Profile Intervals for Odds Ratios and Rate Ratios

Why Profile a Ratio?

Odds ratios and rate ratios are common targets in generalized linear models, but each is usually a function of several fitted coefficients rather than a single coefficient. A profile interval handles that directly: it fixes the target ratio at a proposed value, maximizes the likelihood over all remaining freedom, and retains the values that fit sufficiently well. The resulting interval reflects nuisance-parameter estimation and can be asymmetric.

For logistic regression, a coefficient contrast describes a change in log odds; exponentiating it gives an odds ratio. For Poisson regression with a log link, exponentiating a contrast gives a rate ratio when comparing rates at the specified covariate values. In both settings, it is useful to profile the log ratio first and then express the interval on the ratio scale.

Ratio Targets as Scalar Profile Parameters

Let $\beta$ be the coefficient vector in a regression model, and let $\ell(\beta)$ be its log-likelihood. For a fixed contrast vector $c$, define the scalar log-ratio target $\gamma=c^\mathsf{T}\beta$ and the ratio target $R=\exp(\gamma)$. The profile log-likelihood for $\gamma$ maximizes the full log-likelihood subject to that contrast being fixed.

Definition: For a scalar target $\gamma=c^\mathsf{T}\beta$, its profile log-likelihood is $$ \ell_p(\gamma)=\sup_{\beta:\,c^\mathsf{T}\beta=\gamma}\ell(\beta). $$ If the supremum is attained, this is the largest log-likelihood among coefficient vectors satisfying the constraint. The profile likelihood-ratio confidence set at nominal level $1-\alpha$ is $$ \left\{\gamma:2\bigl[\ell(\hat\beta)-\ell_p(\gamma)\bigr]\leq \chi^2_{1,1-\alpha}\right\}, $$ where $\hat\beta$ is the unrestricted maximum-likelihood estimate and $\chi^2_{1,1-\alpha}$ is the $1-\alpha$ quantile of a chi-squared distribution with one degree of freedom.

This is the usual likelihood-ratio construction for a scalar target. Its chi-squared calibration is an asymptotic one: it requires a correctly specified, identifiable model, an interior true parameter, sufficient smoothness, nonsingular Fisher information, and a regular scalar target with nonzero gradient. Small or sparse samples can make that approximation inaccurate.

How the Constraints Arise

In logistic regression, suppose $Y_i$ is Bernoulli with success probability $p_i$, and $\operatorname{logit}(p_i)=x_i^\mathsf{T}\beta$. Compare covariate settings $x_A$ and $x_B$. The conditional odds ratio for $B$ versus $A$ is $\exp\{(x_B-x_A)^\mathsf{T}\beta\}$. Thus the contrast vector is $c=x_B-x_A$, and fixing a proposed odds ratio $r>0$ imposes the coefficient constraint $c^\mathsf{T}\beta=\log r$.

In Poisson regression, suppose $Y_i$ has a Poisson distribution with mean $\mu_i$, and $\log(\mu_i)=\log(t_i)+x_i^\mathsf{T}\beta$, where $t_i$ is an exposure or observation time. Comparing rates at two covariate settings with the same exposure structure gives rate ratio $\exp\{(x_B-x_A)^\mathsf{T}\beta\}$. If the exposures differ, compare rates $\mu_i/t_i$; the offsets cancel from that rate ratio as well. Fixing the proposed rate ratio $r$ again imposes $c^\mathsf{T}\beta=\log r$.

Result: For either ratio target $R=\exp(c^\mathsf{T}\beta)$, the profile log-likelihood at $R=r>0$ is $$ \ell_{p,R}(r)=\sup_{\beta:\,c^\mathsf{T}\beta=\log r}\ell(\beta) =\ell_p(\log r). $$ Consequently, its likelihood-ratio confidence set is obtained by exponentiating the endpoints of the log-ratio confidence set, if that set is an interval.

