Recovering Risks From Odds Ratios

Why Recover an Absolute Risk?

An odds ratio describes a comparison of odds, not a probability. Yet a genetic result is often easier to interpret when expressed as the probability of the phenotype in each genotype group. To recover that probability for a comparison genotype, we need both its odds ratio relative to a reference genotype and the reference genotype’s risk.

The conversion is especially important when the phenotype is common enough that odds and risks differ appreciably. For a rare phenotype, an odds ratio may be close to a risk ratio, but it is not generally equal to one. Multiplying a baseline risk by an odds ratio therefore does not, in general, give the correct genotype-specific risk.

Reference Risk, Comparison Risk, and Odds Ratio

Definition: Let \(Y=1\) indicate that a person has a specified binary phenotype during a specified time window. For a reference genotype \(g_0\), its risk is \(p_0=P(Y=1\mid G=g_0)\). For a comparison genotype \(g_1\), its risk is \(p_1=P(Y=1\mid G=g_1)\). The odds of risk \(p\) are \(p/(1-p)\). The odds ratio comparing \(g_1\) with \(g_0\) is \(\theta=[p_1/(1-p_1)]/[p_0/(1-p_0)]\).

The genotype labels and comparison direction matter: \(\theta\) here compares the odds for \(g_1\) with the odds for \(g_0\). Reversing the comparison gives the reciprocal odds ratio. The risks also refer to the same defined phenotype and time window. For this conversion, assume \(0<p_0<1\) and a finite, positive \(\theta\); these conditions make the reference odds and the stated odds ratio well-defined.

In a genetic-effects model, the reference genotype establishes the baseline, while the odds ratio describes the comparison genotype’s odds relative to it. The conversion recovers the comparison group’s probability from those two pieces of information.

Deriving the Conversion

Begin with the odds-ratio definition and solve for \(p_1\). The algebra is short, but each step shows why the baseline risk cannot be omitted.

Write the odds ratio.
The comparison genotype’s odds equal \(\theta\) times the reference genotype’s odds.

Substitute the risk expressions.
Express each genotype’s odds using its risk and its probability of not having the phenotype.

Isolate the comparison risk.
Collect the terms involving \(p_1\) on one side and divide by their coefficient.

\begin{aligned} \theta &=\frac{p_1/(1-p_1)}{p_0/(1-p_0)}\\ \theta p_0(1-p_1) &=p_1(1-p_0)\\ \theta p_0 &=p_1(1-p_0+\theta p_0)\\ p_1 &=\frac{\theta p_0}{1-p_0+\theta p_0}. \end{aligned}

Key result: Given the reference risk \(p_0\) and the odds ratio \(\theta\) for \(g_1\) versus \(g_0\), the comparison genotype’s risk is \(p_1=\theta p_0/(1-p_0+\theta p_0)\).

The denominator includes both the probability of not having the phenotype in the reference group, \(1-p_0\), and the odds-adjusted baseline term, \(\theta p_0\). It follows that changing \(p_0\) can change the resulting \(p_1\), even when the odds ratio is held fixed.

This formula also checks out in important special cases. When \(\theta=1\), the comparison and reference risks are equal. When \(\theta\) is greater than 1, \(p_1\) is greater than \(p_0\); when \(\theta\) is between 0 and 1, \(p_1\) is less than \(p_0\). For valid inputs, the formula produces a risk between 0 and 1.

Worked Risk Conversions

Worked Example: A Higher Odds Ratio With a Low Baseline Risk

Hypothetical data. Suppose the risk of a specified phenotype by age 50 is 4% for reference genotype \(g_0\), so \(p_0=0.04\). Suppose the odds ratio for comparison genotype \(g_1\) versus \(g_0\) is \(\theta=2.5\). Find the risk for \(g_1\).

The inputs refer to the same phenotype and age window, and the odds ratio is stated in the direction \(g_1\) versus \(g_0\). Apply the conversion:

p_1=\frac{\theta p_0}{1-p_0+\theta p_0} =\frac{2.5(0.04)}{1-0.04+2.5(0.04)} =\frac{0.10}{0.96+0.10} =\frac{0.10}{1.06} \approx 0.09434.

The estimated risk for genotype \(g_1\) is about 9.43% by age 50. It is greater than the 4% reference risk, consistent with an odds ratio above 1. The answer is not 10%: that would result from multiplying the baseline risk directly by the odds ratio, which treats an odds ratio as though it multiplied risks.

