Profile Likelihood Intervals for Rate Ratios

Why Profile a Rate Ratio?

A rate ratio is often the quantity of scientific interest, even when the fitted Poisson model contains several other coefficients. A Wald interval uses an estimated standard error for the contrast. A profile likelihood interval instead asks, for each proposed rate ratio, how well the model can fit after all remaining coefficients have been re-estimated subject to that proposed value. This can produce an interval that reflects the shape of the likelihood more directly than a symmetric interval on the log-rate-ratio scale.

The essential operation is to hold the contrast fixed, not the nuisance coefficients. For each candidate contrast, maximize the likelihood over every coefficient that is not determined by that constraint. The likelihood-ratio statistic then compares this constrained fit with the unrestricted maximum.

Definition: Suppose independent counts $Y_i$ follow Poisson distributions with means $\mu_i$, and a log-linear model specifies $\log(\mu_i)=o_i+x_i^\top\beta$, where $o_i$ is a known offset. For a contrast $\psi=c^\top\beta$, the profile log-likelihood is $$ \ell_p(\psi)=\sup_{\beta:\,c^\top\beta=\psi}\ell(\beta). $$ If the rate ratio of interest is $r=\exp(\psi)$, write $\ell_p(r)=\ell_p(\log r)$. The profile likelihood-ratio statistic is $$ \Lambda(r)=2\{\ell(\hat\beta)-\ell_p(r)\}, $$ where $\hat\beta$ maximizes the unrestricted log-likelihood.

A contrast represents a log rate ratio when the difference in linear predictors between the two profiles of interest is $c^\top\beta$; known offsets must be accounted for when translating that contrast into a rate ratio. The likelihood may include nuisance coefficients such as baseline rates, stratum effects, or other covariate effects. Profiling re-estimates those coefficients for each fixed $r$, rather than plugging in their unrestricted estimates.

From a Likelihood-Ratio Test to an Interval

For a nominal confidence level $1-\alpha$, the usual large-sample profile likelihood interval contains the candidate values whose likelihood-ratio statistic is no larger than the appropriate chi-square cutoff.

Theorem: Under the regularity conditions for Wilks' theorem, the likelihood-ratio statistic for testing one fixed, identifiable scalar contrast has an asymptotic $\chi^2_1$ distribution when the null value is true. Thus an approximate $1-\alpha$ profile likelihood confidence set is $$ \left\{r>0:\Lambda(r)\leq \chi^2_{1,\,1-\alpha}\right\}. $$ Here $\chi^2_{1,\,1-\alpha}$ is the $1-\alpha$ quantile of a chi-square distribution with one degree of freedom.

The standard approximation assumes a correctly specified, identifiable model with a fixed-dimensional parameter, a true parameter in the interior of the parameter space, and regular likelihood behavior, including nonsingular information for the parameter of interest. In practice, existence of the relevant unrestricted and constrained fits is also important. If the model is sparse, a parameter lies on a boundary, or a maximum is not attained, the chi-square calibration may be unreliable. This interval is a likelihood-ratio approximation, not an exact Poisson interval.

Derivation: For each proposed $r$, the likelihood-ratio test compares the unrestricted maximum $\ell(\hat\beta)$ with the best fit satisfying $c^\top\beta=\log r$, namely $\ell_p(\log r)$. Twice their log-likelihood difference is $\Lambda(r)$. Inverting the test means retaining exactly those null values not rejected by the rule $\Lambda(r)>\chi^2_{1,\,1-\alpha}$. Therefore the retained set is precisely the displayed profile likelihood confidence set. Since $r=\exp(\psi)$ is a one-to-one transformation, this test inversion on the contrast scale gives the corresponding set on the rate-ratio scale. $\square$

A Useful Two-Group Derivation

The calculation is especially transparent for two independent Poisson counts. Let $Y_0\sim\operatorname{Pois}(E_0\lambda_0)$ and $Y_1\sim\operatorname{Pois}(E_1\lambda_1)$, where $E_0,E_1>0$ are known exposures and $r=\lambda_1/\lambda_0$. Write $y_+=y_0+y_1$. For a fixed positive $r$, the constraint is $\lambda_1=r\lambda_0$. Ignoring terms that do not depend on the rates, the log-likelihood under this constraint is $$ \ell(\lambda_0;r)=y_+\log(\lambda_0)+y_1\log(r)-\lambda_0(E_0+rE_1)+\text{constant}. $$

Result: If $y_+>0$, the constrained maximum occurs at $$ \tilde\lambda_0(r)=\frac{y_+}{E_0+rE_1}, \qquad \tilde\lambda_1(r)=\frac{r y_+}{E_0+rE_1}. $$ These are finite, positive constrained maximum-likelihood estimates for every $r>0$.