To verify the result, the constraint $R=r$ is equivalent to $\exp(c^\mathsf{T}\beta)=r$. Because the exponential function is one-to-one from the real line to the positive real numbers, this is equivalent to $c^\mathsf{T}\beta=\log r$. The two constrained sets of coefficient vectors are therefore identical, so their supremum log-likelihoods are identical. The likelihood-ratio statistic is consequently the same at $r$ as at $\gamma=\log r$. Since exponentiation is strictly increasing, it maps any interval $[\gamma_L,\gamma_U]$ to $[\exp(\gamma_L),\exp(\gamma_U)]$, with no reversal of endpoints. This establishes the result. $\square$

Worked Examples

Worked Example: An Adjusted Odds Ratio with an Interaction

Suppose a logistic model has linear predictor $\eta=\beta_0+\beta_1S+\beta_2T+\beta_3ST$, where $S$ indicates smoking and $T$ indicates treatment. Find the smoking odds ratio among treated patients and state the profile constraint for a proposed value $r$. Suppose the fitted coefficients are $\hat\beta_1=0.40$ and $\hat\beta_3=-0.10$.

Among treated patients, $T=1$. Changing smoking status from $S=0$ to $S=1$ changes the linear predictor by $(\beta_1+\beta_3)$. The odds ratio is therefore $\exp(\beta_1+\beta_3)$. At the fitted coefficients, the log odds ratio is $0.40-0.10=0.30$, so the estimated odds ratio is $\exp(0.30)\approx1.350$.

For a proposed odds ratio $r$, fix $\beta_1+\beta_3=\log r$ and maximize the logistic log-likelihood over the remaining coefficient freedom. The contrast vector has entries $0,1,0,1$ in the order $(\beta_0,\beta_1,\beta_2,\beta_3)$. Profiling $\beta_1$ alone would answer a different question: it would describe the smoking odds ratio when $T=0$, not when $T=1$.

Worked Example: A Profile Interval for a Logistic Odds Ratio

Consider two independent groups, each with 10 Bernoulli observations. In group 0, 5 outcomes are successes; in group 1, 5 are successes. Fit a logistic model with an intercept and a group indicator. Find the profile likelihood-ratio interval for the odds ratio of group 1 versus group 0, using the 95% cutoff $3.841$.

Write the success probabilities as $p_0$ and $p_1$. The log odds ratio is $\gamma=\operatorname{logit}(p_1)-\operatorname{logit}(p_0)$, so the ratio is $r=\exp(\gamma)$. The log-likelihood, omitting binomial coefficients that do not depend on the probabilities, is \[ \ell=5\log p_0+5\log(1-p_0)+5\log p_1+5\log(1-p_1). \] For fixed $r$, the constraint is $p_1/(1-p_1)=r\,p_0/(1-p_0)$. The data and constraint are unchanged in form by swapping the groups and replacing $r$ by $1/r$. The constrained maximum therefore has $p_1=1-p_0$. Substituting into the constraint gives $(1-p_0)/p_0=\sqrt r$, or $p_0=1/(1+\sqrt r)$.

The unrestricted maximum is $p_0=p_1=0.5$, with log-likelihood $10\log(0.25)$. At the constrained maximum, $p_1=1-p_0$, and the log-likelihood is $10\log\{p_0(1-p_0)\}$. Thus the profile likelihood-ratio statistic is \[ 2\{\ell(\hat\beta)-\ell_p(\gamma)\} =20\log\left\{\frac{0.25}{p_0(1-p_0)}\right\}. \] Setting this equal to $3.841$ gives $p_0(1-p_0)=0.25\exp(-3.841/20)\approx0.2063$. Solving the quadratic gives the lower probability $p_0\approx0.291$ (and the corresponding upper solution $0.709$). The associated upper odds ratio is \[ r=\left(\frac{1-p_0}{p_0}\right)^2 \approx\left(\frac{0.709}{0.291}\right)^2\approx5.94. \] By symmetry, the lower endpoint is its reciprocal, approximately $1/5.94=0.168$. The approximate 95% profile interval is therefore $[0.168,5.94]$. The endpoints are reciprocal here because the observed success proportions are equal; this symmetry is not a general property of profile intervals.

Worked Example: A Profile Interval for a Poisson Rate Ratio

Suppose two groups have independent Poisson event counts, each observed for one unit of exposure. The counts are $Y_0=5$ and $Y_1=5$. Fit rates $\lambda_0$ and $\lambda_1$, and construct the 95% profile likelihood-ratio interval for $r=\lambda_1/\lambda_0$, again using cutoff $3.841$.

