Combining Fitted Effects Into a Policy-Level Indication

From Fitted Coefficients to a Policy-Level Prediction

A fitted GLM describes how a response changes with the predictors in the model. To use it for a particular policy, identify that policy’s predictor values, assemble the linear predictor, apply the specified link function, and account for exposure if it is part of the model. The result may be a fitted rate, an expected count, or another conditional mean, depending on the response and model specification.

For ratemaking models with a log link, coefficients combine additively on the log scale and multiplicatively on the rate scale. This makes it possible to calculate a policy-level indication from a base rate and the fitted effects that apply to that policy. The calculation is valid only when the link, coefficient coding, predictor values, and offset are interpreted as specified.

Definition: For policy $i$, the linear predictor is the sum of the intercept, the applicable predictor terms, and any specified offset. With a log link and exposure $e_i$, a common count model is $$ \log(\mu_i)=\log(e_i)+\beta_0+\sum_{j=1}^{p}\beta_jx_{ij}, $$ where $\mu_i$ is the conditional expected count, $e_i$ is exposure, $\beta_0$ is the intercept, and $x_{ij}$ are the policy’s predictor values. The fitted rate per exposure unit is $\mu_i/e_i$.

In this model, the offset is $\log(e_i)$, not $e_i$ itself. Its coefficient is fixed at one: it adjusts the expected count in proportion to exposure without estimating a separate exposure coefficient. If the model instead predicts a rate directly, it may have no exposure offset; multiply the fitted rate by exposure only when a total expected count is needed.

For a categorical predictor represented by treatment indicators, a policy receives the coefficient for its selected non-reference level, while the reference level contributes zero. A continuous predictor contributes its coefficient multiplied by its recorded value, measured in the units and relative to the origin used in the model. If an interaction is present, include its term only when the policy has the predictor combination represented by that interaction.

Why Log-Link Effects Multiply

Suppose a policy’s log rate is the intercept plus a set of applicable effects. Exponentiating that sum gives the rate. The exponential of a sum is the product of the exponentials, so each fitted coefficient becomes a multiplicative factor on the rate scale.

\log(r_i)=\beta_0+\beta_1x_{i1}+\beta_2x_{i2} \quad\Longrightarrow\quad r_i=\exp(\beta_0)\exp(\beta_1x_{i1})\exp(\beta_2x_{i2}).

For a binary indicator $x_{ij}$, $\exp(\beta_j)$ is the factor for the indicated level relative to its reference, conditional on the other terms in the model. For a continuous predictor, a one-unit increase multiplies the fitted rate by $\exp(\beta_j)$, provided the other predictors are held fixed and the model has no unaccounted interaction affecting that comparison. If an interaction is included, the relevant effect depends on the other predictor’s value.

This result follows directly from the log-link model; it does not imply that coefficients themselves are rate factors. The coefficient is on the log scale. Exponentiating the coefficient, or the relevant coefficient sum, is what produces a rate-scale multiplier.

A Reliable Calculation Sequence

Identify the target.
Determine whether the requested quantity is a fitted rate, expected count, or another model mean. Note the response and its units.

Translate the policy into predictors.
Record its categorical levels, continuous values and units, and whether each interaction condition applies. Respect the reference coding.

Assemble the specified linear predictor.
Add the intercept and applicable predictor terms. Include $\log(e_i)$ only when exposure is an offset in the count model.

Apply the inverse link and check units.
For a log link, exponentiate the predictor. With an exposure offset, this gives the expected count; remove the exposure factor to report the rate per exposure unit.

Worked Policy-Level Calculations

Worked Example: A Policy’s Fitted Claim Frequency

A hypothetical insurer fits a Poisson model for claim counts with a log link. Counts are conditional on the rating variables and have mean equal to variance. Exposure $e$ is measured in car-years, and the model is $\log(\mu_i)=\log(e_i)+\beta_0+\beta_1Y_i+\beta_2U_i$. Here $Y_i=1$ for a young-driver policy and $U_i=1$ for an urban-territory policy; zero denotes the relevant reference level. The fitted coefficients are $\beta_0=\log(0.06)$, $\beta_1=\log(1.4)$, and $\beta_2=\log(1.25)$. Find the fitted rate and expected count for a young-driver urban policy with $e_i=0.75$ car-years.

