5.4 Interaction Terms & Non-Linear Effects

Theoretical Foundations of Predictor Interactions in GLMs

In standard Generalized Linear Models, rating predictors are assumed to operate additively on the scale of the link function (and multiplicatively on the original loss scale under a log link). This implies that the effect of any single rating variable is constant across all levels of all other variables:

$\eta_i = g(\mu_i) = \beta_0 + \sum_{j=1}^p \beta_j x_{ij}$

However, in physical insurance risk processes, rating variables frequently interact. An interaction occurs when the marginal effect of variable $X_1$ on expected loss cost depends on the specific level or value of variable $X_2$:

$\frac{\partial g(\mu)}{\partial X_1} = f(X_2) \ne \text{constant}$

To capture this joint behavior, the linear predictor must incorporate an interaction product term:

$\eta = g(\mu) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} (X_1 \cdot X_2)$

The Log-Link Multiplicative Nuance

A critical conceptual distinction tested on CAS exams is the difference between multiplicative main effects and interaction terms under a log-link GLM ($g(\mu) = \ln(\mu)$).

                  LOG-LINK GLM: MAIN EFFECTS VS. INTERACTIONS
                                       │
         ┌─────────────────────────────┴─────────────────────────────┐
         ▼                                                           ▼
Main Effects Only Model                                Model With Interaction Term
ln(μ) = β₀ + β₁X₁ + β₂X₂                               ln(μ) = β₀ + β₁X₁ + β₂X₂ + β₁₂X₁X₂
μ = exp(β₀) · exp(β₁X₁) · exp(β₂X₂)                    μ = exp(β₀) · exp(β₁X₁) · exp(β₂X₂) · exp(β₁₂X₁X₂)
Main effects NATURALLY MULTIPLY!                       exp(β₁₂X₁X₂) is the INTERACTION ADJUSTMENT
Independent compounding assumed                       Captures departure from pure multiplicativity

1. In a Main-Effects Log-Link Model: Exponentiating the linear predictor yields: $\mu = \exp(\beta_0 + \beta_1 X_1 + \beta_2 X_2) = \exp(\beta_0) \times \exp(\beta_1 X_1) \times \exp(\beta_2 X_2)$ Because the log link converts addition into multiplication, main effects naturally compound multiplicatively. If being a youthful driver has a relativity of $1.50$ ($\beta_1 = 0.405$) and living in an urban territory has a relativity of $1.40$ ($\beta_2 = 0.336$), the main-effects model assumes the combined risk factor is simply $1.50 \times 1.40 = 2.10$.
2. In an Interaction Log-Link Model: Adding the product term yields: $\mu = \exp(\beta_0) \times \exp(\beta_1 X_1) \times \exp(\beta_2 X_2) \times \exp(\beta_{12} X_1 X_2)$ The term $\exp(\beta_{12} X_1 X_2)$ represents an interaction multiplier that adjusts for departures from pure multiplicativity:
   • If $\beta_{12} = 0$, the relationship is perfectly multiplicative.
   • If $\beta_{12} > 0$, the joint risk is super-multiplicative (the combined risk is greater than the product of the individual relativities).
   • If $\beta_{12} < 0$, the joint risk is sub-multiplicative (the combined risk is less than the product of the individual relativities).

Classic Property & Casualty Actuarial Interactions

In casualty insurance ratemaking, several classic two-way interactions are widely recognized by underwriters and supported by empirical loss experience:

                      CLASSIC P&C ACTUARIAL INTERACTIONS

     Driver Age x Territory                 Building Construction x Fire Protection
     ──────────────────────                 ───────────────────────────────────────
     Inexperienced youth drivers            Masonry homes resist brush fire;
     in congested urban centers             wood frame homes in remote areas
     compound hazard exponentially          (Protection Class 9/10) burn rapidly;
     (Super-multiplicative: β₁₂ > 0)        combines construction with distance to fire hall

     Vehicle Class x Vehicle Use            Deductible x Policy Limit
     ───────────────────────────            ─────────────────────────
     Sports cars driven for                 Deductible loss elimination percentage
     pleasure have moderate risk;           varies dramatically depending on the
     sports cars used for long daily        overall policy limit (truncation of
     commercial commuting explode losses    severity distribution at both tails)

1. Driver Age $\times$ Territory (Personal Auto)

• The Risk Dynamic: Inexperienced drivers (ages 16–20) lack vehicle handling skills and hazard perception. In low-density rural territories, this inexperience leads to a moderate frequency increase. However, in dense urban corridors with complex intersections, heavy traffic congestion, and abundant pedestrian activity, inexperience compounds exponentially.
• Actuarial Effect: An additive log-link model ($1.50 \times 1.40 = 2.10$) severely under-prices youthful drivers in urban centers, where the empirical relativity is $2.60$. An interaction term $\beta_{12} > 0$ generates the necessary super-multiplicative rate adjustment ($\exp(\beta_{12}) = 2.60 / 2.10 = 1.238$, a $+23.8%$ interaction load).

