. A large E-value does not establish correct measurement, overlap, model specification, or temporal ordering.

9. What the E-value does—and does not—address

Use the E-value as one component of a broader sensitivity analysis, not as a universal causal-validity score.

AddressesResidual unmeasured confounding strong enough to move an adjusted RR toward the null.

Does not repairMisclassified smoking, mismeasured CVD, missing outcomes, or other measurement error.

Does not repairPoor treatment overlap, extreme weights, unsupported extrapolation, or model misspecification.

Does not repairReverse causation, incorrect time ordering, selection mechanisms, or interference.

Needs contextCompare the benchmark with scientifically plausible confounders and with measured-covariate associations where possible.

Does not prove causationIt quantifies robustness to one sensitivity model; it does not verify all identifying assumptions.

Export vector graphics and current sensitivity summary