To verify the result, differentiate the constrained log-likelihood with respect to $\lambda_0>0$:

\frac{\partial\ell}{\partial\lambda_0} =\frac{y_+}{\lambda_0}-(E_0+rE_1).

The derivative is zero at the stated value. The second derivative is $-y_+/\lambda_0^2<0$, so this stationary point is the unique maximum. The unrestricted estimates, when both counts are positive, are $\hat\lambda_0=y_0/E_0$ and $\hat\lambda_1=y_1/E_1$. Substituting the constrained and unrestricted fits into the Poisson likelihood gives the profile likelihood-ratio statistic. This derivation also shows why an exposure offset matters: the constrained means divide the pooled count according to $E_0+rE_1$, not simply $1+r$.

If $y_+=0$, the log-rate model has no finite maximum: its likelihood supremum is approached as both rates tend to zero. That supremum is the same for every $r>0$, so the data do not distinguish rate ratios through this likelihood. If the parameter space is explicitly extended to include zero rates, the boundary fit has both means zero. It is not a finite maximum under the usual log-rate parameterization.

Worked Examples

Worked Example: A Two-Group Rate-Ratio Interval

Suppose two groups have equal exposure, with observed counts $y_0=20$ and $y_1=30$. Estimate a 95% profile likelihood interval for the rate ratio $r=\lambda_1/\lambda_0$. With equal exposures, the constrained fitted means at fixed $r$ are $50/(1+r)$ and $50r/(1+r)$. It is convenient to set $p=r/(1+r)$, so the constrained means are $50(1-p)$ and $50p$.

The unrestricted fitted means are 20 and 30. The likelihood-ratio statistic at a candidate $p$ is therefore

\Lambda(p)=2\left\{20\log\left(\frac{0.4}{1-p}\right) +30\log\left(\frac{0.6}{p}\right)\right\}.

The 95% cutoff is $\chi^2_{1,0.95}\approx3.841$. Solving $\Lambda(p)=3.841$ numerically gives $p\approx0.4617$ and $p\approx0.7283$. Because $r=p/(1-p)$, transform both endpoints:

\left(\frac{0.4617}{1-0.4617},\, \frac{0.7283}{1-0.7283}\right) \approx (0.86,\,2.68).

The point estimate is $30/20=1.5$, which lies inside the interval. The interval is asymmetric around 1.5 on the rate-ratio scale because the likelihood is profiled and the transformation from $p$ to $r$ is nonlinear.

Worked Example: Profiling a Contrast in a Three-Group Model

Consider three independent Poisson groups with log rates $\beta_0$, $\beta_0+\beta_1$, and $\beta_0+\beta_2$, respectively. Suppose their counts are 8, 12, and 18, with equal exposures. The contrast $\beta_2-\beta_1$ is the log rate ratio for group 2 versus group 1. At a candidate ratio $r$, the group 2 rate must be $r$ times the group 1 rate.

The group with count 8 is unrestricted by this contrast, so it fits its observed mean and cancels from the likelihood-ratio statistic. The counts 12 and 18 form the constrained pair. Their total is 30, so at fixed $r$ their fitted means are $30/(1+r)$ and $30r/(1+r)$. The unrestricted fitted means are 12 and 18. At $r=2$, the constrained means are 10 and 20, giving

\begin{aligned} \Lambda(2) &=2\left\{12\log\left(\frac{12}{10}\right) +18\log\left(\frac{18}{20}\right)\right\}\\ &\approx 2(2.1879-1.8965) =0.5827. \end{aligned}

Since $0.5827<3.841$, the candidate $r=2$ is retained by the approximate 95% profile likelihood confidence set. The fitted rates for the constrained pair—not the coefficient labels alone—determine which observed counts enter this calculation.

Worked Example: A Common Rate Ratio Across Strata

Suppose two strata have separate baseline rates but share a treatment rate ratio $r$. Within each stratum, treatment and control have equal exposure. The observed control and treatment counts are $(10,15)$ in stratum 1 and $(20,18)$ in stratum 2. The stratum-specific baseline rates are nuisance parameters, so profile them at each candidate $r$.