Method. Evaluate the log-link count model for this policy, then separate the fitted rate from the exposure-adjusted expected count.

Step 1: Substitute the policy values. Both indicators equal one, so the predictor excluding the offset is $\log(0.06)+\log(1.4)+\log(1.25)$.

Step 2: Convert to the fitted rate. Exponentiating the sum gives $0.06\times1.4\times1.25=0.084\times1.25=0.105$ claims per car-year.

Step 3: Apply exposure. The expected count is $\mu_i=0.75\times0.105=0.07875$ claims.

Check. The policy has less than one car-year of exposure, so its expected count, $0.07875$, is below the annualized rate of $0.105$. Rounded only at the final step to four decimal places, the expected count is $0.0788$; the rate is $0.105$ claims per car-year.

Common mistake avoided. Adding the coefficients to the base rate would give $0.06+\log(1.4)+\log(1.25)$, which mixes rate-scale and log-scale quantities. The correction is to exponentiate the sum of the log-scale terms, equivalently multiplying the rate by the two factors.

Worked Example: Applying an Interaction Only to the Joint Level

A hypothetical property insurer models claim counts using a Poisson distribution and log link. Exposure $e_i$ is measured in property-years. Let $A_i=1$ indicate a specified roof material and $B_i=1$ indicate a monitored alarm. The fitted model is $\log(\mu_i)=\log(e_i)+\beta_0+\beta_AA_i+\beta_BB_i+\beta_{AB}A_iB_i$. The base rate is $0.04$ claims per property-year, and the three coefficient exponentials are $\exp(\beta_A)=1.3$, $\exp(\beta_B)=0.9$, and $\exp(\beta_{AB})=1.2$. Find the fitted rate and expected count for a policy with both features and $2.4$ property-years of exposure.

Method. Use the interaction model as specified. The interaction term is active only when both indicators equal one.

Step 1: Evaluate the indicator products. Here $A_i=B_i=1$, so $A_iB_i=1$. The rate includes the base rate, both main-effect factors, and the interaction factor.

Step 2: Calculate the rate. $0.04\times1.3=0.052$, $0.052\times0.9=0.0468$, and $0.0468\times1.2=0.05616$ claims per property-year.

Step 3: Calculate the expected count. $\mu_i=2.4\times0.05616=0.134784$ claims.

Check. The fitted rate is above the $0.04$ base rate because the combined factor is $1.3\times0.9\times1.2=1.404$. Multiplying $0.04$ by $1.404$ also gives $0.05616$. Rounded only at the final step to four decimal places, the expected count is $0.1348$.

Common mistake avoided. Omitting the interaction would give $0.04\times1.3\times0.9=0.0468$, and an expected count of $2.4\times0.0468=0.11232$. That calculation is appropriate only when the interaction indicator is zero; for this policy both conditions hold, so the factor $1.2$ must be included.

Worked Example: Evaluating a Continuous Predictor in Its Stated Units

A hypothetical equipment insurer models counts of specified breakdown events with a Poisson log-link model. Exposure is measured in units of one thousand machine-hours, and $x$ is machine age in ten-year increments above the model’s reference age. The model is $\log(\mu_i)=\log(e_i)+\beta_0+\beta_1x_i$, where $\exp(\beta_0)=0.02$ events per thousand machine-hours at the reference age and $\exp(\beta_1)=1.05$. Find the fitted rate and expected count for equipment four increments above the reference age with $e_i=3.2$.

Method. Use the continuous-predictor coefficient once for each increment in $x$, then apply exposure in the model’s stated units.

Step 1: Determine the continuous term. Four increments mean $x_i=4$. Thus the predictor excluding the offset is $\beta_0+4\beta_1$.

