Confidence Intervals for Logistic Odds Ratios

From Coefficient Uncertainty to Odds-Ratio Uncertainty

In logistic regression, a coefficient describes a change in log odds. Exponentiating that coefficient expresses the same change as a multiplicative odds ratio, which is often easier to interpret. Uncertainty must be transformed as well: a confidence interval for the coefficient becomes an odds-ratio interval by exponentiating both endpoints.

This tutorial develops the Wald interval for a logistic coefficient and its odds-ratio interpretation. The interval is based on the coefficient estimate’s approximate normal sampling distribution. It is therefore an asymptotic procedure, not a promise of exact coverage in every finite sample.

Definition: Suppose a fitted logistic regression has coefficient vector \(\hat{\boldsymbol\beta}\) and estimated covariance matrix \(\widehat{\operatorname{Var}}(\hat{\boldsymbol\beta})\). For coefficient \(\hat\beta_j\), let \(\widehat{\operatorname{SE}}(\hat\beta_j)\) be the square root of the \(j\)th diagonal entry of that matrix. A two-sided Wald confidence interval with confidence level \(1-\alpha\) for \(\beta_j\) is

\hat\beta_j \ \pm\ z_{1-\alpha/2}\widehat{\operatorname{SE}}(\hat\beta_j),

where \(z_{1-\alpha/2}\) is the \(1-\alpha/2\) quantile of the standard normal distribution. The corresponding confidence interval for the odds ratio \(\exp(\beta_j)\) is obtained by exponentiating the lower and upper coefficient limits.

In a logistic model, the logit of the probability of outcome \(1\) is linear in the predictors. Holding the other predictors fixed, increasing predictor \(x_j\) by one unit changes the linear predictor by \(\beta_j\), and multiplies the odds by \(\exp(\beta_j)\). For a binary predictor, this compares the odds at value \(1\) with the odds at value \(0\), with other predictors held fixed. If a predictor is rescaled, the coefficient and its odds ratio refer to that stated unit change.

Constructing the Interval

For a 95% Wald interval, \(z_{0.975}\) is approximately \(1.96\). First calculate the coefficient limits by adding and subtracting \(1.96\) standard errors. Then exponentiate each limit. Because the exponential function is strictly increasing, the lower coefficient limit remains the lower odds-ratio limit.

\left[ \hat\beta_j-1.96\,\widehat{\operatorname{SE}}(\hat\beta_j),\ \hat\beta_j+1.96\,\widehat{\operatorname{SE}}(\hat\beta_j) \right] \quad\longrightarrow\quad \left[ \exp\!\left\{\hat\beta_j-1.96\,\widehat{\operatorname{SE}}(\hat\beta_j)\right\},\ \exp\!\left\{\hat\beta_j+1.96\,\widehat{\operatorname{SE}}(\hat\beta_j)\right\} \right].

The coefficient interval is symmetric around \(\hat\beta_j\), but the odds-ratio interval is generally not symmetric around \(\exp(\hat\beta_j)\). The transformation preserves the ordering and the multiplicative scale; it does not turn the interval into an additive margin around the estimated odds ratio.

Worked Example: Odds Ratio per 10 mg

Suppose a logistic regression uses \(x=\text{dose}/10\), where dose is measured in milligrams. Thus, a one-unit increase in \(x\) means a 10 mg increase in dose. The estimated coefficient is \(\hat\beta=0.120\), with standard error \(0.050\). Calculate and interpret the 95% Wald odds-ratio interval.

Step 1: Calculate the coefficient margin. The margin is \(1.96(0.050)=0.098\). The coefficient limits are \(0.120-0.098=0.022\) and \(0.120+0.098=0.218\), so the interval is \([0.022,0.218]\).

Step 2: Exponentiate the estimate and limits. The estimated odds ratio is \(\exp(0.120)=1.12750\). The limits are \(\exp(0.022)=1.02224\) and \(\exp(0.218)=1.24359\). The odds-ratio interval is therefore approximately \([1.022,1.244]\).

Interpretation. Under the model, a 10 mg increase in dose is associated with estimated odds multiplied by \(1.12750\), or about a 12.75% increase. The interval corresponds to odds increases from about \(100(1.02224-1)=2.22\%\) to \(100(1.24359-1)=24.36\%\). This is an interval for an odds ratio, not for a probability increase.

Worked Example: A Binary Treatment Indicator

Suppose treatment is coded \(1\) for treated and \(0\) for untreated. In a logistic model, its estimated coefficient is \(-0.510\), with standard error \(0.240\). Calculate the 95% Wald interval and interpret its direction.

The coefficient margin is \(1.96(0.240)=0.4704\). The coefficient limits are \(-0.510-0.4704=-0.9804\) and \(-0.510+0.4704=-0.0396\). Exponentiating gives \(\exp(-0.510)=0.6005\), \(\exp(-0.9804)\approx0.375\), and \(\exp(-0.0396)\approx0.961\). Thus, the estimated odds ratio for treated versus untreated is about \(0.60\), with a 95% Wald interval of approximately \([0.375,0.961]\).

The estimated odds in the treated group are about 60% of the untreated odds, conditional on the other model covariates. Since the entire interval is below \(1\), it is consistent with lower odds in the treated group at this confidence level. The conclusion concerns odds; it does not state that treatment reduces risk by 40%.

Why Exponentiating the Endpoints Works

The interval transformation follows from the monotonicity of the exponential function. The coverage statement follows from the large-sample normal approximation used to construct the coefficient interval. These facts explain both how to calculate the interval and what its confidence level means.

