Rate Ratios From Offset Models

Why Compare Rates Through Coefficients?

In an offset Poisson model, exposure determines the scale of the expected count, while regression coefficients describe how the rate changes with covariates. This separation makes comparisons of rates particularly direct: a ratio of rates is an exponential of a difference in linear predictors. The same calculation also shows why the ratio of expected counts is not generally the rate ratio when the exposure amounts differ.

The central task is to identify the two covariate patterns being compared, subtract their coefficient contributions, and use the fitted covariance matrix to quantify uncertainty in that contrast. This tutorial focuses on those comparisons and their confidence intervals.

Rate Ratios as Linear Contrasts

For observation \(i\), let \(Y_i\) be a Poisson count and let \(E_i>0\) be its known exposure. An offset Poisson regression model has conditional mean

Y_i\mid x_i,E_i\sim \operatorname{Poisson}(\mu_i), \qquad \log(\mu_i)=\log(E_i)+x_i^{\mathsf T}\beta.

Here \(x_i\) is the covariate vector and \(\beta\) is the coefficient vector. The corresponding modeled rate per unit of exposure is \(\lambda_i=\mu_i/E_i\). Therefore,

\lambda_i=\exp(x_i^{\mathsf T}\beta).

To compare two covariate patterns \(x_A\) and \(x_B\), define their rate ratio as the rate for pattern \(B\) divided by the rate for pattern \(A\). The order matters: reversing the comparison reciprocates the ratio.

Definition: For two covariate patterns \(x_A\) and \(x_B\), the log rate ratio for \(B\) relative to \(A\) is the linear contrast \(c^{\mathsf T}\beta\), where \(c=x_B-x_A\). The rate ratio is \(\exp(c^{\mathsf T}\beta)\). A linear contrast is a fixed weighted sum of model coefficients.

If both patterns have the same values of all covariates except a binary group indicator, \(c\) selects the coefficient for that indicator. For more involved comparisons, several coefficients can contribute. For instance, the contrast between two non-reference categories is typically a difference between two coefficients.

Result: With the exposure offset specified as above, the ratio of expected counts at patterns \(B\) and \(A\) is \(\frac{E_B}{E_A}\exp\{(x_B-x_A)^{\mathsf T}\beta\}\). The rate ratio is just \(\exp\{(x_B-x_A)^{\mathsf T}\beta\}\), because each expected count is divided by its own exposure.

This distinction is important whenever the two records have different exposure amounts. Equal covariate patterns have equal modeled rates, even if one record has ten times the exposure and thus ten times the expected count.

Worked Example: A Binary Group Comparison

Suppose a fitted model for event counts has the form \(\log(\mu_i)=\log(E_i)+\beta_0+\beta_1G_i\), where \(G_i=0\) denotes group A and \(G_i=1\) denotes group B. The fitted group coefficient is \(\hat\beta_1=0.405\). Find the estimated rate ratio for B relative to A.

For A, the rate is \(\exp(\beta_0)\). For B, it is \(\exp(\beta_0+\beta_1)\). Dividing the B rate by the A rate cancels the intercept: \[ \frac{\exp(\beta_0+\beta_1)}{\exp(\beta_0)} =\exp(\beta_1). \] Thus the estimated rate ratio is \(\exp(0.405)\approx1.499\), or about 1.50. In this model, the fitted rate in B is about 50% higher than the fitted rate in A.

If the two records had exposures of 200 and 500 person-years, respectively, their expected-count ratio would instead be \((500/200)\exp(0.405)\approx2.5(1.499)=3.748\). That is a ratio of expected counts, not a rate ratio.

Uncertainty for a Rate Ratio

A rate ratio is positive and often easier to estimate on the log scale. Let \(\hat\beta\) be the fitted coefficient vector, and let \(\widehat{\operatorname{Var}}(\hat\beta)\) be its estimated covariance matrix. For a specified contrast vector \(c\), the estimated log rate ratio is \(c^{\mathsf T}\hat\beta\), and its estimated standard error is \(\sqrt{c^{\mathsf T}\widehat{\operatorname{Var}}(\hat\beta)c}\). This expression uses the variances and covariances of all coefficients in the contrast.