Step 2: Calculate the fitted rate. $\exp(\beta_0+4\beta_1)=0.02\times1.05^4$. Since $1.05^2=1.1025$, $1.05^4=1.21550625$, and $0.02\times1.21550625=0.024310125$ events per thousand machine-hours.

Step 3: Calculate the expected count. $\mu_i=3.2\times0.024310125=0.0777924$ events.

Check. Four increments produce a factor greater than one, so the fitted rate exceeds the reference rate of $0.02$. The exposure is $3.2$ thousand machine-hours, not $3.2$ individual machine-hours. Rounded only at the final step to four decimal places, the expected count is $0.0778$.

Common mistake avoided. Using $1.05$ just once ignores the predictor value of four. Using $1.05^{40}$ would instead misread the recorded predictor: $x$ is already measured in ten-year increments, and its value in this example is four.

Link Functions, Exposure, and Interpretation

The inverse link determines how a linear predictor becomes a fitted mean. Under a log link, exponentiation produces a positive mean and additive terms become multiplicative effects. Under an identity link, the fitted mean is the linear predictor itself; coefficients add on the response scale. Therefore, do not turn every coefficient into a multiplicative factor. First check which link the model specifies.

An offset also needs to be interpreted in context. In a count model with log exposure as an offset, the fitted count changes proportionally with exposure while the fitted rate per unit of exposure is determined by the remaining terms. In a model that predicts a rate directly, there may be no offset at all. If the task asks for expected total count from that rate, multiply by exposure after predicting the rate.

Key takeaway: A policy-level indication is a prediction from the specified fitted model, not a new estimate of its coefficients. For a log-link model, evaluate the applicable terms, exponentiate to obtain the rate or mean, and apply exposure exactly once when converting a rate to an expected count.

Common Checks Before Reporting an Indication

Keep scales consistent. Add coefficients on the linear-predictor scale; multiply exponentiated effects on the rate scale.
Use the actual coding. A reference category contributes no treatment indicator effect, and an interaction applies only to the represented combination.
Respect predictor units. A continuous coefficient is per one stated unit or increment, not automatically per year, percentage point, or raw measurement unit.
Apply exposure once. With a log-exposure offset, the full inverse-link result is an expected count. Do not multiply by exposure again.
Report the requested quantity. A rate and an expected count have different units even when they come from the same model.

Check Your Understanding

For each problem, use the stated model and report the requested fitted quantity.

A hypothetical count model has a log link, a base rate of $0.05$ claims per delivery-route year, and a route factor of $1.6$ for a selected territory. A policy has $0.5$ route-years of exposure. Find its fitted rate and expected count. (Answer: rate $0.08$ claims per route-year; expected count $0.04$.)
A hypothetical property-count model has a base rate of $0.12$ claims per property-year. A roof factor is $0.75$, a region factor is $1.2$, and exposure is $1.5$ property-years. Find the fitted rate and expected count. (Answer: rate $0.108$ claims per property-year; expected count $0.162$.)
A log-link count model has base rate $0.03$ events per exposure unit, main-effect factors $1.4$ and $1.1$, and an interaction factor $0.8$ when both features apply. For a policy with both features and exposure $2$, find the expected count. (Answer: $0.07392$ events.)
A log-link model has reference rate $0.025$ events per thousand operating-hours and a continuous-predictor factor of $1.08$ per stated increment. A policy is three increments above reference and has exposure $4$ thousand operating-hours. Find its fitted rate and expected count. (Answer: rate $0.0314928$, approximately $0.0315$ events per thousand operating-hours; expected count $0.1259712$, approximately $0.1260$.)
A stated identity-link model predicts annual cost per exposure unit as $m=240+35A-20B$, where $A$ and $B$ are binary indicators. For a policy with $A=1$, $B=1$, and $1.5$ exposure units, find the fitted cost per unit and total expected cost. (Answer: fitted cost $255$ dollars per exposure unit; total expected cost $\$382.50$.)

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