Why Interactions Change Coefficient Interpretation
In a log-linear rating model without interactions, a predictor’s multiplicative effect is constant across the values of other predictors in the model. An interaction changes that interpretation: the effect of one predictor depends on the value of another. A coefficient such as \(\beta_x\) therefore describes an effect at a particular reference value of the interacting predictor, not necessarily the effect for every risk.
This distinction matters when translating a fitted model into rate indications. The model coefficients are on the log scale, while rates and relativities are usually discussed on the multiplicative scale. The reliable approach is to write the linear predictor, identify the comparison being made, and subtract the two linear predictors before exponentiating.
For the examples below, \(\mu\) denotes an expected claim frequency per unit of exposure. If a policy has exposure \(e\), the corresponding expected claim count is \(e\mu\). The exposure offset is \(\log(e)\); it adjusts expected counts for different exposure amounts but does not change the rate relativities calculated below.
Assume a log link and a correctly specified model for the conditional mean, with positive fitted rates. The examples use hypothetical rating factors and rates. For binary predictors \(A\) and \(B\), a model with an interaction is
At \(A=0,B=0\), the rate is \(\exp(\beta_0)\). Moving from \(A=0\) to \(A=1\) at \(B=0\) multiplies the rate by \(\exp(\beta_A)\). At \(B=1\), the same change in \(A\) multiplies the rate by \(\exp(\beta_A+\beta_{AB})\). Thus \(\exp(\beta_A)\) is the effect of \(A\) at the reference value \(B=0\), not automatically its effect at \(B=1\).
Deriving the Conditional Relativity
The general calculation follows directly from the log link. Compare a fitted rate at \(A=1,B=b\) with the rate at \(A=0,B=b\), keeping \(B\) fixed. Subtract the two log rates; terms common to both cancel.
At \(A=1\), the terms involving \(A\) are \(\beta_A+\beta_{AB}b\). At \(A=0\), both of those terms are zero.
The difference is \(\log(\mu_{1,b})-\log(\mu_{0,b})=\beta_A+\beta_{AB}b\).
Because the log link gives \(\log(\mu_{1,b}/\mu_{0,b})=\log(\mu_{1,b})-\log(\mu_{0,b})\), the rate ratio is \(\exp(\beta_A+\beta_{AB}b)\).
For binary \(B\), the interaction multiplier \(\exp(\beta_{AB})\) is the ratio of the conditional \(A\) relativities at \(B=1\) and \(B=0\). It is not itself the \(A\) relativity at \(B=1\); that relativity is \(\exp(\beta_A)\exp(\beta_{AB})\). The same reasoning applies with \(A\) and \(B\) exchanged.
Worked Example: Two Binary Rating Factors
A hypothetical insurer fits a frequency model for two binary indicators: \(A=1\) means a vehicle has an advanced safety package, and \(B=1\) means it is primarily used for commercial deliveries. The baseline rate at \(A=0,B=0\) is 40 claims per 100 exposure units. The fitted multipliers are \(\exp(\beta_A)=1.25\), \(\exp(\beta_B)=0.80\), and \(\exp(\beta_{AB})=1.50\). Find all four fitted rates and the conditional relativity for \(A\) at each value of \(B\).
Method. Use the baseline rate and multiply by the factor for each active main effect and by the interaction factor when \(A=B=1\). The model is \(\mu=40(1.25)^A(0.80)^B(1.50)^{AB}\).
At \(A=0,B=0\): Neither indicator is active, so \(\mu=40\).
At \(A=1,B=0\): The \(A\) main effect applies, while the \(B\) main effect and interaction do not. Thus \(\mu=40(1.25)=50\).
At \(A=0,B=1\): Only the \(B\) main effect applies. Thus \(\mu=40(0.80)=32\).
At \(A=1,B=1\): Both main effects and the interaction apply: \(\mu=40(1.25)(0.80)(1.50)=40(1.20)(1.50)=60\).