Under the usual large-sample conditions for maximum likelihood estimation—independent observations, a correctly specified Poisson mean model with fixed known positive exposures, an identifiable parameter, and a finite interior true parameter with nonsingular information—the estimator is asymptotically normal. A Wald interval for the log rate ratio is consequently formed by adding and subtracting a standard-normal critical value times its estimated standard error. Exponentiating both endpoints gives an interval for the rate ratio.

Wald confidence interval: For a nominal \(100(1-\alpha)\%\) interval, let \(z_{1-\alpha/2}\) be the \(1-\alpha/2\) quantile of the standard normal distribution and let \(s_c=\sqrt{c^{\mathsf T}\widehat{\operatorname{Var}}(\hat\beta)c}\). The interval for the rate ratio is \[ \left( \exp\{c^{\mathsf T}\hat\beta-z_{1-\alpha/2}s_c\}, \, \exp\{c^{\mathsf T}\hat\beta+z_{1-\alpha/2}s_c\} \right). \]

Deriving the Contrast Interval

The interval follows from the variance formula for a linear combination and the approximate normal distribution of the fitted coefficients. The covariance terms cannot in general be discarded: two coefficient estimates from the same model may be correlated.

Form the contrast.
For the chosen patterns, set \(c=x_B-x_A\). The target log rate ratio is \(c^{\mathsf T}\beta\), estimated by \(c^{\mathsf T}\hat\beta\).

Calculate its estimated variance.
For any random vector with covariance matrix \(\Sigma\), the variance of \(c^{\mathsf T}\hat\beta\) is \(c^{\mathsf T}\Sigma c\). Estimate it by \(c^{\mathsf T}\widehat{\operatorname{Var}}(\hat\beta)c\).

Make the normal-scale interval.
Use the estimated standard error and the chosen normal critical value to form an interval for the log rate ratio.

Transform to the rate-ratio scale.
Exponentiation is increasing, so exponentiating the lower and upper log-scale endpoints preserves their order and gives the interval for the rate ratio.

For completeness, the variance identity follows by expanding the centered contrast. If \(\Sigma=\operatorname{Var}(\hat\beta)\), then

\begin{aligned} \operatorname{Var}(c^{\mathsf T}\hat\beta) &=E\!\left[\{c^{\mathsf T}(\hat\beta-E[\hat\beta])\}^2\right]\\ &=E\!\left[c^{\mathsf T}(\hat\beta-E[\hat\beta]) (\hat\beta-E[\hat\beta])^{\mathsf T}c\right]\\ &=c^{\mathsf T}\Sigma c. \end{aligned}

The final equality holds because \(c\) is fixed, so it can be taken outside the expectation. To see the interval construction, asymptotic normality gives \((c^{\mathsf T}\hat\beta-c^{\mathsf T}\beta)/s_c\) approximately standard normal when \(s_c\) consistently estimates the contrast's standard deviation. Thus the approximate probability that the true log ratio lies between the two Wald endpoints is \(1-\alpha\). Since exponentiation is strictly increasing, the same coverage approximation holds after transforming the endpoints. This is an asymptotic, not an exact finite-sample, confidence interval. \(\square\)

Worked Example: Comparing Two Non-Reference Groups

Suppose group A is the reference category, and the model includes indicators for groups B and C. The fitted coefficients are \(\hat\beta_B=0.25\) and \(\hat\beta_C=-0.20\). Their estimated variances are \(0.0400\) and \(0.0625\), and their estimated covariance is \(0.0100\). Find the estimated rate ratio for C relative to B and a 95% Wald interval.

The log rate for C minus the log rate for B is \(\beta_C-\beta_B\), so the estimated log ratio is \(-0.20-0.25=-0.45\). The estimated rate ratio is \(\exp(-0.45)\approx0.638\).

For the contrast \(\hat\beta_C-\hat\beta_B\), the estimated variance is \[ 0.0625+0.0400-2(0.0100)=0.0825. \] Therefore the standard error is \(\sqrt{0.0825}\approx0.2872\). The 95% log-scale interval is \[ -0.45\pm1.96(0.2872) \approx(-1.0130,\;0.1130). \] Exponentiating the endpoints gives approximately \[ (\exp(-1.0130),\exp(0.1130))\approx(0.363,\;1.120). \] The point estimate suggests a lower rate in C than in B, but this interval includes 1, the value corresponding to equal rates.

