Converting Odds Ratios Before Computing E-Values

Why an Odds Ratio Needs a Baseline Risk

An odds ratio and a risk ratio describe different comparisons. When an outcome is uncommon in both exposure groups, their numerical values may be close. When the outcome is common, the odds ratio can be substantially farther from 1 than the risk ratio. That difference matters when an E-value is to be calculated: the standard risk-ratio E-value formula takes a risk ratio as its input, not an odds ratio.

Converting an odds ratio therefore requires more information than the odds ratio itself. In particular, we need the outcome risk in the unexposed group, measured for the same outcome, follow-up period, and target population as the comparison. A calculation using a mismatched baseline risk can produce a precise-looking result on the wrong scale.

Definition: Let \(p_0\) be the risk of a binary outcome among unexposed individuals and \(p_1\) its risk among exposed individuals, over a specified period. The risk ratio is \(RR=p_1/p_0\). The odds ratio is \(\{p_1/(1-p_1)\}/\{p_0/(1-p_0)\}\). When \(p_0\) is known, an odds ratio can be converted to a risk ratio for that baseline risk.

The Conversion and Its Derivation

Write the odds ratio as \(\theta\). Starting from its definition, solve for the exposed-group risk \(p_1\). This derivation also shows why the conversion depends on \(p_0\).

\theta = \frac{p_1/(1-p_1)}{p_0/(1-p_0)} \quad\Longrightarrow\quad \theta p_0(1-p_1)=p_1(1-p_0)

Collecting the terms involving \(p_1\) gives \(\theta p_0=p_1(1-p_0+\theta p_0)\). Thus the implied exposed risk and the risk ratio are

p_1=\frac{\theta p_0}{1-p_0+\theta p_0}, \qquad RR=\frac{p_1}{p_0} =\frac{\theta}{1-p_0+\theta p_0} =\frac{\theta}{1+p_0(\theta-1)}.

For \(0<p_0<1\), the denominator is positive. If \(\theta>1\), then \(1+p_0(\theta-1)>1\), so the converted \(RR\) is less than \(\theta\) but greater than 1. If \(0<\theta<1\), the converted \(RR\) is greater than \(\theta\) but less than 1. In either direction, the conversion moves the comparison toward the null value 1. At \(p_0\) close to zero, the denominator is close to 1, which explains the rare-outcome approximation \(RR\approx\theta\). The approximation is not a substitute for the conversion when the baseline risk is appreciable.

Assumptions for this conversion: The odds ratio and \(p_0\) must refer to the same binary outcome, follow-up period, and target population. The comparison must use the same exposure groups, and \(p_0\) must be a valid risk between 0 and 1. The algebra converts a specified odds ratio into a risk ratio; it does not establish that either association is causal.

From the Converted Risk Ratio to an E-Value

For a risk ratio \(R\geq 1\), the E-value calculation used here is \(R+\sqrt{R(R-1)}\). For a protective association, first express its magnitude as the reciprocal risk ratio, so the input is again at least 1. A risk ratio exactly equal to 1 has E-value 1. The key point for this tutorial is that the input must be the risk ratio obtained on the appropriate scale, not an odds ratio treated as though it were automatically a risk ratio.

Worked Example: Converting a Harmful Odds Ratio

Suppose a hypothetical analysis reports an odds ratio of \(2.5\) for an outcome, and the unexposed risk for the same population and follow-up period is \(p_0=0.20\). Convert the odds ratio, then calculate the E-value.

The conversion denominator is \(1-p_0+\theta p_0=0.80+(2.5)(0.20)=1.30\). Therefore,

p_1=\frac{(2.5)(0.20)}{1.30} =\frac{0.50}{1.30} =\frac{5}{13} =0.3846153846\ldots

The implied risk ratio is \(p_1/p_0\). Using the unrounded \(p_1\), the result is

RR=\frac{5/13}{1/5} =\frac{25}{13} =1.923076923\ldots

Now use \(R=25/13\) in the E-value formula:

E = \frac{25}{13} + \sqrt{\frac{25}{13}\left(\frac{25}{13}-1\right)} = \frac{25}{13}+\frac{\sqrt{300}}{13} = 3.2554.

Thus, based on the converted risk ratio, the E-value is approximately 3.255. The calculation uses the same hypothetical population and baseline risk at every step; rounding \(p_1\) before dividing would slightly change the displayed arithmetic.

Worked Example: A Common Outcome Makes the Difference Visible

In a second hypothetical setting, let the odds ratio be \(3.0\) and the unexposed risk be \(p_0=0.40\). This baseline risk is not rare, so the approximation of treating the odds ratio as a risk ratio is especially questionable.

The exposed risk is

p_1 = \frac{(3.0)(0.40)}{1-0.40+(3.0)(0.40)} = \frac{1.20}{1.80} = 0.6667.

Consequently, using the rounded exposed risk, \(0.6667/0.40=1.66675\) (about 1.6668). Using the unrounded risk \(2/3\), \(RR=(2/3)/0.40=5/3\approx1.6667\). The corresponding E-value is

E = \frac{5}{3} + \sqrt{\frac{5}{3}\left(\frac{5}{3}-1\right)} = \frac{5}{3}+\frac{\sqrt{10}}{3} = 2.7208.

If the odds ratio were incorrectly used as the risk ratio, the calculation would instead be \(3+\sqrt{3(3-1)}=5.449\). That is a different calculation based on the wrong input scale. Here the conversion changes the risk-ratio input from 3.0 to 1.6667 and produces an E-value of 2.7208.

