Reference Levels and Reconstructing the Base Rate

Why Reference Levels Matter

A fitted rating model often reports an intercept and a set of coefficients rather than a table of rates for every combination of rating characteristics. To use the model, an actuary must determine which levels are represented by the intercept and how each displayed coefficient changes the fitted rate from that baseline. The omitted category in a treatment-coded factor is not missing information: it is the reference level, represented by a zero indicator.

This tutorial focuses on reconstructing the base rate and fitted cell rates in a log-link generalized linear model (GLM) with categorical predictors. The same interpretation applies whether the response is modeled as a rate directly or as a count with an exposure offset, provided the model and its offset are identified correctly. These calculations are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: In treatment coding, one level of each categorical factor is selected as its reference level. For a factor with \(k\) levels, the model uses \(k-1\) indicator variables. Each indicator equals 1 when an observation is at its corresponding nonreference level and 0 otherwise. The reference level has all of that factor’s indicators equal to 0.

The Intercept and the Base Rate

Consider a GLM with a log link and categorical factors. For a rate response, write the fitted model as

\log(\widehat{r}_i)=\beta_0+\sum_{j=1}^{J}\sum_{k\ne r_j}\beta_{jk}x_{ijk}, \qquad \widehat{r}_i=\exp\!\left(\beta_0+\sum_{j=1}^{J}\sum_{k\ne r_j}\beta_{jk}x_{ijk}\right).

Here \(\widehat{r}_i\) is the fitted rate for observation \(i\), \(\beta_0\) is the intercept, \(r_j\) denotes the reference level of factor \(j\), and \(x_{ijk}\) is the indicator for nonreference level \(k\) of factor \(j\). The corresponding coefficient is \(\beta_{jk}\). This expression assumes the stated model contains main effects only; it does not include interaction terms.

When every factor is at its reference level, all indicator variables equal zero. The linear predictor is therefore \(\beta_0\), so the fitted rate for that all-reference cell is \(\exp(\beta_0)\). This is the base rate under the specified coding and model. It is not necessarily a portfolio-wide average or a rate for the most common policy type.

For a count response with exposure \(e_i\), a common specification is a log link with an exposure offset:

\log\!\left(E[N_i]\right)=\log(e_i)+\beta_0+\sum_{j=1}^{J}\sum_{k\ne r_j}\beta_{jk}x_{ijk}.

The offset \(\log(e_i)\) has coefficient fixed at 1. Thus the fitted expected count is \(e_i\) times the fitted rate. The base rate remains \(\exp(\beta_0)\), while the expected count for a reference-level risk with exposure \(e_i\) is \(e_i\exp(\beta_0)\).

Key result: Under treatment coding and a log link, the intercept exponentiated is the fitted rate for an observation at every factor’s reference level, with any exposure offset set to zero on the log scale (equivalently, unit exposure). A nonreference coefficient changes the rate by the factor \(\exp(\beta_{jk})\), holding the other modeled predictors fixed.

Why Exponentiating Reconstructs the Rate

The reconstruction follows directly from the link function and indicator values. For the all-reference cell, every indicator is zero, so the log fitted rate is \(\beta_0\). Applying the inverse of the log function gives \(\exp(\beta_0)\). If one factor is instead at a nonreference level \(k\), its indicator is 1, and the log fitted rate increases by \(\beta_{jk}\). Taking the difference between the two log rates and exponentiating shows the multiplicative change:

\frac{\widehat{r}_{\text{nonreference}}}{\widehat{r}_{\text{reference}}} = \frac{\exp(\beta_0+\beta_{jk})}{\exp(\beta_0)} = \exp(\beta_{jk}).

With several main-effect factors at nonreference levels, their indicator contributions add on the log scale. Exponentiation turns that sum into a product of rate multipliers. This reasoning depends on the stated log-link model. A coefficient from a model using a different link cannot automatically be interpreted or reconstructed this way.

Read the model specification.
Confirm the response, link, included factors, reference levels, and whether an exposure offset is present.

Set the indicators.
For the requested cell, assign 1 to each applicable nonreference-level indicator and 0 to the others.

Build the linear predictor.
Add the intercept and the coefficients whose indicators equal 1. Include the log exposure only when calculating a count-model mean.

