Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 37 / 2

The data do not cross every age with every gender: only age 21 has both genders, while age 30 has only women and age 40 only men. Thus a gender effect separate from age can be estimated under an additive model, but age-by-gender interactions cannot all be estimated or checked. There is only one observation in each recorded combination of age, gender, policy, and points, so there is no within-cell replication for separating pure error from lack of fit. No sampling method or additional risk characteristics are supplied, limiting comparisons between policyholder categories.

There is also a source discrepancy: the printed table gives $268$ for one comprehensive premium, whereas the R data vector gives $368$ at the corresponding position. The supplied regression output is treated as referring to the R data. This should be resolved against the original data before substantive use.

Let $Y_{agpq}$ be the premium at age $a\in \{21,30,40\}$, gender $g\in \{F,M\}$, policy $p\in \{3rd,\mathrm{comp}\}$, and points $q\in \{0,3,6,9\}$, for the combinations actually observed. The additive normal linear model isIts corner-point constraints are ${\alpha }_{21}={\gamma }_F={\delta }_{3rd}={\eta }_0=0$. Thus $\mu$ is the reference-category mean; the other coefficients are additive contrasts. The model assumes a common variance and excludes all interactions. It has $1+2+1+1+3=8$ estimable mean parameters and $32-8=24$ residual degrees of freedom. The variance estimate is

The reduced model combines the 0-, 3-, and 6-point categories. Relative to the full model, the null hypothesis is ${\eta }_3={\eta }_6=0$, leaving the 9-point contrast unrestricted. These are two linear restrictions on an otherwise unchanged normal linear model.

For nested normal models, the reduction in residual sum of squares divided by ${\sigma }^2$ has a chi-squared distribution with degrees of freedom equal to the number of restrictions. It is independent of the full model's residual variance estimate. Hence the F-test statistic isIt is below the supplied 5% critical value $3.402826$, so do not reject equal effects for 0, 3, and 6 points. There is insufficient evidence that separating those categories improves this additive model; this does not prove that their true premiums are identical.

Let $h(q)=1$ for $q=0,3,6$ and $h(9)=2$. The new normal linear model allows an age-by-policy interaction:The corner-point constraints are ${\alpha }_{21}={\delta }_{3rd}={\gamma }_F={\xi }_1=0$ and ${\kappa }_{21,p}={\kappa }_{a,3rd}=0$. Compared with the six-parameter reduced additive model, there are two additional interaction coefficients. Its F-test givesThus the interaction model significantly improves the reduced additive model. Estimated premiums decrease with age. Men have an estimated premium $94.375$ higher than women at fixed other characteristics, and 9 points adds $41.708$ relative to 0, 3, or 6. Comprehensive cover costs more than third-party cover, but its increment depends on age: $175.375$ at 21, $148.500$ at 30, and $79.500$ at 40. These are model-based comparisons, including extrapolations into the unobserved age-gender combinations.

For a 40-year-old woman with comprehensive cover and 6 points, the gender and points contrasts vanish. The fitted premium is