Rate Predictions At New Exposure Levels

From a Fitted Rate to a New Count Mean

An offset Poisson model can answer two related questions about a new setting: what event rate does the fitted model assign to its covariates, and how many events does it therefore expect over a specified exposure? Those answers are connected, but they are not the same quantity. The rate is attached to the covariate pattern; the expected count also depends on how much exposure is observed.

For example, a model may assign the same annual event rate to two people with the same modeled characteristics, even if one is followed for three months and the other for two years. Their expected counts differ because their exposure durations differ. A careful prediction states both the covariate pattern and the exposure scale, then labels the result as a rate or a count mean.

Definition: Consider an offset Poisson model for a count \(Y_i\), with positive exposure \(t_i\) and covariate vector \(x_i\). Its conditional mean is \(E[Y_i\mid x_i,t_i]=\mu_i=t_i\lambda_i\), where the event rate per unit exposure is \(\lambda_i=\exp(x_i^{\mathsf T}\beta)\). Equivalently, \(\log(\mu_i)=\log(t_i)+x_i^{\mathsf T}\beta\), where \(\log(t_i)\) is a fixed offset. For a fitted coefficient vector \(\hat{\beta}\), the fitted rate at a new covariate vector \(x_0\) is \(\hat{\lambda}_0=\exp(x_0^{\mathsf T}\hat{\beta})\), and the fitted expected count at exposure \(t_0>0\) is \(\hat{\mu}_0=t_0\exp(x_0^{\mathsf T}\hat{\beta})=t_0\hat{\lambda}_0\).

Here, \(x_0\) includes the intercept component if the model has an intercept, along with the encoded values of all other covariates. Use the same coding and units used to fit the model. The exposure \(t_0\) must be expressed in the unit used by the fitted rate. In what follows, predictions are plug-in estimates: they substitute \(\hat{\beta}\) for the unknown coefficient vector. They do not say that a realized future count must equal its conditional mean.

Calculating the Two Predictions

The calculation has a reliable order. First evaluate the linear predictor \(x_0^{\mathsf T}\hat{\beta}\). Exponentiating it gives the fitted rate per unit exposure. Only then multiply by the new exposure to obtain the fitted expected count. This order makes the role of each input visible and helps prevent a count mean from being mislabeled as a rate.

Specify the new covariate pattern.
Record every predictor value and apply the model's existing coding. Assemble them into \(x_0\), including the intercept component when appropriate.

Calculate the linear predictor.
Compute \(\hat{\eta}_0=x_0^{\mathsf T}\hat{\beta}\). This is on the log-rate scale.

Exponentiate to get the fitted rate.
Set \(\hat{\lambda}_0=\exp(\hat{\eta}_0)\). Attach the model's rate unit, such as events per person-year.

Multiply by the new exposure.
For exposure \(t_0\), calculate \(\hat{\mu}_0=t_0\hat{\lambda}_0\). This is an expected count over that exposure, not a rate.

Worked Example: A New Follow-Up Duration

Suppose a fitted model for event counts has rate \(\hat{\lambda}(x)=\exp\{\log(0.02)+x\log(2)\}\) events per person-year, where \(x=0\) denotes the reference group and \(x=1\) the comparison group. Find the fitted rate and expected count for a person in the comparison group followed for 3 person-years.

Step 1: Evaluate the linear predictor. With \(x=1\), \(\hat{\eta}=\log(0.02)+\log(2)=\log(0.04)\).

Step 2: Exponentiate for the rate. \(\hat{\lambda}=\exp\{\log(0.04)\}=0.04\) events per person-year.

Step 3: Multiply by exposure for the expected count. \(\hat{\mu}=3(0.04)=0.12\) events over 3 person-years.

The value \(0.12\) is an expected count, not the predicted rate. If the same covariate pattern were followed for 7.5 person-years, the fitted rate would remain \(0.04\) events per person-year, while the expected count would be \(7.5(0.04)=0.30\) events.