2. Vehicle Class $\times$ Vehicle Use (Commercial / Personal Auto)

• The Risk Dynamic: A high-performance sports car driven $3,000$ miles per year strictly for pleasure poses a modest loss cost. A high-performance sports car used for daily commercial delivery or long-distance business commuting represents an extreme physical hazard. Conversely, a family minivan exhibits almost no change in frequency when moving from pleasure to commute use.
• Actuarial Effect: The vehicle use factor must vary depending on the vehicle performance class.

3. Construction Type $\times$ Protection Class (Homeowners Property)

• The Risk Dynamic: ISO Public Protection Class (PPC) ranges from Class 1 (superior municipal water supply and fire department within 5 road miles) to Class 10 (unprotected rural area). A wood frame home in Protection Class 10 is at catastrophic risk of total destruction upon ignition. A fire-resistive reinforced masonry building in Protection Class 10 can withstand fire significantly longer, making it far less sensitive to fire station travel distance.
• Actuarial Effect: Sub-multiplicative interaction ($\beta_{12} < 0$) for masonry construction in poor protection classes.

4. Deductible $\times$ Policy Limit (Commercial Liability & Property)

• The Risk Dynamic: The percentage of loss eliminated by a $5,000 deductible depends heavily on the policy limit. In a commercial property policy with a $100,000 limit, a $5,000 deductible eliminates a massive fraction of expected claims. In a policy with a $50,000,000 limit, a $5,000 deductible eliminates a negligible percentage of total expected dollars. Deductibles and limits interact nonlinearly across the severity distribution.

Methods for Identifying Interactions

With $p$ rating variables, there are $\binom{p}{2} = \frac{p(p-1)}{2}$ potential pairwise interaction terms. If $p = 40$, evaluating all $780$ pairwise combinations directly in a GLM creates extreme computational drag, consumes hundreds of degrees of freedom, and causes severe multicollinearity. Actuaries utilize four systematic screening methods:

1. Two-Way Likelihood Ratio Tests (LRT)

Each candidate interaction term $X_j \cdot X_k$ is added individually to the baseline main-effects GLM. The drop in deviance $\Delta D$ is evaluated against a $\chi^2$ distribution with $\Delta df$ degrees of freedom:

$\Delta D = D_{\text{main}} - D_{\text{main} + (X_j \cdot X_k)} \sim \chi^2_{\Delta df}$

If predictors are categorical with $L_j$ and $L_k$ levels, the interaction adds $(L_j - 1)(L_k - 1)$ degrees of freedom. Interactions yielding large, statistically significant deviance drops are flagged for actuarial review.

2. Multi-Way Residual Diagnostic Plots

To detect an interaction between a continuous predictor $X_1$ and a categorical predictor $X_2$:

1. Extract deviance residuals $r_i^D$ from the main-effects GLM.
2. Plot $r_i^D$ against $X_1$, stratified across distinct panels or colors for each level of $X_2$.
3. Fit a smoothed loess curve through the residuals in each stratum:
   • No Interaction: The residual loess curves are flat and horizontal around zero across all levels of $X_2$.
   • Interaction Present: The residual slopes diverge—sloping upward for one level of $X_2$ and downward for another. This systematic divergence indicates that the main-effects model underpredicts at high $X_1$ for some groups and overpredicts for others.

                      RESIDUAL DIAGNOSTIC INTERACTION PLOT

         No Interaction: Parallel/Flat             Interaction Present: Divergent Slopes
     Deviance Residual                         Deviance Residual
         ▲                                         ▲
         │  ────── Level 1 (Flat)                  │   ╱ Level 1 (Positive Slope)
       0 ┼───────────────────────                0 ┼──╳───────────────────
         │  - - -  Level 2 (Flat)                  │   ╲ Level 2 (Negative Slope)
         └───────────────────────► X₁              └───────────────────────► X₁

3. Shallow Decision Trees on GLM Residuals

An innovative, powerful actuarial technique involves fitting a shallow decision tree (CART) of depth 2 or 3 directly to the deviance residuals of the main-effects GLM:

$\text{Target for Tree} = r_i^D = \text{sign}(y_i - \hat{\mu}_i) \sqrt{d_i}$

• If the main-effects GLM captures all systematic variation, the residuals represent pure white noise, and the decision tree will fail to find any split that significantly reduces variance.
• If an interaction exists between $X_j$ and $X_k$, the decision tree will naturally identify it: the root node will split on $X_j$, and the child nodes will immediately split on $X_k$.
• Consecutive splits on two different predictors isolate the multi-dimensional risk pocket where the additive GLM failed, providing an automated screen for interaction candidates.

