Two Sensitivity Questions, Not One Scale
Matching can make treated and untreated participants similar on measured covariates, but it cannot guarantee that they are also comparable on unmeasured factors. Rosenbaum Gamma bounds and E-values both examine sensitivity to such residual differences. They do not, however, quantify the same thing. Gamma limits how much an unmeasured factor may alter treatment-assignment odds within matched sets and is used to bound inference under a specified test. An E-value summarizes how strongly an unmeasured confounder would need to be associated with treatment and outcome, on a risk-ratio scale, to explain away an estimated association or confidence limit.
The distinction matters in matched studies. A small worst-case p-value at a stated Gamma says that the chosen test remains significant under a particular assignment-odds bound. A large E-value says that confounding associations of a certain strength would be needed, under its bounding framework, to move a risk-ratio estimate to the null. Neither statement establishes that confounding is absent, and neither metric can be converted to the other by comparing their numerical values.
What Gamma Bounds in a Matched-Pair Test
For a matched pair, let \(p\) be the conditional probability that one specified member receives treatment, given that exactly one member is treated. The treatment-assignment odds are \(p/(1-p)\). If the two members' assignment odds can differ by no more than a factor \(\Gamma\), the probability for either member is bounded by
At \(\Gamma=1\), treatment assignment is equally likely to either member of the pair under this model. Larger values allow greater imbalance in assignment odds. This interpretation concerns treatment assignment within matched sets; Gamma is not a bound on an outcome risk ratio or on the association between a confounder and the outcome.
A simple illustration uses discordant matched pairs with a binary outcome. Under the sharp null that treatment has no effect on any individual, the outcome values within a discordant pair are fixed, and the treatment assignment determines which member's outcome is attributed to treatment. For a prespecified one-sided direction, the probability of a treatment-favorable sign in any discordant pair is at most \(q=\Gamma/(1+\Gamma)\). If there are \(m\) independent discordant pairs and \(S\) favorable signs, the worst-case upper-tail probability is bounded by the corresponding binomial tail with success probability \(q\). This result is specific to this paired sign-test setup; other matched designs and test statistics require their appropriate sensitivity calculations.
Worked Example: A Worst-Case Matched-Pair P-Value
Suppose a hypothetical matched study has 10 discordant pairs, of which 9 have an outcome difference in the prespecified treatment-favorable direction. Assume the pairs are independent, the sharp null is the target of the test, and there are no ties among these discordant pairs. Calculate the worst-case one-sided p-value at \(\Gamma=1.5\).
The maximum probability of a favorable sign under this sensitivity model is \(q=\Gamma/(1+\Gamma)=1.5/2.5=0.60\). Thus the worst-case tail probability is bounded by \(P(X\geq 9)\) for \(X\sim\text{Binomial}(10,0.60)\):
The worst-case one-sided p-value is about 0.0464. At a prespecified \(\alpha=0.05\), this test remains significant at \(\Gamma=1.5\), subject to the stated model and test assumptions. At \(\Gamma=2\), \(q=2/3\), giving \(P(X\geq9)=10(2/3)^9(1/3)+(2/3)^{10}=6144/59049\approx0.104\). The same test would not be significant at the 0.05 level under that larger bound. This comparison describes robustness of this test result, not the size of a treatment effect.
What an E-Value Summarizes
For a risk ratio \(RR\geq1\), the E-value is \(RR+\sqrt{RR(RR-1)}\). For a protective estimate below one, first take its reciprocal and apply the formula. An E-value of \(E\) has a bounding interpretation: to explain away the estimate, an unmeasured confounder would need associations with treatment and outcome that are each at least \(E\) on the risk-ratio scale, conditional on measured covariates, in the equal-strength formulation. Other combinations of the two associations can also suffice; the E-value is not a claim that both associations actually have a particular value.
For an interval estimate, the corresponding E-value is calculated from the confidence limit closest to the null, provided that limit is beyond the null. It summarizes the confounding strength that could move that limit to the null. If the interval already includes the null, this confidence-limit E-value is 1, so it offers no additional robustness threshold beyond the interval itself.
Worked Example: E-Values for an Estimate and Its Confidence Limit
A hypothetical matched cohort analysis reports a risk ratio of 1.80 with a 95% confidence interval from 1.20 to 2.70. Calculate the E-value for the point estimate and for the confidence limit closest to the null. Assume the reported quantities are risk ratios and that the interval is on the same scale.
For the point estimate:
For the lower confidence limit, which is the limit closer to 1:
Under the E-value framework, associations of at least 3.00-fold each, in the equal-strength formulation, would be required to explain away the point estimate, conditional on measured covariates. Associations of about 1.69-fold each could move the lower confidence limit to the null. These are sensitivity summaries, not probabilities that the result is biased or causal.