In a stratum with total count $m_s$, the constrained fitted control and treatment means are $m_s/(1+r)$ and $m_sr/(1+r)$. This follows by maximizing over that stratum's baseline rate, just as in the two-group derivation. Here the totals are 25 and 38. At candidate $r=1.5$, the fitted means are $(10,15)$ in stratum 1 and $(15.2,22.8)$ in stratum 2. The first stratum matches its observed counts exactly. The unrestricted fit uses the observed means in both strata, so

\begin{aligned} \Lambda(1.5) &=2\left\{ 10\log\left(\frac{10}{10}\right) +15\log\left(\frac{15}{15}\right) +20\log\left(\frac{20}{15.2}\right) +18\log\left(\frac{18}{22.8}\right) \right\}\\ &=2\left\{20\log\left(\frac{20}{15.2}\right) +18\log\left(\frac{18}{22.8}\right)\right\}\\ &\approx2(5.4887-4.2550)=2.4675. \end{aligned}

This is below 3.841, so $r=1.5$ belongs to the approximate 95% profile confidence set. Profiling has allowed each stratum's baseline to adjust at the proposed common ratio; it has not forced a common baseline rate.

Computation and Interpretation

In a general log-linear model, there may be no closed-form expression for the constrained maximum. A practical calculation repeats the following optimization for each candidate contrast.

Fit the unrestricted model.
Find the maximum log-likelihood $\ell(\hat\beta)$ and the estimated contrast $c^\top\hat\beta$.

Choose a candidate ratio.
Set the contrast to $\psi=\log r$, or equivalently impose $c^\top\beta=\log r$.

Re-maximize over nuisance coefficients.
Compute $\ell_p(\log r)$ under the constraint; do not keep nuisance coefficients fixed at their unrestricted estimates.

Compare with the cutoff.
Calculate $2\{\ell(\hat\beta)-\ell_p(\log r)\}$ and retain $r$ if it is no larger than the chosen chi-square quantile.

The retained values usually form an interval, but one should check the profile rather than assume its shape. A numerical routine can locate the endpoints by solving $\Lambda(r)=\chi^2_{1,1-\alpha}$ on either side of the maximum. For numerical stability, it is often preferable to search over $\psi=\log r$, then exponentiate the endpoints. If the profile is flat, irregular, or affected by boundary estimates, the reported result should explain that limitation rather than imply ordinary asymptotic precision.

Common Mistakes

Freezing nuisance coefficients. That evaluates the likelihood at one arbitrary nuisance fit, not the profile likelihood. Re-optimize the remaining coefficients for each fixed contrast.
Using a Wald cutoff or standard error in place of the likelihood ratio. A profile interval is obtained from the constrained and unrestricted log-likelihoods; it is not simply a standard-error calculation.
Forgetting the scale transformation. If the profile is computed for $\psi=\log r$, exponentiate its endpoints to report a rate-ratio interval.
Calling a boundary supremum a finite estimate. With zero total count in a Poisson rate model using finite log-rates, the likelihood supremum is approached as rates tend to zero; it is not attained by finite coefficients.
Interpreting the chi-square interval as exact. The usual cutoff is justified asymptotically under regularity conditions. Sparse data or boundary behavior can make it a poor approximation.

Key takeaway: A profile likelihood interval for a rate ratio is built by fixing each candidate log-rate contrast, maximizing over the remaining coefficients, and retaining candidates whose likelihood-ratio statistic is below the specified cutoff. The nuisance parameters are re-estimated at every candidate.

Check Your Understanding

Answer these questions using the definitions and calculations above.

In a model with contrast $\psi=c^\top\beta$, what is maximized when computing $\ell_p(\psi)$, and what is held fixed?
For equal-exposure Poisson groups with counts 14 and 21, write the constrained fitted means at a candidate rate ratio $r$.
Why does a profile likelihood interval for a rate ratio generally require exponentiating endpoints found on the log-ratio scale?
In a stratified model with separate baseline rates and a shared rate ratio, why must the stratum-specific baselines be re-estimated at each candidate ratio?
What is the correct interpretation of the likelihood supremum when both observed counts are zero under a finite log-rate parameterization?