Apply the inverse link and check units.
Exponentiate to obtain a rate or, if the offset was included, an expected count. Compare the result with the baseline and the coefficient multipliers.

Worked Example: One Factor and a Fitted Base Rate

A hypothetical property rating model uses a log link for claim frequency. Construction class has levels Standard, Reinforced, and Specialty, with Standard as the reference. The fitted model has intercept \(\log(600)\), a Reinforced coefficient \(\log(1.30)\), and a Specialty coefficient \(\log(0.75)\). Find the base rate and each fitted class rate. State the method and a mistake to avoid.

Method. Use treatment-coded indicators and exponentiate the linear predictor. Since this is a frequency rate model, the results are fitted claim frequencies per unit of exposure.

For Standard, both nonreference indicators are 0. The linear predictor is \(\log(600)\), so

\widehat{r}_{\text{Standard}}=\exp(\log(600))=600.

For Reinforced, its indicator is 1 and the Specialty indicator is 0:

\widehat{r}_{\text{Reinforced}} =\exp(\log(600)+\log(1.30)) =600(1.30)=780.

For Specialty, its indicator is 1 and the Reinforced indicator is 0:

\widehat{r}_{\text{Specialty}} =\exp(\log(600)+\log(0.75)) =600(0.75)=450.

The fitted base rate is 600 for Standard construction. As a check, the Specialty multiplier is below 1, so its fitted rate, 450, is below the base rate. Common mistake avoided: treating the intercept \(\log(600)\) itself as the rate. It is the log of the base rate; exponentiating it gives 600.

Combining Factors and Handling Exposure

For multiple main-effect factors, the all-reference cell is defined by taking the reference level of every factor simultaneously. A cell’s fitted rate is obtained by adding the relevant coefficients to the intercept on the log scale, or equivalently multiplying the base rate by the exponentiated coefficients. The rate is not changed by a count-model exposure offset; the offset scales the expected count for the given exposure.

Worked Example: Two Factors With an Exposure Offset

A hypothetical auto frequency model has two factors: vehicle type, with Compact as the reference and SUV as a nonreference level; and territory, with Rural as the reference and Urban as a nonreference level. The intercept is \(\log(420)\), the SUV coefficient is \(\log(1.20)\), and the Urban coefficient is \(\log(1.50)\). The count model includes offset \(\log(e_i)\). Find the base rate, the fitted rate for an Urban SUV, and its fitted expected count at exposure \(e_i=0.8\).

Method. First reconstruct the rate from the intercept and applicable factor coefficients. Then multiply that rate by exposure to obtain the expected count.

For the Compact Rural reference cell, the fitted rate is

\widehat{r}_{\text{Compact,Rural}}=\exp(\log(420))=420.

For an Urban SUV, both nonreference indicators equal 1. Its rate is

\widehat{r}_{\text{SUV,Urban}} =\exp(\log(420)+\log(1.20)+\log(1.50)) =420(1.20)(1.50)=756.

At exposure 0.8, the fitted expected count is

\widehat{E[N_i]}=0.8(756)=604.8.

The offset changes the expected count from the unit-exposure rate; it does not change the reconstructed rate of 756. As a check, the two multipliers combine to 1.80, and \(420(1.80)=756\). Common mistake avoided: multiplying the base rate by exposure before identifying the rate factors can obscure the distinction between a rate and an expected count. Apply both components, but keep their roles separate.

Reading a Coefficient Table and Checking the Cell

A coefficient table usually displays the intercept and coefficients for nonreference levels. The reference levels are inferred from the model’s factor specification or coding: they are the levels without their own indicator coefficient. They cannot always be identified from a list of coefficient estimates alone if the factor levels or coding scheme are not supplied. Before reconstructing a rate, verify the stated reference levels rather than guessing from coefficient size or from which level seems most common.

A useful check is to write the requested cell’s indicators explicitly. This prevents adding a coefficient for a level that is not present. It also distinguishes a cell at the reference level of one factor from the all-reference cell, which must be at the reference level of every factor.

