Rare Outcomes and the Odds-Risk Approximation

Why Compare Odds Ratios With Risk Ratios?

An odds ratio is the natural effect measure in binary logistic regression, but risks are often easier to interpret. When an outcome is rare, the two measures can be numerically close. That approximation is useful, but it is not an identity: its accuracy depends on both the baseline risk and the size and direction of the odds ratio.

We will compare two risks for the same population or covariate profile. Let \(p_0\) be the risk in a reference group and \(p_1\) the risk in a comparison group. Assume \(0<p_0<1\), \(0<p_1<1\), and define the odds ratio \(\mathrm{OR}\) as the comparison-group odds divided by the reference-group odds. The risk ratio is \(\mathrm{RR}=p_1/p_0\). These are descriptive contrasts; a causal interpretation requires additional assumptions about the study and comparison.

The Exact Relationship

Starting with an odds ratio and a baseline risk, we can recover the comparison risk exactly. This conversion also gives the exact risk ratio, so there is no need to treat the rare-outcome approximation as a guess.

Proof. By the definition of the odds ratio,

Multiplying by \(p_0(1-p_1)\) gives \(\mathrm{OR}\,p_0(1-p_1)=p_1(1-p_0)\). Collect the terms involving \(p_1\):

Since \(1-p_0+\mathrm{OR}\,p_0>0\), division gives the stated expression for \(p_1\). Dividing that expression by \(p_0>0\) gives the formula for \(\mathrm{RR}\). \(\square\)

The denominator \(1+(\mathrm{OR}-1)p_0\) is positive: it is also \(1-p_0+\mathrm{OR}\,p_0\), a sum of two positive terms. The exact formula makes the main point visible. The odds ratio equals the risk ratio when \(\mathrm{OR}=1\), but otherwise the difference depends on the baseline risk as well as the odds ratio.

Quantifying the Approximation Error

To say precisely how close the measures are, compare the risk ratio with the odds ratio relative to the odds ratio. The exact formula implies

For \(\mathrm{OR}>1\), the denominator exceeds 1, so \(\mathrm{RR}<\mathrm{OR}\): the odds ratio overstates the risk ratio. For \(0<\mathrm{OR}<1\), the denominator is less than 1 but remains positive, so \(\mathrm{RR}>\mathrm{OR}\): the odds ratio is numerically smaller than the risk ratio. The relative discrepancy, measured against the odds ratio, has an exact expression.

Proof. Subtract \(\mathrm{OR}\) from the exact risk-ratio formula and combine terms:

Take absolute values and divide by \(\mathrm{OR}>0\). The denominator is positive, as established above, so the result is the stated expression. For fixed \(\mathrm{OR}>0\), the numerator tends to zero as \(p_0\) tends to zero, while the denominator tends to 1. Hence the relative error tends to zero. \(\square\)

This result gives a practical accuracy check. For \(\mathrm{OR}>1\), the relative error is \[ \frac{(\mathrm{OR}-1)p_0}{1+(\mathrm{OR}-1)p_0}. \] For example, to keep this error at or below a chosen tolerance \(\varepsilon\), where \(0<\varepsilon<1\), it is necessary and sufficient that \((\mathrm{OR}-1)p_0\leq \varepsilon/(1-\varepsilon)\). Thus “rare” should be judged in relation to the effect size, not by a baseline-risk cutoff alone.

Effect Direction and Baseline Risk

The same dependence on baseline risk holds for protective associations, but the direction of the numerical discrepancy reverses. The odds ratio is then below the risk ratio. The next example checks the conversion directly.

What “Rare” Does—and Does Not—Guarantee

For a fixed odds ratio, making the baseline risk small makes the risk ratio approach the odds ratio. But a small baseline risk does not by itself guarantee a small discrepancy if the odds ratio is very large. The relevant quantity for an odds ratio above 1 is \((\mathrm{OR}-1)p_0\). For instance, if \(p_0=0.01\) and \(\mathrm{OR}=20\), then

which is noticeably below 20 despite a 1% reference risk. The comparison risk in this case is approximately \(0.1681\), so the outcome is not rare in the comparison group. The exact conversion avoids relying on a verbal label such as “rare” when the baseline risk or effect size makes the approximation questionable.

In a logistic regression, odds ratios may be interpreted for a particular covariate profile or contrast. For such a contrast, the relevant \(p_0\) is the reference risk at that profile; a single odds ratio need not correspond to one universal risk ratio across profiles with different baseline risks. The algebra here applies to each pair of risks individually. It does not turn an association into a causal effect, nor does it justify treating an odds ratio as a risk ratio when the baseline risk is unknown.

Common Mistake

A common mistake is to say, “The outcome is rare, so an odds ratio of 2 means the risk is doubled,” and then report \(\mathrm{RR}=2\) without checking the baseline risk. But if \(p_0=0.20\) and \(\mathrm{OR}=2\), the exact conversion gives

The correction is to distinguish odds from risks, identify the baseline risk for the contrast, and use \(\mathrm{RR}=\mathrm{OR}/[1+(\mathrm{OR}-1)p_0]\) when the risk ratio is needed. If the baseline risk is unavailable, an exact risk ratio cannot be recovered from the odds ratio alone.