Conditional \(A\) relativities. At \(B=0\), the relativity is \(50/40=1.25\). At \(B=1\), it is \(60/32=1.875\). Their ratio is \(1.875/1.25=1.50\), exactly the interaction multiplier. This check confirms that the interaction changes the \(A\) effect by a factor of 1.50.
Common mistake avoided: Do not omit the interaction multiplier in the \(A=1,B=1\) cell. Multiplying only the main effects would give \(40(1.25)(0.80)=40\), not the model’s fitted rate of 60.
Interactions With Continuous Predictors
For two continuous predictors \(x\) and \(z\), consider
A one-unit increase in \(x\), from \(x\) to \(x+1\), at a fixed value \(z\) changes the linear predictor by \(\beta_x+\beta_{xz}z\). Its conditional relativity is therefore \(\exp(\beta_x+\beta_{xz}z)\). For an increase of \(\Delta x\) units, the relativity is \(\exp\{\Delta x(\beta_x+\beta_{xz}z)\}\). Units and reference values matter: a one-unit change means one unit in the predictor as it is actually coded.
Worked Example: A Binary Factor Interacting With a Continuous Predictor
A hypothetical model predicts expected claims per 100 exposure units. The baseline rate at \(T=0,x=0\) is 200. Here \(T=1\) indicates a selected equipment package, and \(x\) is the number of years since installation. The fitted multipliers are \(\exp(\beta_x)=1.05\), \(\exp(\beta_T)=1.20\), and \(\exp(\beta_{Tx})=0.90\). Find the fitted rate at \(T=1,x=2\), and compare the one-year \(x\) relativity with and without the package.
Method. The model is \(\mu=200(1.05)^x(1.20)^T(0.90)^{Tx}\). The interaction multiplier is applied once for each unit of the product \(Tx\).
Fitted rate. At \(T=1,x=2\), the main-effect factors are \((1.05)^2\) and \(1.20\), and \(Tx=2\), so the interaction factor is \((0.90)^2\). The calculation is \(1.05^2=1.1025\), \(0.90^2=0.81\), and \(200(1.1025)(1.20)(0.81)=200(1.323)(0.81)=200(1.07163)=214.326\). The fitted rate is 214.33 claims per 100 exposure units, rounded to the nearest cent.
One-year relativity without the package. At \(T=0\), the interaction contributes no change, so the relativity is \(1.05\).
One-year relativity with the package. At \(T=1\), the interaction modifies the \(x\) effect. The relativity is \(1.05(0.90)=0.945\). Thus, at the same package status, increasing \(x\) by one year corresponds to a 5.5% decrease in fitted rate under this model.
Check. The rate at \(T=1,x=0\) is \(200(1.20)=240\). Applying the conditional one-year relativity twice gives \(240(0.945)^2=240(0.893025)=214.326\), matching the direct calculation.
Common mistake avoided: The package main effect \(1.20\) is not the whole package effect at every \(x\). At \(x=2\), the total package-to-no-package rate ratio is \(1.20(0.90)^2=0.972\), because the interaction also contributes.
When both predictors are continuous, the same difference-in-linear-predictors method applies. The main-effect multiplier \(\exp(\beta_x)\) gives the one-unit \(x\) relativity at \(z=0\); at a different \(z\), the interaction modifies it. If the model uses centered predictors, then zero corresponds to the centering reference, and the coefficient interpretation is conditional on that value.
Worked Example: Two Continuous Predictors
A hypothetical property model has a baseline rate of 75 claims per 100 exposure units at \(x=0,z=0\). The predictors are measured in the units used to fit the model. Its fitted coefficients satisfy \(\exp(\beta_x)=1.04\), \(\exp(\beta_z)=1.03\), and \(\exp(\beta_{xz})=1.02\). Find the fitted rate at \(x=2,z=3\), and the relativity for increasing \(x\) by one unit at \(z=3\).
Method. The fitted rate is \(75(1.04)^x(1.03)^z(1.02)^{xz}\). This form follows by exponentiating each term of the log-linear predictor.
