E-Values After Covariate Adjustment

Why Adjustment Changes the E-Value Question

Covariate adjustment aims to account for measured differences between exposed and unexposed groups. An E-value calculated after that adjustment asks a narrower question than one calculated from a crude estimate: how strong would associations involving an unmeasured confounder need to be, beyond the measured covariates already handled by the analysis, to explain away the adjusted association? The answer depends on using an effect estimate that represents the analysis’s intended risk-ratio estimand.

An adjusted coefficient is not automatically a risk ratio. A logistic regression coefficient is on the log-odds scale, for example, and exponentiating it gives an odds ratio. A risk-ratio E-value should use a risk ratio, not an odds ratio inserted as if the two measures were interchangeable. The estimate must also refer to the outcome, follow-up period, population, and adjustment strategy being interpreted.

For example, a conditional estimate might compare risks at the same values of age and baseline severity. A standardized estimate might compare risks averaged over the age and severity distribution of a defined target population. These need not be numerically identical, particularly when effects vary across covariate strata or the standardization targets a different population. Do not choose between them by asking which gives the larger E-value: choose the one that answers the study’s prespecified question.

What the Adjusted E-Value Quantifies

Suppose the exposure \(A\) is binary, the outcome \(Y\) is binary, \(C\) denotes measured covariates, and \(U\) is an unmeasured confounder. For an adjusted risk ratio above 1, let \(a\) describe the association between \(U\) and exposure conditional on \(C\), and let \(b\) describe the association between \(U\) and outcome conditional on both exposure and \(C\). In the standard risk-ratio bias bound, these are maximum risk-ratio associations over the relevant levels of the conditioning variables and, for the outcome association, across exposure levels. The conditioning on exposure in the definition of \(b\) is essential.

Under the standard confounding bias-bound conditions on the risk-ratio scale, the factor by which such confounding can distort the adjusted risk ratio is bounded by \(B=ab/(a+b-1)\). This bound concerns residual confounding under the specified association strengths and conditioning; it does not automatically cover selection bias, measurement error, model misspecification, or other sources of bias.

The E-value is the minimum common strength \(E\) that the two associations would each need to reach, on the risk-ratio scale, to move an observed risk ratio to the null, under this bound. To see the calculation, set \(a=b=E\), so the maximum bound at equal strengths is \(E^2/(2E-1)\). Setting that bound equal to an observed risk ratio \(R>1\) gives \(E^2-2RE+R=0\). The solution at least 1 is:

This is a benchmark for joint confounder strength, not a claim that the two associations must actually be equal. It is often described as the minimum strength 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 association under the bound.

Selecting the Estimate Before Calculating

A sound calculation begins by checking what the reported estimate represents. Use the adjusted risk ratio from the analysis that corresponds to the question of interest. If the fitted model reports a log-risk coefficient, exponentiating that coefficient can yield a risk ratio under the model’s specification. A log-binomial model is one possible approach; a Poisson model with a robust standard error is another commonly used approach for estimating risk ratios with binary outcomes. A standardization procedure can instead produce population-averaged risks and their ratio.

If the analysis reports only an adjusted odds ratio or hazard ratio, do not label it an adjusted risk ratio. The odds ratio may require conversion using baseline risk assumptions, and the hazard ratio is a different measure; use a valid risk-ratio estimate for the intended outcome and time horizon before applying the risk-ratio formula. Similarly, a crude risk ratio is not a substitute for the adjusted estimate when the purpose is to assess residual confounding after measured-covariate adjustment.

For a protective association \(R<1\), orient the comparison so that the ratio is at least 1 by using \(1/R\), then calculate the E-value from that reciprocal. For a confidence interval, use the confidence limit nearest the null when the interval excludes 1. If the interval includes 1, its confidence-limit E-value is 1: the interval already includes the null without invoking additional unmeasured confounding.

Assumptions, Diagnostics, and Failure Modes

The numerical formula is simple, but its interpretation depends on the analysis that produced the adjusted estimate. A causal reading requires, among other things, a well-defined exposure and outcome, consistency, no interference relevant to the estimand, adequate positivity across measured covariate patterns, and control of confounding by the measured covariates for the target comparison. The adjusted estimate must also be adequately estimated: serious model misspecification or sparse data can make it a poor representation of the intended risk ratio. An E-value does not repair violations of these conditions.

Useful diagnostics therefore begin before the E-value calculation. Check that adjustment addresses the prespecified measured confounders, that the exposure groups have adequate overlap across covariate patterns, and that the model or standardization procedure estimates risks appropriately. Review whether the reported ratio is conditional or marginal and whether it targets the population relevant to the scientific question. If estimates change substantially across reasonable model specifications, that instability should be reported rather than concealed by a single E-value.

The sensitivity parameters also need context. The E-value does not say that a single confounder has one association “with the effect”; it concerns two associations: \(U\) with exposure given measured covariates, and \(U\) with outcome given exposure and measured covariates. Both must be considered. A confounder that is strongly related to exposure but weakly related to outcome may have limited capacity to explain the association under the bound, as may the reverse. The E-value also does not measure how common the confounder is.

For estimates below 1, use the reciprocal to orient the association before calculation. For an interval that includes 1, do not report a confidence-limit E-value greater than 1 as though the interval excluded the null. And if the input is an odds ratio, do not use the risk-ratio formula without a valid conversion or a separately justified approximation. These errors can produce precise-looking sensitivity numbers that answer the wrong question.

How This Sensitivity Analysis Fits Among Alternatives

An E-value summarizes the joint strength of exposure–confounder and outcome–confounder associations needed to move an adjusted risk ratio to the null under a risk-ratio bias bound. It is compact and can be useful when detailed information about a particular unmeasured confounder is unavailable. It is not a replacement for careful covariate adjustment, overlap assessment, or subject-matter judgment.

If plausible ranges for a specific unmeasured confounder’s associations are available, a quantitative bias analysis can use those ranges directly and show how the estimate changes across scenarios. Rosenbaum bounds address a different sensitivity framework for hidden bias in matched observational designs. These approaches answer related but distinct questions; numerical values from them should not be treated as interchangeable. The E-value’s advantage is a concise threshold, while its limitation is that a single number suppresses details about confounder prevalence, direction, and the separate strengths of its two associations.