Worked Example: Reconstructing a Cell From Multiple Coefficients

A hypothetical small-commercial GLM models loss cost with a log link and no exposure offset in its cell-rate equation. The factors are coverage plan (Bronze, Silver, Gold) and region (North, South). Bronze and North are the references. The fitted intercept is \(\log(900)\); the Silver coefficient is \(\log(4/3)\), the Gold coefficient is \(\log(5/3)\), and the South coefficient is \(\log(0.60)\). Find the all-reference base rate, the fitted Gold-North rate, and the fitted Gold-South rate.

Method. Translate each named level into its indicator value, add only the corresponding coefficients to the intercept, and exponentiate.

For Bronze-North, all indicators are 0. Thus

\widehat{r}_{\text{Bronze,North}}=\exp(\log(900))=900.

For Gold-North, the Gold indicator is 1 and the South indicator is 0:

\widehat{r}_{\text{Gold,North}} =\exp(\log(900)+\log(5/3)) =900\left(\frac{5}{3}\right)=1500.

For Gold-South, both the Gold and South indicators are 1:

\widehat{r}_{\text{Gold,South}} =\exp(\log(900)+\log(5/3)+\log(0.60)) =900\left(\frac{5}{3}\right)(0.60)=900.

The reconstructed Gold-South rate returns to 900 because \((5/3)(0.60)=1\). This equality is a consequence of the specified coefficients, not a claim that the two cells are the same in every respect. Common mistake avoided: adding the Bronze coefficient or using a South coefficient for North. Both reference levels have zero indicators and contribute no coefficient.

Assumptions, Limits, and Exam Traps

The calculations above require the model specification to be understood correctly. In particular, the base rate interpretation assumes a log link and treatment coding with an intercept. If a different coding scheme is used, the intercept may represent a different quantity. If the model has interactions, a cell’s predictor may also include interaction terms; the main-effects-only calculation cannot then be used without those terms. For a count model, an offset must be included when calculating the fitted mean count, but not mistaken for a rating-factor coefficient.

A coefficient table can also be presented on a transformed scale or may show standard errors alongside estimates. Use the coefficient estimates in the linear predictor; do not exponentiate a standard error as though it were a rate multiplier. Keep the rate’s units attached to the base rate, and report expected counts in the units implied by the exposure.

Question	Correct interpretation	Frequent error
What is the base rate?	Exponentiate the intercept for the all-reference cell.	Report the log-scale intercept itself as a rate.
Which coefficients enter a cell calculation?	Use coefficients whose nonreference indicators equal 1.	Add every displayed coefficient in the table.
What does an exposure offset do?	Scales the rate to a fitted expected count for the given exposure.	Treat exposure as another rating relativity or omit it from a count mean.
Can this main-effects calculation always be used?	Only when it matches the specified model; account for any additional modeled terms.	Ignore interactions or assume the intercept has the same meaning under all coding schemes.

Key takeaway: Identify the coding and references first. Under an intercept-and-main-effects log-link model, exponentiate the intercept for the all-reference base rate, multiply by the exponentiated coefficients for the selected nonreference levels, and apply exposure separately when calculating a count mean.

Check Your Understanding

For each problem, identify the relevant reference cell and distinguish a rate from an expected count where needed.

A log-link frequency model has one factor, deductible group, with Low as the reference. Its intercept is \(\log(250)\), and the High coefficient is \(\log(1.40)\). What are the base rate and the High-group rate? (Answer: Low reference base rate \(250\); High-group rate \(350\).)
A main-effects model has a base rate of 800. The Urban coefficient is \(\log(0.90)\), and the Heavy-vehicle coefficient is \(\log(1.25)\). Both indicators are 1 for a particular cell. What is its fitted rate? (Answer: \(800(0.90)(1.25)=900\).)
A count model has a base rate of 500, coefficients \(\log(1.10)\) for a nonreference territory and \(\log(0.70)\) for a nonreference class, and offset \(\log(e_i)\). Find the expected count when both indicators equal 1 and exposure is 2.4. (Answer: Rate \(500(1.10)(0.70)=385\); expected count \(2.4(385)=924\).)
A two-factor model has an intercept of \(\log(1200)\). Factor A has reference level A0 and coefficient \(\log(1.50)\) for A1. Factor B has reference level B0 and coefficient \(\log(2/3)\) for B1. What is the all-reference base rate and the A1-B1 fitted rate? (Answer: Base rate \(1200\); A1-B1 rate \(1200(1.50)(2/3)=1200\).)

What You'll Learn