Contrasts for Continuous Predictors and Interactions

A contrast need not compare categories. If a continuous predictor \(z\) enters the model linearly with coefficient \(\beta_z\), increasing \(z\) by \(\Delta\) units while holding the other covariates fixed gives log rate ratio \(\Delta\beta_z\). The estimated standard error is \(|\Delta|\) times the standard error of \(\hat\beta_z\). This makes the unit of comparison explicit: a one-unit contrast and a ten-unit contrast are different rate ratios.

With an interaction, the group contrast can depend on the value of another predictor. For example, if the linear predictor includes \(\beta_GG+\beta_{Gz}Gz\), the log rate ratio for \(G=1\) versus \(G=0\) at a fixed \(z\) is \(\beta_G+z\beta_{Gz}\), not just \(\beta_G\). Its variance must include the covariance between the two coefficient estimates.

Worked Example: A Five-Unit Increase

Suppose a model has a continuous predictor \(z\) with fitted coefficient \(\hat\beta_z=0.080\) and estimated standard error \(0.025\). Find the rate ratio and 95% Wald interval for an increase of five units, holding other predictors fixed.

The contrast is \(5\beta_z\), so the estimated log rate ratio is \(5(0.080)=0.400\), and its standard error is \(5(0.025)=0.125\). The estimated rate ratio is \(\exp(0.400)\approx1.492\).

The log-scale interval is \[ 0.400\pm1.96(0.125)=0.400\pm0.245 =(0.155,\;0.645). \] Exponentiating gives \((\exp(0.155),\exp(0.645))\approx(1.168,\;1.906)\). Thus the fitted rate for a five-unit increase is about 1.49 times as large, with a 95% Wald interval from about 1.17 to 1.91.

Common Pitfalls and Interpretation

A rate ratio is a comparison of modeled rates for two specified covariate patterns. It is not automatically a comparison of population-average rates across groups: the latter may require averaging over a covariate distribution, a different task. Here the calculation is for the patterns stated in the contrast.

Using the wrong order. A contrast \(x_B-x_A\) gives B relative to A. Reversing the order changes the sign of the log ratio and takes the reciprocal of the ratio.
Dropping covariance terms. For a contrast involving multiple coefficients, use \(c^{\mathsf T}\widehat{\operatorname{Var}}(\hat\beta)c\). Adding variances alone is generally incorrect.
Comparing counts when the question asks for rates. The expected-count ratio includes the exposure ratio. Divide each expected count by its exposure before interpreting a rate ratio.
Reporting a symmetric interval around the ratio. The Wald interval is symmetric on the log scale; after exponentiation it is generally asymmetric on the rate-ratio scale.
Ignoring interactions or the comparison increment. State the covariate patterns and units. With an interaction, the relevant contrast may change with the fixed covariate value.

The interval relies on the large-sample normal approximation for the coefficient estimator and on the fitted model being suitable for the data. Its endpoints quantify uncertainty in a particular coefficient contrast; they do not account for uncertainty in an exposure that was treated as known, or for model misspecification.

Key takeaway: Specify the two covariate patterns, subtract them to obtain \(c\), and calculate the log rate ratio \(c^{\mathsf T}\hat\beta\). Use the full estimated covariance matrix for its standard error, then exponentiate the log-scale confidence limits.

Check Your Understanding

For each question, identify the contrast before interpreting the result.

In a model with a binary group indicator, what coefficient contrast gives the rate ratio for the indicator value 1 relative to 0?
Two covariate patterns have exposures of 300 and 600 units. How does their expected-count ratio relate to their rate ratio?
For a log rate ratio estimated as \(0.30\) with standard error \(0.10\), write the 95% Wald interval on the log scale before exponentiating.
Why does the variance of \(\hat\beta_2-\hat\beta_1\) include a covariance term?
In a model with a group-by-continuous-predictor interaction, which quantities must be specified to define the group rate ratio?

What You'll Learn