The joint likelihood, omitting factors that do not depend on the rates, is proportional to $\lambda_0^5\lambda_1^5\exp(-\lambda_0-\lambda_1)$. Under the constraint $\lambda_1=r\lambda_0$, the constrained log-likelihood, up to a constant, is $10\log\lambda_0+5\log r-(1+r)\lambda_0$. Differentiating with respect to $\lambda_0$ gives $10/\lambda_0-(1+r)=0$, so the constrained maximum is $\hat\lambda_0=10/(1+r)$, with $\hat\lambda_1=10r/(1+r)$.

Equivalently, let $q=\lambda_1/(\lambda_0+\lambda_1)=r/(1+r)$. The unrestricted maximum has both rates equal to 5, so its total rate is 10 and $q=0.5$. After profiling the total rate, the log-likelihood depending on $q$ is $5\log q+5\log(1-q)$, up to a constant. Hence the profile likelihood-ratio statistic is \[ 10\log\left\{\frac{0.25}{q(1-q)}\right\}. \] Equating this to $3.841$ gives $q(1-q)=0.25\exp(-3.841/10)\approx0.1703$. The smaller solution is $q\approx\{1-\sqrt{1-4(0.1703)}\}/2\approx0.218$. Thus the lower rate-ratio endpoint is $q/(1-q)\approx0.278$, and the upper endpoint is its reciprocal, approximately $3.59$. The approximate 95% profile interval is $[0.278,3.59]$. The calculation profiles out the baseline rate rather than treating its estimate as fixed.

Interpreting and Reporting the Interval

The interval is obtained by comparing the best fit under each proposed ratio with the unrestricted best fit. A ratio is retained when this loss in log-likelihood is no larger than the chosen likelihood-ratio cutoff. The procedure does not hold the other regression coefficients at their estimates: it re-optimizes them under the ratio constraint. That feature is what accounts for nuisance-parameter uncertainty.

For a practical analysis, report the ratio definition and comparison, the point estimate, the confidence level, and the profile likelihood method. State the reference covariate settings when interactions are present. For example, with an interaction, an odds ratio may vary with treatment status; reporting only “the odds ratio” can conceal which contrast was profiled.

Common Mistakes

Profiling a coefficient that is not the target contrast. With an interaction, a conditional odds ratio can involve a sum of coefficients. Constrain that sum to $\log r$, not just one coefficient.
Constraining the ratio to $r$ on the log scale. The log-ratio is $\log r$, so the coefficient constraint is $c^\mathsf{T}\beta=\log r$, not $c^\mathsf{T}\beta=r$.
Holding nuisance coefficients fixed. That evaluates the likelihood at one chosen nuisance value; a profile likelihood maximizes over nuisance parameters subject to the target constraint.
Exponentiating the wrong quantity. Construct the likelihood-ratio interval for the log ratio, then exponentiate its endpoints. The estimate on the ratio scale is $\exp(c^\mathsf{T}\hat\beta)$.
Assuming the interval is symmetric around the ratio estimate. Likelihood curvature can differ on the two sides of the maximum. Symmetry on the log-ratio scale, when it happens, becomes reciprocal endpoints on the ratio scale, not equal additive distances from the estimate.

Key takeaway: An odds ratio or rate ratio formed by comparing two covariate settings is the exponential of a coefficient contrast. Fix the log contrast, maximize over the remaining coefficients, apply the scalar likelihood-ratio cutoff, and exponentiate the resulting endpoints.

Check Your Understanding

For each question, identify the contrast or profiling step that determines the answer.

In a logistic model with an exposure-by-treatment interaction, which coefficient contrast gives the exposure odds ratio among treated individuals?
For a proposed rate ratio $r$, what equality constraint should be imposed on a Poisson regression coefficient contrast?
Why must nuisance coefficients be re-optimized when computing a profile likelihood?
If a profile interval for a log rate ratio is $[-0.4,0.7]$, what is the corresponding interval for the rate ratio?
Why might a profile interval for an odds ratio fail to be symmetric around its point estimate?