Worked Example: Interpreting a Protective Risk Ratio
In a separate hypothetical matched study, the estimated risk ratio is 0.70. Find its E-value and state what it addresses. The estimate is below the null value of 1, so first reverse its direction: \(1/0.70=1.428571\).
The E-value is about 2.21. In the equal-strength interpretation, unmeasured confounding associations of about 2.21-fold each with treatment and outcome could be needed to move this point estimate to the null, conditional on measured covariates. The calculation does not say that such a confounder exists, and it does not assess whether treatment assignment within pairs was sensitive to unmeasured factors in the way Gamma does.
Comparing the Inferential Targets
| Feature | Rosenbaum Gamma bounds | E-values |
|---|---|---|
| Sensitivity parameter | Maximum ratio of treatment-assignment odds within a matched set | Strength of confounder associations with treatment and outcome on a risk-ratio scale |
| Primary inputs | Matched-set design, outcomes, a test statistic, and a chosen Gamma | Risk-ratio estimate or a confidence limit beyond the null |
| Typical target | Worst-case p-value under an assignment model and a null hypothesis; confidence sets may be obtained by test inversion | Confounding strength that could explain away an estimate or move a confidence limit to the null |
| Useful question | How large could assignment-odds differences be before this test loses its conclusion? | How strong would confounding associations need to be to account for this risk-ratio result? |
The two procedures can complement one another when the design and effect measure support both, but they answer different questions. In particular, a Gamma of 1.5 is not “weaker” or “stronger” than an E-value of 1.5. Gamma describes a treatment-assignment odds ratio constraint; an E-value describes a joint confounding-strength threshold on a risk-ratio scale. Their numerical scales have no direct conversion.
Assumptions, Diagnostics, and Failure Modes
A Gamma analysis relies on the matched design being correctly represented, on the chosen assignment model being a defensible sensitivity model, and on an appropriate test statistic and null hypothesis. In the matched-pair sign-test example, independence across pairs and the sharp null are essential to the stated binomial-tail calculation. For more complex sets or outcomes, the relevant assignment probabilities and test distribution must match that design and statistic. Check the number and sizes of matched sets, missing outcomes, ties or discarded pairs, the test direction, and whether the analysis actually computes a worst-case bound over allowed assignments. A single paired formula should not be applied mechanically to a different matched design.
An E-value relies on a risk-ratio interpretation and the confounding bound that underlies its formula. Its usual interpretation is conditional on measured covariates and addresses unmeasured confounding; it does not automatically account for selection bias, measurement error, model misspecification, or other sources of bias. Before calculating it, verify that the estimate and relevant confidence limit are on a risk-ratio scale and identify the limit closest to the null. If an estimate is protective, reverse its direction before applying the formula. An E-value is a compact summary, not a substitute for assessing plausible confounders in the study context.
A common mistake is to read “significant up to \(\Gamma=1.5\)” and “E-value 1.5” as equivalent evidence. The first is a claim about a particular test's worst-case p-value under an assignment-odds constraint. The second is a threshold involving two confounder-outcome and confounder-treatment associations on the risk-ratio scale. The correction is to report each measure with its target and assumptions, then judge whether those assumptions and scales are relevant to the scientific question.
Use Gamma bounds when the target is robustness of matched-study inference under constrained treatment-assignment odds. Use an E-value when the target is a confounding-strength summary for a risk-ratio estimate or confidence limit.
For Gamma, confirm the matched-set structure, outcome, test, and direction. For an E-value, confirm the risk-ratio scale and whether the point estimate or nearest-null confidence limit is being summarized.
Give the sensitivity parameter, the resulting bound or threshold, and the assumptions that give it meaning. Do not compare the raw numbers across metrics.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a matched-pair analysis, what does \(\Gamma=1\) mean for treatment-assignment odds?
- For 10 discordant pairs with 9 favorable signs, what is the worst-case one-sided tail probability at \(\Gamma=1.5\), and what is the conclusion at \(\alpha=0.05\)?
- What is the E-value for a risk ratio of 2.00?
- Why can a Gamma of 2 not be described as equivalent to an E-value of 2?
- If a risk-ratio confidence interval includes 1, what does its confidence-limit E-value indicate?
Solutions. (1) The two members of each pair have equal treatment-assignment odds under the model. (2) \(q=1.5/2.5=0.60\), so \(P(X\geq9)=10(0.60)^9(0.40)+(0.60)^{10}=0.04636\). It is below 0.05, so this one-sided test remains significant under the stated assumptions at this Gamma. (3) \(2+\sqrt{2(1)}=3.414\). (4) Gamma bounds treatment-assignment odds within matched sets, while the E-value summarizes confounder associations on a risk-ratio scale; they have different meanings and targets. (5) The nearest-null limit is at or across the null, so the confidence-limit E-value is 1; this summary does not establish robustness of the interval to unmeasured confounding.