Worked Example: An Odds Ratio Below 1

Hypothetical data. For a specified binary phenotype during a one-year period, suppose reference genotype \(g_0\) has risk \(p_0=0.25\), and comparison genotype \(g_1\) has an odds ratio \(\theta=0.60\) relative to \(g_0\). Calculate the risk for \(g_1\).

The reference risk is strictly between 0 and 1, and the comparison direction is specified. Substitute:

p_1=\frac{0.60(0.25)}{1-0.25+0.60(0.25)} =\frac{0.15}{0.75+0.15} =\frac{0.15}{0.90} \approx 0.1667.

The comparison genotype’s risk is approximately 16.7% over the one-year period. This is below the reference risk of 25%, as expected for an odds ratio below 1. The odds ratio describes the direction and size of the odds comparison; the converted risk gives a probability in the original time window.

Worked Example: The Same Odds Ratio at a Different Baseline

Hypothetical data. Suppose a different phenotype has reference risk \(p_0=0.20\) for genotype \(g_0\), and the odds ratio for genotype \(g_1\) versus \(g_0\) is again \(\theta=2.5\). Find the comparison risk.

The conversion uses the baseline risk for this phenotype, rather than carrying over the 4% baseline from the first example:

p_1=\frac{2.5(0.20)}{1-0.20+2.5(0.20)} =\frac{0.50}{0.80+0.50} =\frac{0.50}{1.30} \approx 0.3846.

The risk for \(g_1\) is approximately 38.5%. This differs from the 9.43% risk in the first example even though the odds ratio is the same: the reference risks are different. The odds ratio alone does not determine an absolute risk.

Conditions for a Meaningful Conversion

The arithmetic is valid only when the inputs describe a compatible comparison. Before interpreting a converted risk, check the following:

Same outcome definition. The reference risk and odds ratio must concern the same phenotype, including its definition and observation period.
Same reference and comparison groups. Use the risk for the genotype that is actually the reference in the odds ratio. If the comparison direction is reversed, the odds ratio must also be reversed.
Compatible population and covariates. A conditional odds ratio for people with a particular covariate profile should be combined with a reference risk for that same profile. Combining an adjusted odds ratio with an unrelated marginal reference risk need not produce a meaningful marginal genotype-specific risk.
Known reference risk. The formula needs \(p_0\), not just \(\theta\). If \(p_0\) is estimated, the resulting \(p_1\) is also an estimate.

In a logistic model with a binary genotype contrast, the odds ratio is often \(\exp(\beta)\), where \(\beta\) is the coefficient for that contrast. The reference risk supplies the intercept information needed to express a probability. This agrees with the conversion: the odds for the comparison genotype are the reference odds multiplied by \(\exp(\beta)\), after which odds are converted back to a probability.

The formula converts point values. If the odds ratio or reference risk is uncertain, the resulting risk is uncertain too. A point conversion does not by itself provide an uncertainty interval; the uncertainty in the inputs must be accounted for when making interval estimates.

Common Mistakes and Checks

Multiplying risk by the odds ratio. The reasoning “risk is 4% and the odds ratio is 2.5, so the risk is 10%” is generally incorrect. The odds ratio multiplies odds. Convert the baseline risk to odds, apply the odds ratio, and convert the result back to a risk—or use the derived formula.
Using the wrong baseline. A baseline from another phenotype, time window, population, or covariate profile may not be compatible with the odds ratio. Match the reference risk to the comparison being interpreted.
Reversing the genotype contrast. An odds ratio for \(g_1\) versus \(g_0\) cannot be used as though it described \(g_0\) versus \(g_1\). Keep the genotype order explicit in both the odds ratio and the risk statement.
Calling an odds ratio a risk multiplier. An odds ratio above 1 indicates higher odds, not that risk is that many times higher. Report the converted risk as a probability, with its phenotype and time window.

A useful final check is to compare the converted risk with the reference risk: its direction should agree with whether the odds ratio is above or below 1, and the computed value should lie between 0 and 1. If either check fails, revisit the arithmetic and the inputs.

Check Your Understanding

Use the conversion formula and state each answer in context.

A reference genotype has risk 10%, and the odds ratio for a comparison genotype is 2. Find the comparison risk.
Why is multiplying a reference risk by an odds ratio generally not the correct conversion?
If the odds ratio for a comparison genotype is 1, what does the formula imply about its risk relative to the reference risk?
What must be checked before combining a covariate-adjusted odds ratio with a reference risk?
Two hypothetical settings have the same odds ratio but different reference risks. Should their converted comparison risks necessarily be the same? Explain.