Interpreting Per-Allele Odds Ratios

From a Logistic Coefficient to an Odds Ratio

In a case-control association analysis, a logistic-regression coefficient is naturally reported on the log-odds scale. That scale is useful for fitting the model, but a coefficient such as $0.40$ is not usually the most direct description of the genetic association. Exponentiating the coefficient gives an odds ratio, which describes the multiplicative change in the odds of the outcome associated with one additional copy of the effect allele under an additive model.

The distinction between odds and probability matters. An odds ratio of $1.5$ means 50% higher odds, not necessarily 50% higher probability or risk. The relationship between odds and probability depends on the starting probability, so the same odds ratio can correspond to different changes in probability in different settings.

Definition: Let $G_i$ be the dosage of the effect allele for individual $i$, coded from 0 to 2, and let $Y_i$ indicate the binary outcome. In an additive logistic-regression model, $$ \log\left(\frac{P(Y_i=1\mid G_i,\mathbf{Z}_i)}{1-P(Y_i=1\mid G_i,\mathbf{Z}_i)}\right) =\beta_0+\beta_G G_i+\boldsymbol{\gamma}^{T}\mathbf{Z}_i, $$ where $\mathbf{Z}_i$ denotes modeled covariates. The per-allele odds ratio is $\exp(\beta_G)$: the ratio of the outcome odds for two individuals whose dosages differ by one effect allele, with the modeled covariates held fixed.

The additive assumption is specifically that each one-unit increase in dosage adds the same amount, $\beta_G$, to the log odds. Thus the model implies a constant multiplicative change in odds for each additional effect allele. It does not assert that the probability changes by a constant amount, and it does not require genotype dosage to be limited to observed values of exactly 0, 1, or 2. With imputed genotype data, a dosage can be fractional and still represents an expected allele count.

For the coefficient and interval to support this interpretation, the logistic model must be appropriately specified for the association being described, and the estimate and its standard error must be usable. The usual Wald confidence interval also relies on the coefficient estimate being approximately normally distributed. These approximations are most credible when the data contain enough information to estimate the model stably; a very large standard error or an extreme estimate is a warning to inspect the fit rather than report a precise-sounding interpretation.

Confidence Intervals on the Odds-Ratio Scale

A standard Wald interval is first calculated for $\beta_G$, on the log-odds scale. For a 95% confidence interval, the estimate is combined with its standard error using the standard normal critical value, approximately 1.96. Exponentiating the estimate gives the odds ratio, and exponentiating the two interval endpoints gives the corresponding interval for the odds ratio.

\widehat{\beta}_G \ \pm\ 1.96\,SE(\widehat{\beta}_G) \quad\longrightarrow\quad \left[ \exp\{\widehat{\beta}_G-1.96\,SE(\widehat{\beta}_G)\}, \exp\{\widehat{\beta}_G+1.96\,SE(\widehat{\beta}_G)\} \right].

This transformation is valid because the exponential function is strictly increasing: values below the lower log-scale endpoint remain below the transformed lower endpoint, and likewise for the upper endpoint. The resulting interval is generally asymmetric around the reported odds ratio. A confidence interval describes uncertainty in the estimated odds ratio under the model and the interval procedure; it is not a statement that the fixed parameter has a 95% probability of lying in this particular interval.

Worked Example: An Odds Ratio Above One

Suppose a hypothetical GWAS logistic-regression fit estimates an effect-allele coefficient of $0.40$ with standard error $0.15$. Assume the dosage is coded as 0, 1, or 2 effect alleles, the model includes the stated covariates, and the coefficient estimate is sufficiently well behaved for a Wald interval.

Estimate. Exponentiate the coefficient: $$ \widehat{OR}=\exp(0.40)=1.492. $$

95% confidence interval. First calculate the log-scale margin: $$ 1.96(0.15)=0.294. $$ The log-scale endpoints are $0.40-0.294=0.106$ and $0.40+0.294=0.694$. Exponentiating gives $$ [\exp(0.106),\exp(0.694)]=[1.112,2.002]. $$

Interpretation. Under this fitted model, one additional copy of the effect allele is associated with estimated outcome odds 1.492 times as large, or about 49.2% higher, at the same modeled covariate values. The 95% Wald interval corresponds to odds from about 1.112 to 2.002 times as large, or approximately 11.2% to 100.2% higher odds. This is an association estimate, not by itself evidence that the allele causes the outcome.

Interpreting Odds Ratios Below One

An odds ratio below one indicates lower odds for the comparison as coded: the numerator group has lower odds than the reference group. For example, an odds ratio of $0.70$ means the odds are 70% as large, or 30% lower. It does not mean that the probability is 30 percentage points lower. A percentage comparison of probabilities requires the baseline probability as well as the odds ratio.

It is often useful to express a below-one odds ratio as a reduction in odds. For an odds ratio $r<1$, the relative reduction in odds is $100(1-r)\%$. The direction must be tied to the effect allele and the comparison in the model; an unlabeled statement such as “the allele is protective” goes beyond what the number alone establishes.

Worked Example: An Odds Ratio Below One

In a separate hypothetical analysis, the fitted effect-allele coefficient is $-0.357$, with standard error $0.14$. Suppose the model assumptions are considered reasonable and the coefficient estimate is approximately normal.

Estimate. The per-allele odds ratio is $$ \widehat{OR}=\exp(-0.357)=0.6997. $$ Thus each additional effect allele is associated with about 70.0% as large outcome odds, or about 30.0% lower odds, conditional on the modeled covariates.

