Risk-Difference Estimates and E-Values

Why a Risk Difference Needs a Baseline Risk

A risk difference describes an absolute contrast: the difference between two risks. An E-value, by contrast, is calculated from a risk ratio. A risk difference alone does not determine a risk ratio. To make the conversion, one must also specify the risk in the comparator group, and that baseline risk must refer to the same outcome, population, follow-up period, and estimand as the risk difference.

This matters because a fixed absolute change can represent a modest relative change when the baseline risk is high, or a much larger relative change when the baseline risk is low. The resulting E-values can therefore differ even though the risk difference is unchanged. Treating a risk difference as though it were already a risk ratio, or choosing a baseline risk from a different population, can produce a misleading sensitivity summary.

For a harmful association, \(d>0\) and \(RR>1\), so \(q=RR\). For a protective association, \(d<0\) and \(0<RR<1\), so \(q=1/RR\). The reciprocal is used to express the strength of the association in the direction away from the null; it does not change the original risk contrast. The formula presumes a finite, positive risk ratio. The zero-risk boundaries require separate attention.

Calculation and Interpretation

The calculation has two distinct stages. First, use the specified baseline risk to recover the two risks and their ratio. Then apply the risk-ratio E-value formula. The E-value is on the risk-ratio scale: under the standard E-value interpretation, it is the minimum strength of association, on the risk-ratio scale, that an unmeasured confounder would need to have with both exposure and outcome, conditional on the measured covariates, to explain away the observed association, under the relevant assumptions. It is not a probability that confounding exists, nor a guarantee that bias is absent.

Why the Baseline Risk Changes the Result

The dependence on baseline risk follows directly from the conversion. For a fixed positive difference \(d\), \(RR=1+d/b\). Increasing \(b\) makes \(d/b\) smaller, so the risk ratio moves toward 1. For a fixed negative difference with feasible positive risks, \(RR=1+d/b\) also moves toward 1 as \(b\) increases; its reciprocal \(q\) consequently moves toward 1. The E-value increases with \(q\) for \(q\geq1\), so in either direction the E-value decreases as the relative contrast approaches the null.

This is not a separate property of the risk-difference scale: it is the consequence of converting an absolute contrast into a relative one. The conversion and E-value calculation can be shown in sequence:

Here \(p_1\) denotes the exposed-group risk. The formula also gives a short justification for why the baseline risk cannot be omitted: with \(d\) fixed, changing \(b\) changes \(d/b\), hence \(RR\), \(q\), and \(E\). For example, the two harmful-effect examples above have the same \(d\), but their risk ratios differ because their values of \(b\) differ.

Assumptions, Diagnostics, and Boundary Cases

The arithmetic requires a coherent pair of risks, not merely a numerical baseline chosen for convenience. The baseline must be the comparator risk for the same target contrast. If a risk difference is adjusted or standardized, a crude baseline risk may not be compatible with it. A mismatch can make the resulting ratio a calculation between unlike quantities rather than a meaningful relative effect for the stated estimand.

The risks must also be feasible: \(0\leq b\leq1\) and \(0\leq b+d\leq1\). For a finite risk ratio, additionally require \(b>0\). For a finite reciprocal-based E-value when \(RR<1\), also require \(b+d>0\). If \(b+d=0\) while \(b>0\), then \(RR=0\) and \(1/RR\) is undefined. The usual reciprocal-based formula therefore gives no finite E-value; under a limiting convention as \(RR\) approaches zero from above, the E-value tends to infinity. If \(b=0\), the risk ratio is undefined; if both risks are zero, there is no relative risk contrast to calculate. These cases should not be silently repaired by inserting an arbitrary small number.

For a positive difference, \(b+d\leq1\) is also essential. For example, \(b=0.95\) and \(d=0.10\) imply an exposed risk of 1.05, which is impossible. A computed E-value from that pair would not be a valid sensitivity summary, even if the algebra could be carried out.

Useful diagnostics are simple but important: verify the signs and units of the risk difference, reconstruct both risks, check probability bounds, confirm the baseline’s target population and estimand, and independently verify whether the risk ratio is above or below 1 before orienting it. If the E-value is unexpectedly large or small, inspect the baseline-risk choice before interpreting the result. Rounding the risks too early can also move a near-zero risk to zero, creating an artificial boundary problem.

What This Summary Does and Does Not Compare

A risk difference is often useful because it expresses an absolute change in percentage points, while a risk ratio describes a relative change. The E-value is not a replacement for either measure: it summarizes sensitivity to unmeasured confounding after a risk-ratio-scale association has been specified. Reporting the risk difference, baseline risk, implied risk ratio, and E-value together makes the scale conversion transparent.

An alternative is to analyze and report the risk ratio directly when that is the target estimand. In that case, the risk-difference conversion is unnecessary. Another option is to report sensitivity results for a range of plausible baseline risks when no single baseline is clearly justified. This is more informative than presenting an arbitrary single value, but it does not remove uncertainty about which risk corresponds to the target population. A risk-difference sensitivity analysis and a risk-ratio sensitivity analysis are not automatically interchangeable: the former needs a baseline-risk specification to produce the latter’s E-value.

The E-value also does not establish that the observed association is causal, quantify the probability of residual confounding, or show that a confounder with the required strength exists. Its standard interpretation depends on the causal estimate and assumptions relevant to the analysis, including the treatment of measured covariates and the use of a risk ratio for the target association. It is a benchmark for sensitivity, not a test that validates the study design.

Common Mistakes

• Calculating an E-value directly from a risk difference. A difference of 0.08 is not a risk ratio of 1.08. Specify \(b\), calculate \(b+d\), and divide the two risks first.
• Using an unmatched baseline risk. A baseline from a different population or estimand can yield a mathematically valid but scientifically incoherent conversion. Check the target of both quantities.
• Forgetting to orient a protective ratio. When \(0<RR<1\), use \(1/RR\) in the E-value formula, while still describing the original association as protective.
• Applying the reciprocal at \(RR=0\). Division by zero is undefined. State that the finite reciprocal-based calculation does not apply; if using the limiting convention, make that explicit.
• Interpreting a larger E-value as proof of no bias. It indicates a stronger confounder association would be needed under the stated framework to explain away the estimate; it does not prove that such confounding is absent.