Why Interactions Change Relativities
In a multiplicative rating model without interactions, the relativity for one factor is the same at every level of another factor. An interaction allows that relativity to vary. For example, a hypothetical coverage feature might change expected claim frequency differently for two vehicle-use categories. The fitted main effect alone does not describe either conditional comparison until the interaction is included.
The interpretation below uses a log-link generalized linear model with two categorical rating factors. The response may be a frequency or another positive mean; the calculations concern the fitted conditional mean. For count data, a common model assumes conditionally independent Poisson observations with the stated mean. When exposure varies, an exposure offset is included; it cancels from comparisons between cells with the same exposure basis. These are hypothetical models, not claims about actual insurance data.
The interaction term is added on the log-mean scale, so its effect is multiplicative on the mean scale. In particular, \(\exp(\delta_{ij})\) is the factor by which the relativity for \(A=i\) versus its reference changes at \(B=j\), compared with the relativity when \(B\) is at its reference level. It is generally not the relativity for \(A=i\) by itself.
Deriving Conditional Relativities
To compare two levels of \(A\) while holding \(B=j\) fixed, subtract their fitted log means. Any terms that are the same in both cells cancel. Exponentiating the difference gives a ratio of fitted means, which is the multiplicative relativity.
This cancellation is why the effect must be described as conditional on \(B=j\): the interaction contribution depends on that level. Comparing the same \(A\) contrast at two levels \(j\) and \(k\) gives
When \(k=0\), the reference-level interaction is zero, and the ratio of conditional relativities is simply \(\exp(\delta_{ij})\). Thus, an interaction coefficient of \(\log(0.8)\) means the specified \(A\) relativity is multiplied by \(0.8\) at that \(B\) level relative to its value at the reference level of \(B\). It does not mean that the relativity equals \(0.8\).
The same logic works when comparing two non-reference levels of \(A\). At \(B=j\), the relativity for \(A=i\) versus \(A=k\) is \(\exp(\alpha_i-\alpha_k+\delta_{ij}-\delta_{kj})\). Consequently, the change in that contrast between two levels of \(B\) depends on the difference between the two interaction terms, not either interaction term alone.
Worked Example: A Binary Interaction and Its Conditional Effects
A hypothetical insurer models claim counts with an independent Poisson log-link GLM and equal exposure in each cell. Let \(A=0,1\) denote two vehicle-use categories and \(B=0,1\) denote two coverage options. Suppose the fitted mean for the reference cell is 200 claims, the main-effect multipliers are \(1.20\) for \(A=1\) and \(1.50\) for \(B=1\), and the interaction multiplier is \(0.80\). Find the fitted mean in every cell, the conditional relativity for \(A=1\) at each \(B\) level, and how that relativity changes across \(B\). All quantities are hypothetical; no interest or mortality assumptions apply.
Method. The corresponding log coefficients are \(\alpha_1=\log(1.20)\), \(\gamma_1=\log(1.50)\), and \(\delta_{11}=\log(0.80)\). With a log link, the fitted mean in each cell is the reference mean multiplied by the applicable main-effect and interaction multipliers.
At \(A=0,B=0\), both factors are at reference, so \(\mu_{00}=200\).
At \(A=1,B=0\), there is no interaction contribution because \(B\) is at reference: \(\mu_{10}=200(1.20)=240\).
At \(A=0,B=1\), only the \(B\) main effect applies: \(\mu_{01}=200(1.50)=300\).
At \(A=1,B=1\), include both main effects and the interaction: \(\mu_{11}=200(1.20)(1.50)(0.80)=288\).
The relativity for \(A=1\) versus \(A=0\) at \(B=0\) is \(240/200=1.20\). At \(B=1\), it is \(288/300=0.96\), equivalently \(1.20(0.80)=0.96\). The factor’s relativity therefore falls from \(1.20\) to \(0.96\). The ratio of these relativities is \(0.96/1.20=0.80\), exactly the interaction multiplier.