95% confidence interval. The margin on the coefficient scale is $1.96(0.14)=0.2744$. The endpoints are $-0.357-0.2744=-0.6314$ and $-0.357+0.2744=-0.0826$. Therefore, $$ [\exp(-0.6314),\exp(-0.0826)]=[0.532,0.921]. $$

The interval indicates that the data are compatible, under the model and Wald procedure, with per-allele odds ranging from about 53.2% to 92.1% of the comparison odds. Equivalently, this is about 7.9% to 46.8% lower odds. The interval is entirely below one, but its interpretation remains in terms of odds, not risk.

Odds Are Not Probabilities

For an outcome probability $p$, the odds are $p/(1-p)$. If the odds are multiplied by an odds ratio $r$, the new probability is not generally the original probability multiplied by $r$. To see the conversion, let the baseline probability be $p_0$. Its odds are $p_0/(1-p_0)$; multiplying by $r$ gives the comparison odds. Converting those odds back to a probability yields

p_1 =\frac{r\,p_0/(1-p_0)}{1+r\,p_0/(1-p_0)} =\frac{r p_0}{1-p_0+r p_0}.

Consequently, an odds ratio alone does not determine the difference in probabilities. In particular, an odds ratio should not be reported as a risk ratio unless the context and additional information justify that approximation. When the outcome is common, odds ratios and risk ratios can differ substantially.

Worked Example: The Same Odds Ratio, Not the Same Risk Ratio

Consider a hypothetical covariate stratum in which the modeled outcome probability for dosage $G=0$ is $p_0=0.10$. Suppose the fitted per-allele coefficient is $\log(1.5)=0.4055$, so the per-allele odds ratio is $1.5$.

At dosage $G=0$, the odds are $0.10/(1-0.10)=0.1111$. At dosage $G=1$, the odds are $1.5(0.1111)=0.1667$, which correspond to probability $$ p_1=\frac{0.1667}{1+0.1667}=0.1429. $$ The probability ratio is $0.1429/0.10=1.429$, not 1.5. The odds are 50% higher, while the probability is about 42.9% higher in this comparison.

Under the same model and covariate values, dosage $G=2$ has odds $1.5^2(0.1111)=0.25$, corresponding to probability $0.25/(1+0.25)=0.20$. The two-copy odds ratio relative to dosage zero is $1.5^2=2.25$, while the probability ratio is $0.20/0.10=2.00$. The odds ratio compounds multiplicatively by dosage; the probability ratio need not equal it.

Changing the Effect Allele and Checking the Result

The odds ratio depends on which allele is designated as the effect allele. If the alternate allele is replaced by the other allele, dosage changes from $G$ to $2-G$ for a diploid genotype. The coefficient changes sign, so the odds ratio becomes its reciprocal. The confidence interval endpoints also become reciprocals, with their order reversed.

\widehat{\beta}_{\text{reversed}}=-\widehat{\beta}_G, \qquad \widehat{OR}_{\text{reversed}}=\frac{1}{\widehat{OR}}, \qquad [L,U]\ \longrightarrow\ [1/U,1/L].

For example, an odds ratio of $0.70$ for one allele corresponds to $1/0.70=1.429$ for the opposite allele under the reversed comparison. This means about 42.9% higher odds for the reversed comparison, not 30% higher odds. Keep the effect-allele label with the coefficient, odds ratio, and interval so that this reversal is not mistaken for a contradictory result.

Several checks help identify errors or overinterpretation:

Exponentiating a positive coefficient must give an odds ratio above one; exponentiating a negative coefficient must give one below one.
The odds-ratio interval must have positive endpoints and contain the point estimate when the coefficient-scale interval contains its estimate.
If an interval crosses one, the data and this interval procedure do not exclude the null odds ratio of one at the corresponding two-sided confidence level.
A very wide interval signals substantial uncertainty. Extreme estimates or very large standard errors can indicate an unstable fit, so the transformed number should not be presented as precise evidence.
Confirm the effect-allele coding, dosage scale, outcome coding, and covariates before comparing reported odds ratios across analyses.

Common mistake: “The odds ratio is 1.5, so the probability of the outcome is 50% higher.” This treats an odds ratio as a risk ratio. The correction is to say that the odds are 50% higher; to calculate a probability or risk comparison, the baseline probability is also needed.

A per-allele odds ratio is most directly compared with another estimate when the effect allele, outcome definition, covariate conditioning, and additive dosage model agree. A change in any of these can change the numerical interpretation. Even when they agree, an odds ratio summarizes an association under the fitted model; it does not establish a biological mechanism or a causal effect.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A fitted coefficient is $0.20$. Is its odds ratio above or below one, and what is its approximate value?
A per-allele odds ratio is $0.80$. State the interpretation in terms of odds, and explain why “20% lower probability” is not justified by that number alone.
A coefficient estimate is $-0.30$ with standard error $0.10$. What is the 95% Wald interval on the odds-ratio scale?
If the effect allele is reversed and the original odds ratio is $1.25$ with interval $[1.05,1.49]$, what are the reversed estimate and interval?
At a baseline probability of $0.20$, an odds ratio of $2$ is applied. What is the resulting probability, and what is the probability ratio?

Solutions. (1) $\exp(0.20)=1.221$, above one, so the estimated odds are about 22.1% higher per effect allele. (2) The odds are 80% as large, or 20% lower; a probability comparison requires the baseline probability. (3) The coefficient interval is $-0.30\pm1.96(0.10)=[-0.496,-0.104]$. Exponentiating gives approximately $[0.609,0.901]$. (4) The reversed estimate is $1/1.25=0.800$, and the reversed interval is $[1/1.49,1/1.05]=[0.671,0.952]$, rounded. (5) The new probability is $2(0.20)/(1-0.20+2(0.20))=0.40/1.20=0.333$. The probability ratio is $0.333/0.20=1.667$, not 2.