4. Friedman's $H$-Statistic

Jerome Friedman introduced the $H$-statistic to quantify the strength of interaction between predictors based on partial dependence functions:

$H_{jk}^2 = \frac{\sum_{i=1}^n \left[ \bar{f}_{jk}(x_{ij}, x_{ik}) - \bar{f}_j(x_{ij}) - \bar{f}_k(x_{ik}) \right]^2}{\sum_{i=1}^n \bar{f}_{jk}^2(x_{ij}, x_{ik})}$

Where $\bar{f}_{jk}(x_j, x_k)$ is the two-way partial dependence function, and $\bar{f}_j(x_j), \bar{f}k(x_k)$ are the one-way marginal partial dependence functions. $H{jk} \in [0, 1]$ measures the proportion of variance in the joint prediction that cannot be explained by the sum of individual main effects:

• If $H_{jk} = 0$, the relationship is purely additive (no interaction).
• If $H_{jk} > 0.10$, a meaningful interaction exists between $X_j$ and $X_k$.

Regulatory, Implementation & Credibility Considerations

Incorporating interaction terms into a production rating structure introduces serious regulatory and technical challenges that require strict adherence to actuarial standards.

The Statistical Hierarchy Principle

Under the hierarchy principle (also known as marginality), if an interaction term $X_1 \cdot X_2$ is included in a model, all lower-order main effects ($X_1$ and $X_2$) must be retained in the model, regardless of their individual p-values:

$\eta = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} (X_1 X_2)$

Dropping a main effect (e.g., forcing $\beta_1 = 0$) while retaining $\beta_{12}$ causes the model to become origin-dependent: shifting the baseline or zero-point of $X_1$ alters the value of $\beta_{12}$ and changes the fitted values of the model. Retaining all main effects ensures scale invariance and statistical coherence.

Credibility Across Multi-Way Risk Cells

While main-effects categories typically contain thousands of earned exposures, cross-classifying two categorical predictors shatters the portfolio into a vast grid of multi-way cells. In personal auto:

• $10$ Driver Age classes $\times 50$ Territories $\times 8$ Vehicle Use classes $= 4,000$ cells.

Many multi-way cells will contain very few earned exposures (e.g., Age 17 drivers in Rural Territory 48 driving for Commercial Delivery, with $n = 1.2$ car-years). Pricing actuaries cannot file unadjusted, volatile interaction relativities on thin cells. Techniques to preserve credibility include:

1. Collapsing Adjacent Levels: Combining small territories or age brackets before forming interaction terms.
2. Credibility Shrinkage: Shrinking raw cell interaction factors toward the baseline multiplicative expectation using Bühlmann credibility.
3. Constrained Coefficients: Setting $\beta_{12} = 0$ for cells that fail a minimum exposure threshold ($n_{\text{cell}} < 100$).

Regulatory Defensibility & State Rate Filings

State insurance departments (e.g., California Department of Insurance, New York Department of Financial Services, Texas Department of Insurance) scrutinize interaction terms heavily:

• Proxy Discrimination Concerns: Regulators examine whether an interaction between a facially neutral variable (e.g., credit score tier) and territory acts as a disguised proxy for prohibited demographic characteristics (e.g., race, redlining).
• Rate Inversions: Actuaries must verify that interaction multipliers do not cause rate inversions—such as making an experienced 40-year-old driver pay more than an otherwise identical 18-year-old driver in the same territory.
• Filing Presentation: Actuaries must display interaction terms clearly in rate filings by submitting complete two-way relativity matrices showing: (1) Main effect factors, (2) Interaction multipliers, (3) Final combined relativities, and (4) Earned exposures per cell.

Rating Engine Architecture

Legacy policy administration systems frequently support only 1D table lookups:

$\text{Premium} = \text{Base Rate} \times F_{\text{Age}}(\text{Age}) \times F_{\text{Terr}}(\text{Territory})$

To implement a two-way interaction without overhauling legacy software, actuaries construct a composite rating factor table:

1. Concatenate the two predictors into a compound key: Age_Territory_Code = Concat(Age_Bracket, "_", Territory_ID).
2. Populate the single 1D lookup table with the combined product: $F(\text{Age}, \text{Terr}) = \exp(\beta_{\text{age}}) \times \exp(\beta_{\text{terr}}) \times \exp(\beta_{\text{age}\cdot\text{terr}})$.

This translates complex statistical interactions seamlessly into standard commercial rating engines.