Check. Applying the conditional relativity to the \(B=1\) reference mean gives \(300(0.96)=288\), matching the fitted cell mean. The common mistake is to report \(1.20\) as the \(A\) relativity at both \(B\) levels; the interaction makes the \(B=1\) relativity \(0.96\).
Interactions With More Than Two Levels
With more than two levels, reference coding still gives a direct calculation for each non-reference level: add that level’s main-effect coefficient to its interaction coefficient for the specified level of the other factor, then exponentiate. Each non-reference level can have a different interaction and therefore a different conditional relativity.
Worked Example: Conditional Effects for a Three-Level Factor
Consider a hypothetical independent Poisson count model with a log link and equal exposure across cells. Factor \(A\) has reference level \(A=0\) and levels \(A=1,2\); factor \(B\) has reference level \(B=0\) and level \(B=1\). The fitted mean at \((0,0)\) is 500. The main-effect multipliers are 1.30 for \(A=1\), 0.80 for \(A=2\), and 1.40 for \(B=1\). At \(B=1\), the interaction multipliers are 0.90 for \(A=1\) and 1.25 for \(A=2\). Find the conditional relativities for both \(A\) levels at both \(B\) levels and the four fitted means for \(A=1,2\). All data are hypothetical; interest and mortality assumptions are not applicable.
Method. At \(B=0\), the interaction contribution is one, so the \(A\) relativities are their main-effect multipliers. At \(B=1\), multiply each main-effect multiplier by its corresponding interaction multiplier.
For \(A=1\), the relativity at \(B=0\) is \(1.30\). At \(B=1\), it is \(1.30(0.90)=1.17\). For \(A=2\), the relativity at \(B=0\) is \(0.80\). At \(B=1\), it is \(0.80(1.25)=1.00\).
First calculate the reference-\(A\) mean at \(B=1\): \(\mu_{01}=500(1.40)=700\). At \(B=0\), the two requested means are \(\mu_{10}=500(1.30)=650\) and \(\mu_{20}=500(0.80)=400\). At \(B=1\), they are \(\mu_{11}=700(1.17)=819\) and \(\mu_{21}=700(1.00)=700\).
Check. The cell ratios are \(650/500=1.30\), \(819/700=1.17\), \(400/500=0.80\), and \(700/700=1.00\), agreeing with the four conditional relativities. The interaction multipliers are also recovered: \(1.17/1.30=0.90\) for \(A=1\), and \(1.00/0.80=1.25\) for \(A=2\). The common mistake is to apply the \(A=1\) interaction to \(A=2\), or vice versa; reference-coded interaction terms are specific to their level combination.
Interactions also matter when the requested contrast is between two non-reference levels. In that case, both levels’ terms appear in the comparison, so it is not enough to use the interaction coefficient for just one of them.
Worked Example: Comparing Two Non-Reference Levels Across Groups
A hypothetical independent Poisson model with a log link and equal exposure has reference mean 360. Factor \(A\) has reference level \(A=0\) and levels \(A=1,2\); factor \(B\) has reference level \(B=0\) and level \(B=1\). The main-effect multipliers for \(A=1,A=2,B=1\) are \(1.50,0.75,1.25\), respectively. The interaction multipliers at \(B=1\) are \(0.80\) for \(A=1\) and \(1.20\) for \(A=2\). Compare \(A=1\) with \(A=2\) at each level of \(B\), and state how the comparison changes. All values are hypothetical; interest and mortality assumptions do not apply.
Method. At \(B=0\), the interaction terms are at reference, so the \(A=1\) versus \(A=2\) relativity is \(1.50/0.75=2.00\).
At \(B=1\), the conditional relativity for \(A=1\) versus \(A=0\) is \(1.50(0.80)=1.20\). For \(A=2\) versus \(A=0\), it is \(0.75(1.20)=0.90\). Dividing these gives the requested comparison: \(1.20/0.90=1.3333\), to four decimal places.