Why Exposure Changes the Count Mean, Not the Rate

The separation between rate and expected count follows directly from the offset model. For any fixed covariate pattern, the fitted rate depends on the linear predictor and fitted coefficients. Exposure enters the mean as a multiplier. The next result makes this distinction explicit.

Result: In the offset Poisson model, fix a covariate vector \(x\) and let \(t_1,t_2>0\) be two exposure amounts in the same unit. The fitted rate at \(x\) is the same for both exposure amounts, while the fitted expected counts satisfy \(\hat{\mu}(x,t_2)=(t_2/t_1)\hat{\mu}(x,t_1)\).

Proof. By the model definition, the fitted rate at \(x\) is \(\hat{\lambda}(x)=\exp(x^{\mathsf T}\hat{\beta})\), which contains no exposure term. Thus it is unchanged when \(t_1\) is replaced by \(t_2\). The fitted expected count at exposure \(t_j\) is \(\hat{\mu}(x,t_j)=t_j\hat{\lambda}(x)\), for \(j=1,2\). Therefore, \(\hat{\mu}(x,t_2)=t_2\hat{\lambda}(x)=(t_2/t_1)t_1\hat{\lambda}(x)=(t_2/t_1)\hat{\mu}(x,t_1)\). This is the claimed relationship. \(\square\)

Worked Example: A Continuous Covariate and a Binary Predictor

Suppose a model uses age centered at 50 and a binary smoking indicator \(s\), with fitted rate \(\hat{\lambda}=\exp\{\log(0.01)+(a-50)\log(1.03)+s\log(1.5)\}\) events per person-year. Predict the rate and expected count for a smoker aged 60 followed for 2.5 person-years.

Step 1: Insert the covariates. Here \(a-50=10\) and \(s=1\), so \(\hat{\eta}=\log(0.01)+10\log(1.03)+\log(1.5)\).

Step 2: Exponentiate the sum. Using the product rule for exponentials, \(\hat{\lambda}=0.01(1.03)^{10}(1.5)\). Since \((1.03)^{10}\approx1.343916\), \(\hat{\lambda}\approx0.01(1.343916)(1.5)=0.0201587\) events per person-year.

Step 3: Calculate the mean count. \(\hat{\mu}=2.5(0.0201587)\approx0.0503968\) events over 2.5 person-years.

The covariates determine the fitted rate, and the follow-up duration scales that rate to the expected count. In reporting, retain the distinction between approximately \(0.02016\) events per person-year and approximately \(0.0504\) expected events over 2.5 person-years.

Changing Covariates and Exposure Together

A new prediction may change both the covariate pattern and the exposure. In that case, calculate the new rate from the new covariates before applying the new exposure. Exposure alone does not change the modeled rate, but changing a predictor can change it because the linear predictor changes.

Result: For two new scenarios \((x_1,t_1)\) and \((x_2,t_2)\), with \(t_1,t_2>0\), the fitted expected counts are \(\hat{\mu}_1=t_1\exp(x_1^{\mathsf T}\hat{\beta})\) and \(\hat{\mu}_2=t_2\exp(x_2^{\mathsf T}\hat{\beta})\). Thus each expected count is obtained using the rate for its own covariate pattern and the exposure for its own scenario.

Derivation. The offset model gives \(\log(\mu_j)=\log(t_j)+x_j^{\mathsf T}\beta\) for scenario \(j\). Substituting the fitted coefficients gives the fitted log mean \(\log(\hat{\mu}_j)=\log(t_j)+x_j^{\mathsf T}\hat{\beta}\). Exponentiating both sides yields \(\hat{\mu}_j=\exp\{\log(t_j)+x_j^{\mathsf T}\hat{\beta}\}=t_j\exp(x_j^{\mathsf T}\hat{\beta})\). This applies separately to each specified scenario. \(\square\)

Worked Example: Predicting at a New Score and Duration

Suppose a fitted rate model is \(\hat{\lambda}(z,g)=0.05\exp(0.4g-0.02z)\) events per person-year, where \(z\) is a score and \(g=1\) indicates membership in a group. Calculate the fitted rate and expected count for \(z=5\), \(g=1\), and exposure of 4 months, expressed as \(1/3\) of a year.