Result: If a coefficient confidence interval is \([L,U]\), then the corresponding interval for \(\exp(\beta_j)\) is \([\exp(L),\exp(U)]\). For any fixed \(\beta_j\), the event that the coefficient interval contains \(\beta_j\) is exactly the event that the exponentiated interval contains \(\exp(\beta_j)\).

Proof. Since the exponential function is strictly increasing, \(L\leq\beta_j\leq U\) holds if and only if \(\exp(L)\leq\exp(\beta_j)\leq\exp(U)\). The two coverage events are therefore identical, which proves the claim. \(\square\)

Result: Under the usual regularity conditions for maximum-likelihood estimation in logistic regression, the Wald interval has asymptotic coverage \(1-\alpha\) for \(\beta_j\), and its exponentiated interval has the same asymptotic coverage for \(\exp(\beta_j)\).

Proof. Suppose the model is correctly specified and identifiable, the true parameter is finite and in the interior of the parameter space, and the observations satisfy the regularity conditions for maximum-likelihood asymptotic normality. In particular, the relevant Fisher information is nonsingular, and the covariance estimator used for the standard error is consistent. Under these conditions,

\frac{\hat\beta_j-\beta_j} {\widehat{\operatorname{SE}}(\hat\beta_j)} \xrightarrow{d}N(0,1).

The standard normal distribution is continuous, so the probability that the absolute value of this statistic is at most \(z_{1-\alpha/2}\) converges to \(1-\alpha\). That event is precisely that the Wald coefficient interval contains \(\beta_j\). By the preceding result, this is exactly the event that the exponentiated interval contains \(\exp(\beta_j)\). Hence the odds-ratio interval also has asymptotic coverage \(1-\alpha\). This argument establishes asymptotic coverage; it does not establish exact finite-sample coverage. \(\square\)

In practice, the covariance estimate must match the analysis. For example, if inference uses a robust covariance estimator, its standard error should be used in the Wald interval. The asymptotic justification still depends on the estimator and covariance estimate being appropriate for the sampling design and model assumptions.

Interpreting the Scale and the Null Value

The odds ratio’s null value is \(1\): multiplying odds by \(1\) leaves them unchanged. On the coefficient scale, the same null is \(0\), because \(\exp(0)=1\). Thus, a Wald interval that includes \(0\) before exponentiation includes \(1\) after exponentiation. The interval does not change its conclusion about whether it contains the null when moved between these scales.

Worked Example: An Adjusted Odds Ratio Near the Null

A multiple-covariate logistic regression estimates a coefficient of \(0.350\) for a binary exposure, with standard error \(0.180\). The coefficient compares exposure \(1\) with exposure \(0\), holding the other included covariates fixed. Find the 95% Wald odds-ratio interval.

The margin is \(1.96(0.180)=0.3528\). The coefficient limits are \(0.350-0.3528=-0.0028\) and \(0.350+0.3528=0.7028\). The estimated adjusted odds ratio is \(\exp(0.350)\approx1.419\). Exponentiating the limits gives \(\exp(-0.0028)\approx0.997\) and \(\exp(0.7028)\approx2.019\). The interval is approximately \([0.997,2.019]\).

The point estimate corresponds to about 1.42 times the odds for exposure \(1\) relative to exposure \(0\), conditional on the included covariates. Because the interval includes \(1\), it also includes the no-association value on the odds-ratio scale. Its broad range allows values close to no association as well as substantially higher odds; the point estimate alone does not convey this uncertainty.

An odds ratio is not a risk ratio or a probability difference. If a baseline risk is \(p_0\) and the odds ratio is \(r\), the corresponding risk is \(rp_0/(1-p_0+rp_0)\), not \(rp_0\) in general. Therefore, an odds-ratio interval cannot be read directly as a confidence interval for a change in probability. Uncertainty for fitted risks requires transforming uncertainty for the relevant linear predictor, a separate calculation.

Common Mistakes

Exponentiating only the coefficient estimate. This gives the estimated odds ratio but not its interval. Calculate both coefficient limits first, then exponentiate each limit.
Adding and subtracting a standard error on the odds-ratio scale. The Wald construction is symmetric on the coefficient scale. Exponentiation usually produces an asymmetric interval on the odds-ratio scale.
Using \(0\) as the odds-ratio null. The coefficient null is \(0\); the odds-ratio null is \(1\). A value of zero odds is not the null of no association.
Interpreting an odds ratio as a percentage-point change in risk. An odds ratio describes a multiplicative change in odds. The corresponding probability change depends on the starting risk.
Forgetting the predictor unit or comparison. State whether the interval refers to a one-unit increase, a specified unit such as 10 mg, or a particular category relative to its reference group.

Key takeaway: Form the Wald interval on the logistic coefficient scale, then exponentiate both endpoints. Interpret the resulting limits as plausible multiplicative changes in odds for a clearly stated comparison and predictor unit; \(1\), not \(0\), is the odds-ratio null.

Check Your Understanding

Use these questions to check the calculation and interpretation.

A logistic coefficient estimate is \(0.30\) with standard error \(0.10\). Write the 95% Wald interval on the coefficient scale, then give the exponentiated interval to three decimal places.
Why does exponentiating both limits preserve the interval’s coverage event for the odds ratio?
A 95% coefficient interval includes zero. What must be true of its exponentiated interval relative to the odds-ratio null?
Explain why a 1.5 odds ratio does not, by itself, imply a 50-percentage-point increase in probability.
A continuous predictor is measured in milligrams, but the reported coefficient is for a 10 mg increase. What comparison does its odds-ratio interval describe?