Age Confounding in a Smoker Mortality Comparison
An insurer is reviewing mortality in two underwriting classes. Class A (smokers) has 20,000 lives and 120 deaths in the year; Class B (non-smokers) has 18,000 lives and 80 deaths. The crude death rates are 0.0060 and 0.00444, and a chi-square test on the crude counts gives p = 0.04, so the pricing team proposes a 35% mortality loading for Class A. A reviewer notes that Class A has 8,000 lives aged 60+ and 12,000 aged 40–59, while Class B has 4,500 aged 60+ and 13,500 aged 40–59. Age-specific death rates are: Class A, 40–59: 0.00333, 60+: 0.01000; Class B, 40–59: 0.00333, 60+: 0.00778. The standard population for direct standardisation is 50% in each age band. The actuary must test whether the crude difference is real and advise on the loading. The portfolio is too small for a full GLM with many covariates.
1. Calculate directly standardised death rates for both classes using the 50/50 standard population and compare them with the crude rates. What does this show?
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Direct standardisation uses the same age weights for both classes. With a 50/50 standard population:
Class A: 0.5 × 0.00333 + 0.5 × 0.01000 = 0.006665. Class B: 0.5 × 0.00333 + 0.5 × 0.00778 = 0.005555.
The crude rates were 0.00600 for A and 0.00444 for B. Standardisation raises both rates because the standard population is older than either class, but it narrows the gap. The crude ratio is 0.00600 / 0.00444 = 1.35; the standardised ratio is 0.006665 / 0.005555 = 1.20. About 29% of the crude excess is due to Class A’s older age distribution. The 35% loading suggested by the crude comparison is therefore too high before any credibility adjustment.
2. Compute the standardised mortality ratio of Class A using Class B’s age-specific rates, with an approximate 95% confidence interval, and state whether the difference is significant.
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Using Class B’s age-specific rates as the reference, expected deaths in Class A are:
40–59: 12,000 × 0.00333 = 39.96. 60+: 8,000 × 0.00778 = 62.24. Total expected = 102.20.
Observed deaths = 120, so SMR = 120 / 102.20 = 1.174. An approximate 95% confidence interval for the SMR uses SE(log SMR) ≈ 1/√120 = 0.0913:
1.174 × exp(±1.96 × 0.0913) = (0.983, 1.398).
The interval includes 1, so the difference is not statistically significant at the 5% level. The crude chi-square p = 0.04 is not adjusted for age; after standardisation the evidence for a real mortality difference is weak.
3. Advise the pricing team on the mortality loading, including limitations and a possible next step.
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The pricing team should not apply a 35% loading. The standardised ratio suggests about a 20% excess, and the SMR confidence interval is consistent with no difference. A credibility-weighted loading is more defensible: with small numbers, shrink the 20% towards the portfolio’s overall smoker differential, and disclose the uncertainty. Limitations: only two age bands, no sex or duration breakdown, and possible data-quality differences between classes. The next step is to collect more years and fit a Poisson or logistic regression with age, sex, and smoking status, or use indirect standardisation on a larger standard table. Until then, use the 20% as an indication, not a final rate, and test the profit impact of a range such as 10% to 25%.