Worked Example: Converting a Protective Odds Ratio

Suppose a hypothetical study reports an odds ratio of \(0.50\), with an unexposed risk of \(p_0=0.20\). Convert it first, then use the reciprocal of the protective risk ratio in the E-value formula.

The implied exposed risk is

p_1 = \frac{(0.50)(0.20)}{1-0.20+(0.50)(0.20)} = \frac{0.10}{0.90} = \frac{1}{9} = 0.1111.

Thus \(RR=(1/9)/(1/5)=5/9=0.5556\). The reciprocal magnitude is \(1/RR=9/5=1.8\), so

E = 1.8+\sqrt{1.8(1.8-1)} = 1.8+\sqrt{1.44} = 3.000.

The protective estimate must be put on the risk-ratio scale before taking its reciprocal. Using the reciprocal of the odds ratio instead would skip the baseline-risk conversion and need not give the same E-value.

When the Rare-Outcome Approximation Misleads

The approximation \(RR\approx\theta\) follows directly from the conversion denominator \(1+p_0(\theta-1)\): when \(p_0\) is small, this denominator is close to 1. But “small” must be judged in relation to the odds ratio and the precision needed for the analysis. The approximation can be misleading if it is carried into an E-value calculation without checking how much the risk ratio changes.

Worked Example: Checking a Rare-Outcome Approximation

Return to a hypothetical odds ratio of \(2.5\), but now let the unexposed risk be \(p_0=0.005\), or 0.5%. The exact conversion gives

RR = \frac{2.5}{1+0.005(2.5-1)} = \frac{2.5}{1.0075} = 2.481390.

The exact E-value is approximately \(2.481390+\sqrt{2.481390(1.481390)}=4.399\). If the rare-outcome approximation is used, the odds ratio is substituted directly as \(R=2.5\), giving \(2.5+\sqrt{2.5(1.5)}=4.436\). Here the approximation is close, but it is not exact. With the same odds ratio and a 20% baseline risk, the converted risk ratio is 1.923077 and its E-value is 3.255, as calculated earlier. The comparison illustrates why the baseline risk should be checked rather than assumed negligible.

Diagnostics, Scope, and Common Pitfalls

Before reporting a converted risk ratio or E-value, check the information and scale used in the calculation. A defensible calculation should make it possible to identify the odds ratio, the baseline risk, and the population and time period to which both apply. The following checks are particularly important:

Confirm the outcome definition and period. A baseline risk over one year is not interchangeable with a risk over five years.
Check the baseline population. The unexposed risk should correspond to the target population and comparison underlying the reported odds ratio, not merely to a convenient external rate.
Check the outcome scale. The calculation above is for an odds ratio defined from risks for a binary outcome. Do not silently treat a hazard ratio or another measure as an odds ratio.
Keep precision through intermediate steps. Calculate from the stated \(p_0\) and \(\theta\), then round the reported result. Rounded intermediate risks can make a displayed division appear inconsistent.
Keep association separate from causation. The conversion is algebra. It does not remove confounding, validate an outcome model, or make an observational association causal.

An odds ratio from a covariate-adjusted model may be conditional on covariates, while a reported baseline risk may be marginal over a population. Combining those quantities without a justified link can yield a converted number that does not represent either the model’s conditional contrast or the population’s marginal contrast. The conversion formula is mathematically valid for a particular odds ratio and baseline risk, but choosing compatible inputs is a substantive modeling decision.

A common mistake is to reason, “The outcome is binary, so the odds ratio is the risk ratio,” or “The outcome seems uncommon, so I can always use the odds ratio in the E-value formula.” The correction is to compute \(RR=\theta/\{1+p_0(\theta-1)\}\) using an appropriate \(p_0\), and then calculate the E-value from that risk ratio. If \(p_0\) is unavailable or is not compatible with the odds ratio, the conversion is not determined by the odds ratio alone; report that limitation rather than implying an exact risk-ratio E-value.

Key takeaway: An odds ratio does not uniquely determine a risk ratio. Use the unexposed baseline risk to convert it first, then calculate the E-value from the converted risk ratio (or its reciprocal for a protective association). The rare-outcome approximation is a checkable approximation, not a general identity.

Check Your Understanding

Try each calculation, then compare it with the solutions below.

A hypothetical odds ratio is \(2.0\), and the unexposed risk is \(0.10\). Find the converted risk ratio and E-value.
A hypothetical odds ratio is \(1.5\), and the unexposed risk is \(0.40\). Find the converted risk ratio and E-value. Is it appropriate to substitute the odds ratio directly as the risk ratio?
A hypothetical odds ratio is \(0.40\), and the unexposed risk is \(0.25\). Find the converted risk ratio and E-value.
Explain why an odds ratio alone is not enough to determine a risk ratio, and name one important compatibility check for the baseline risk.

Solutions. (1) \(RR=2/\{1+0.10(2-1)\}=2/1.10=1.8182\). Thus \(E=1.8182+\sqrt{1.8182(0.8182)}\approx3.038\). (2) \(RR=1.5/\{1+0.40(1.5-1)\}=1.5/1.20=1.25\). Hence \(E=1.25+\sqrt{1.25(0.25)}=1.809\). Direct substitution of 1.5 would not be exact, and the 40% baseline risk makes the approximation questionable. (3) \(RR=0.40/\{1+0.25(0.40-1)\}=0.40/0.85=0.4706\). Its reciprocal is \(2.125\), so \(E=2.125+\sqrt{2.125(1.125)}\approx3.671\). (4) The conversion denominator contains \(p_0\), so different baseline risks produce different risk ratios for the same odds ratio. The baseline risk must refer to the same outcome, period, exposure comparison, and target population as the odds ratio.