Rate at \(x=2,z=3\). The powers are \(x=2\), \(z=3\), and \(xz=6\). Evaluate the factors: \(1.04^2=1.0816\), \(1.03^3=1.092727\), and \(1.02^6=1.126162419264\). Therefore \(1.0816(1.092727)=1.1818935232\), and \(1.1818935232(1.126162419264)\approx1.3310040694\). Multiplying by the baseline gives \(75(1.3310040694)\approx99.8253052\), or 99.83 claims per 100 exposure units to the nearest cent.
One-unit \(x\) relativity at \(z=3\). It is \(\exp(\beta_x+3\beta_{xz})=1.04(1.02)^3\). Since \(1.02^3=1.061208\), the relativity is \(1.04(1.061208)=1.10365632\), or 1.1037 to four decimal places. This represents an increase of about 10.37% for a one-unit increase in \(x\) at \(z=3\), according to the fitted model.
Check. At \(z=3\), the rate factor associated with \(x=2\) is the square of the one-unit relativity: \(1.10365632^2\approx1.218057\). The \(x=0,z=3\) rate is \(75(1.03)^3=81.954525\); multiplying by \(1.218057\) gives approximately 99.83, consistent with the direct fitted rate.
Common mistake avoided: Using \(1.04\) alone for the one-unit \(x\) relativity at \(z=3\) incorrectly treats the interaction as if \(z\) were zero.
Interpreting Coefficients Carefully
For any two predictor settings, the general procedure is to calculate the difference between their linear predictors and exponentiate it. This avoids relying on an informal reading of a coefficient label. It also makes the direction of the comparison explicit: reversing the numerator and denominator gives the reciprocal relativity.
where \(\eta\) is the model’s linear predictor excluding the exposure offset when comparing rates at equal exposure. If the compared policies have different exposures, expected counts also reflect the exposure ratio; that is separate from the rate relativity.
The coefficient interpretation depends on the design and coding used in the model. With reference-coded binary predictors, zero is the reference level. For continuous predictors, zero is the value at which the main-effect coefficient is interpreted; it may be a meaningful value or a centered reference value. Do not silently change the coding or reference values when translating coefficients into rates.
For work aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus, show the comparison being made, identify the values held fixed, and include the interaction term in the rate ratio. A numerical check using fitted rates is useful: divide the two rates and confirm that the result agrees with the conditional relativity derived from the coefficients.
Check Your Understanding
For each problem, calculate the requested quantity from the stated model and reference values.
- In a model with \(\exp(\beta_A)=1.12\) and \(\exp(\beta_{AB})=0.95\), \(B\) is continuous and enters as \(AB\). Find the relativity for increasing binary \(A\) from 0 to 1 at \(B=2\). (Answer: \(1.12(0.95)^2=1.0108\))
- A model has baseline rate 80, with binary-factor multipliers \(\exp(\beta_A)=1.30\), \(\exp(\beta_B)=0.90\), and interaction multiplier \(\exp(\beta_{AB})=1.40\). Find the rate at \(A=1,B=1\). (Answer: \(80(1.30)(0.90)(1.40)=131.04\))
- For \(\log(\mu)=\beta_0+\beta_xx+\beta_zz+\beta_{xz}xz\), let \(\exp(\beta_x)=1.06\) and \(\exp(\beta_{xz})=1.03\). Find the relativity for increasing \(x\) by 2 units at \(z=4\). (Answer: \([1.06(1.03)^4]^2\approx1.4233\))
- A binary factor \(T\) interacts with continuous \(x\). The baseline rate at \(T=0,x=0\) is 120; \(\exp(\beta_x)=1.04\), \(\exp(\beta_T)=1.15\), and \(\exp(\beta_{Tx})=0.92\). Find the fitted rate at \(T=1,x=3\). (Answer: \(120(1.04)^3(1.15)(0.92)^3\approx120.88\))
- The conditional relativity for a one-unit increase in \(x\) is 1.08 at \(B=0\) and 1.35 at \(B=1\), where \(B\) is binary. Find the interaction multiplier and explain what it compares. (Answer: \(1.35/1.08=1.25\); it is the ratio of the \(x\) relativities at \(B=1\) and \(B=0\))