The fitted means provide a direct check. At \(B=0\), they are \(\mu_{10}=360(1.50)=540\) and \(\mu_{20}=360(0.75)=270\), with ratio \(540/270=2.00\). At \(B=1\), the reference-\(A\) mean is \(360(1.25)=450\); the other means are \(\mu_{11}=450(1.20)=540\) and \(\mu_{21}=450(0.90)=405\). Their ratio is \(540/405=1.3333\), to four decimal places.
The ratio of the two conditional comparisons is \(1.3333/2.00=0.6667\), to four decimal places. From the interaction multipliers, the same change is \(0.80/1.20=0.6667\). Thus, the interaction contrast for comparing \(A=1\) with \(A=2\) is the difference of their log interaction coefficients, or equivalently the ratio of their interaction multipliers.
Check. Both the coefficient calculation and the fitted-mean ratios agree. A common error is to call \(0.80\) the interaction for the \(A=1\)-versus-\(A=2\) comparison. That comparison uses both interaction terms: \(0.80/1.20\).
Interpretation Checks and Common Traps
Before reporting an interaction interpretation, identify the exact two levels being compared and the level at which the other factor is held. Then use the fitted mean ratio or the corresponding coefficient contrast. These steps prevent ambiguity about reference levels and avoid treating a coefficient as though it described every subgroup.
Name the numerator and denominator levels of the factor whose relativity is requested.
Identify its level; the interaction contribution can change with that level.
Subtract the appropriate fitted linear predictors, including each applicable interaction term.
The exponential is a multiplicative ratio. When possible, verify it by dividing the relevant fitted means.
The coefficient interpretation depends on the coding convention. Under reference coding, the main-effect coefficient for \(A=i\) gives its log relativity at the reference level of \(B\), and the interaction coefficient adjusts that effect at another \(B\) level. With another contrast coding, individual coefficients may not have this direct interpretation, although fitted cell means and contrasts between them remain interpretable. The general safeguard is to construct the requested linear-predictor difference first and exponentiate it second.
An interaction is a feature of the specified model for the conditional mean. It does not, by itself, establish that changing a rating characteristic causes a change in risk. Nor should a fitted cell mean be confused with an observed count: the former is the model’s expected response for the cell, given the predictors and model assumptions.
Check Your Understanding
For each hypothetical log-link model, calculate the requested conditional effect or interaction interpretation.
- The main-effect multiplier for \(A=1\) versus \(A=0\) is 1.10, and the interaction multiplier at \(B=1\) is 1.30. What is the \(A\) relativity at \(B=0\) and at \(B=1\), and by what factor does it change? (Answer: 1.10 at \(B=0\), 1.43 at \(B=1\); the relativity is multiplied by 1.30.)
- For \(A=2\) versus reference \(A=0\), the main-effect multiplier is 0.85 and the interaction multiplier at \(B=1\) is 1.20. Find the conditional \(A=2\) relativity at \(B=1\). (Answer: \(0.85(1.20)=1.02\).)
- At \(B=0\), the main-effect multipliers for \(A=1\) and \(A=2\) are 1.40 and 0.70. At \(B=1\), their interaction multipliers are 0.90 and 1.10. Find the \(A=1\)-versus-\(A=2\) relativity at each \(B\) level and its ratio at \(B=1\) to that at \(B=0\). (Answer: 2.00 at \(B=0\), 1.6364 at \(B=1\), and a change factor of \(0.90/1.10=0.8182\), to four decimal places.)
- A reference cell has fitted mean 240. The main-effect multipliers are 1.25 for \(A=1\) and 1.60 for \(B=1\), and the interaction multiplier is 0.75 for the combination \(A=1,B=1\). Find the fitted mean for that combination and the conditional relativity for \(A=1\) at \(B=1\). (Answer: fitted mean \(240(1.25)(1.60)(0.75)=360\); conditional \(A\) relativity \(1.25(0.75)=0.9375\).)
- The \(A=1\)-versus-reference relativity is 1.30 at \(B=0\) and 1.04 at \(B=1\). Find the interaction multiplier for this contrast and its log-scale coefficient, rounded to four decimal places. (Answer: \(1.04/1.30=0.80\); coefficient \(\log(0.80)=-0.2231\).)