Step 1: Calculate the covariate contribution. \(0.4(1)-0.02(5)=0.4-0.1=0.3\).

Step 2: Calculate the rate. \(\hat{\lambda}=0.05\exp(0.3)\approx0.05(1.349859)=0.0674929\) events per person-year.

Step 3: Calculate the count mean for 4 months. \(\hat{\mu}=(1/3)(0.0674929)\approx0.0224976\) expected events over 4 months.

If this same covariate pattern were observed over 18 months, the exposure would be \(1.5\) years. The fitted rate would still be \(0.0674929\) events per person-year, and the expected count would be \(1.5(0.0674929)\approx0.101239\) events. The arithmetic makes clear which quantity is held fixed and which changes with duration.

What a Prediction Does—and Does Not—Say

A fitted expected count is the model's conditional mean for the specified covariates and exposure, after substituting the estimated coefficients. It is not necessarily an integer, because a mean of a discrete random variable need not be an observed count. For example, an expected count of \(0.30\) does not mean that a fraction of an event will be observed. It summarizes the model's average count under the specified conditions.

This calculation also does not, by itself, describe uncertainty in the estimated rate or the variation of a future count around its mean. Those are separate questions. Here the task is to calculate the point prediction and label its scale correctly; a complete analysis should not imply that the point prediction is known without error.

Common Mistakes

Calling the expected count a rate. The wrong reasoning is, “The model gives \(0.12\), so the rate is \(0.12\).” The correction is to check the calculation: if \(0.12=3(0.04)\), it is the expected count over 3 person-years. The rate is \(0.04\) events per person-year.
Multiplying exposure into the linear predictor after exponentiating. In the offset model, the count mean is \(t\exp(x^{\mathsf T}\hat{\beta})\). Equivalently, the log mean is \(\log(t)+x^{\mathsf T}\hat{\beta}\). Exposure multiplies the rate on the mean scale; it is not an ordinary covariate effect to be interpreted as part of the rate.
Changing the rate when only duration changes. If the covariates are held fixed, the fitted rate stays the same. The expected count changes in proportion to exposure.
Using a mismatched exposure unit. A rate per person-year must be multiplied by exposure in person-years. The numerical product is not a valid expected count if the exposure is entered in a different unit without conversion.
Ignoring how the new covariates are encoded. A prediction must use the model's original reference categories, centering, and units. Inserting an uncentered age into a model fitted with age centered at 50 produces a different, incorrect linear predictor.

Key takeaway: At a new covariate pattern \(x_0\), first calculate the fitted rate \(\exp(x_0^{\mathsf T}\hat{\beta})\). Then multiply by the positive exposure \(t_0\), in matching units, to obtain the fitted expected count \(t_0\exp(x_0^{\mathsf T}\hat{\beta})\). The rate describes events per exposure unit; the expected count describes the modeled total over the specified exposure.

Check Your Understanding

For each question, identify whether the requested quantity is a rate or an expected count, and show the relevant calculation.

A fitted rate for a covariate pattern is \(0.08\) events per person-year. What is the fitted expected count over 2 person-years?
For the same covariates, how does the fitted rate change if exposure increases from 2 to 5 person-years? How does the expected count change?
A model gives a linear predictor of \(\log(0.03)\) for a new covariate pattern. Find the fitted rate and the expected count at exposure \(0.5\) person-years.
Explain why an expected count of \(0.06\) can be a valid prediction even though observed counts must be whole numbers.
A fitted rate is expressed per person-year, but the new exposure is given in months. What must be checked before multiplying the rate by the exposure?