Confidence Intervals That Include the Null

Why the Interval and Point Estimate Can Tell Different Sensitivity Stories

An E-value for a point estimate summarizes how strong unmeasured confounding would need to be, on the risk-ratio scale, to explain away that estimate. An E-value for a confidence interval answers a related but distinct question: how much confounding strength would be needed to move the confidence limit nearest the null to the null. These summaries can differ sharply. In particular, if the confidence interval already includes the null, its E-value is one, even when the point-estimate E-value is appreciably larger than one.

This result can seem surprising if the point estimate is clearly away from the null. The key is that the two E-values refer to different features of the estimate. The point-estimate E-value reflects the estimated association; the interval E-value reflects whether the confidence interval already reaches the null. It is not a second calculation of the point-estimate E-value, and it does not mean the point estimate has no apparent association.

For an interval wholly above 1, its nearest-null limit is the lower limit \(L>1\), and the interval E-value is \(L+\sqrt{L(L-1)}\). For an interval wholly below 1, its nearest-null limit is the upper limit \(U<1\); use \(1/U\) in the formula. When the interval includes 1, neither endpoint is the relevant limit beyond the null: the interval itself already reaches the null.

Why an Interval That Includes the Null Has E-Value One

The rule follows from the definition of the interval E-value. Consider a confidence interval \([L,U]\) for a risk ratio. If \(L\leq 1\leq U\), then the null value is already inside the interval. No additional movement of the interval limit is needed to make the interval include the null. The amount of confounding strength required for that purpose is therefore at its minimum, represented by an E-value of 1.

The same point can be expressed using distance from the null on a multiplicative scale. Define the distance of a risk ratio \(x\) from 1 as \(|\log(x)|\). The nearest point in an interval containing 1 has distance zero, because \(\log(1)=0\). Converting this zero distance back to a risk ratio gives \(r=\exp(0)=1\). Substitution into the E-value formula gives:

This argument explains why it would be incorrect to choose an interval endpoint outside the null and calculate its E-value as if that endpoint were the confidence limit nearest the null. If the interval crosses 1, the null itself is closer than either endpoint. The endpoint-based formula is used only when the entire interval lies on one side of 1.

Protective Estimates and the Reciprocal Rule

For a protective association, the estimated risk ratio is below 1. The point-estimate E-value formula is applied to the reciprocal, which puts the association on the same scale as a risk ratio above 1. For example, \(RR=0.70\) corresponds to \(r=1/0.70\), not to \(r=0.70\). For an interval wholly below 1, the upper confidence limit is nearer the null, so it is the reciprocal of that upper limit that enters the interval calculation. If the interval reaches or crosses 1, however, its interval E-value is 1.

Boundary Cases and a Contrast

The rule includes intervals that touch the null at an endpoint. If a confidence interval is \([1.00,2.05]\), its lower limit is exactly the null; it does not lie wholly above the null. Its interval E-value is 1. The same applies to an interval such as \([0.60,1.00]\). By contrast, if an interval is wholly above 1, even by a small amount, the lower limit is above the null and its E-value is greater than 1.

Interpretation, Diagnostics, and Common Mistakes

An interval E-value of 1 is a statement about the interval’s relation to the null, not a verdict about whether confounding exists. It does not say that the data prove no effect, that the point estimate has E-value 1, or that the null hypothesis is true. A confidence interval including 1 indicates that the interval procedure has not excluded the null value at its stated confidence level. It does not assign a probability to the null or to the parameter.

Before reporting an interval E-value, check the following:

• Confirm the effect scale. The formulas here are for a risk ratio. Do not apply them directly to an odds ratio or hazard ratio as if it were a risk ratio.
• Locate the null. For a risk ratio, the null is 1. Check the full interval, not only the point estimate.
• Choose the correct limit. If the interval excludes 1, use the endpoint nearest 1. If it includes 1, report an interval E-value of 1.
• Keep the two summaries separate. Calculate the point-estimate E-value from the point estimate; calculate the interval E-value from the confidence interval.

A common error is to compute the E-value for the point estimate and label it the E-value for the confidence interval. For the first worked example, that would report 2.58 as the interval E-value, despite a confidence interval that includes 1. Another error is to select whichever endpoint looks closest numerically to 1 without checking whether 1 is inside the interval. The correction in both cases is to apply the definition: an interval that includes the null has interval E-value 1.

This summary complements rather than replaces a confidence interval, a p-value, or a sensitivity analysis such as Rosenbaum bounds. A confidence interval describes sampling uncertainty under its model; a p-value summarizes data compatibility with a null under a specified test; the interval E-value summarizes a particular sensitivity threshold on the risk-ratio scale. None alone measures every source of bias or uncertainty. The result also depends on having a meaningful risk-ratio estimate and an appropriate confidence interval; it does not repair confounding, selection bias, measurement error, or model misspecification.