Converting Odds Ratios Into Individual Risks

Why an Odds Ratio Is Not an Individual Risk

A genetic association is often reported as an odds ratio, while a clinical question asks for an absolute probability: for example, the chance of an outcome over a specified period for someone with a particular genetic score. These quantities are related, but they are not interchangeable. To turn an odds ratio into an absolute risk, one also needs a suitable baseline risk.

The conversion is useful only when the odds ratio and baseline risk refer to the same outcome, time horizon, target population, and comparison. In particular, when an odds ratio is conditional on other predictors, the baseline risk must refer to the corresponding values of those predictors. A population-average incidence is not automatically the right baseline for every individual.

For a binary outcome measured over a fixed period, a logistic regression model writes the conditional probability as

Here \(Y=1\) denotes the outcome, \(G\) is a genetic predictor, and \(Z\) denotes other predictors held fixed in the comparison. Under this model, comparing \(G=g_1\) with \(G=g_0\), at the same \(Z\), gives an odds ratio \(\exp\{\beta(g_1-g_0)\}\). This interpretation relies on the logistic model being appropriate for the stated conditional probabilities and contrast. It does not make the odds ratio a risk ratio.

Deriving the Absolute-Risk Conversion

Let \(p_0\) be the baseline probability and \(p_1\) the probability after the genetic contrast. If the odds ratio for that contrast is \(R\), then the definition of an odds ratio gives \(\{p_1/(1-p_1)\}/\{p_0/(1-p_0)\}=R\). Rearranging this equality yields the conversion.

To see the algebra directly, the odds at baseline are \(p_0/(1-p_0)\). Multiplication by \(R\) gives the comparison odds. Solving for the probability:

The conversion has sensible boundary behavior: if \(R=1\), then \(p_1=p_0\); if \(R>1\), then \(p_1>p_0\) for \(0<p_0<1\); and if \(R<1\), then \(p_1<p_0\). It always returns a probability between zero and one when \(0<p_0<1\) and \(R>0\).

Why Multiplying Risk by the Odds Ratio Can Mislead

A tempting shortcut is \(p_1\approx R p_0\), as if the odds ratio were a risk ratio. The exact formula shows when that approximation is reasonable and how it fails. The risk ratio implied by the conversion is

When \(p_0\) is small, the denominator is close to one, so the risk ratio may be close to \(R\). As baseline risk increases, the distinction can matter. For a harmful contrast with \(R>1\), the denominator exceeds one, so the risk ratio is smaller than the odds ratio. For a protective contrast with \(R<1\), the denominator is less than one, so the risk ratio is closer to one than the odds ratio.

The numerical difference is not a defect in logistic regression. Odds and probabilities use different scales: odds have no upper bound, while probability cannot exceed one. A fixed multiplier on odds therefore does not produce the same fixed multiplier on probability.

Contrasts Across Genetic-Score Values

For a continuous score, the reported odds ratio is often attached to a specified increment, such as one standard deviation. Under a logistic model with a linear score term, the odds ratio for a change of \(\Delta G\) is \(\exp(\beta\Delta G)\). If the odds ratio for a one-unit increase is \(R_{\text{unit}}\), then the odds ratio for a \(\Delta G\)-unit increase is \(R_{\text{unit}}^{\Delta G}\). That combined odds ratio can then be converted using the baseline risk for the reference score and the same other covariate values.

Matching the Baseline to the Odds Ratio

The arithmetic is simple; choosing compatible inputs is often the harder part. Before converting, identify what the baseline probability describes and what the odds ratio compares.

• Match the outcome and horizon. A risk for one outcome or follow-up period cannot be paired uncritically with an odds ratio for another.
• Match the population. Baseline incidence can differ across target populations. A baseline risk from one population may not anchor an odds ratio in another.
• Match the conditioning. If the odds ratio is conditional on age and other predictors, use a baseline risk at the corresponding predictor values. Pairing a conditional odds ratio with a marginal, population-average risk may not yield the risk for a particular profile.
• Use the right source for absolute risk. An odds ratio by itself does not supply the baseline probability. In a case-control sample, the sampled proportion with the outcome generally does not estimate population incidence, so that proportion alone cannot serve as the baseline risk.

These requirements matter even when all quantities are described as belonging to the same disease. A broad population incidence averages across people with different predictors; a model-based conditional odds ratio describes a comparison at specified predictor values. Those are not automatically compatible inputs. The conversion is mathematically exact once the inputs match, but it cannot repair a mismatched baseline or an unsuitable odds ratio.

Common Pitfall

A related mistake is applying a reported odds ratio to a baseline risk for a different reference group or population and interpreting the result as an individual’s personalized probability. The calculation may be numerically correct but its interpretation is not supported unless the baseline, comparison, and model conditions align.