**Nicole Gray and Madeleine Reid examine two methods for estimating the life expectancy of those with multiple impairments**

Personal injury claims rely on estimates of life expectancy (LE) to calculate a claimant’s total compensation award. Under tort law, this award should place successful claimants in the financial situation they would have been in had the tort not occurred. To adequately compensate, then, a claimant’s future pecuniary losses must be calculated. These include loss of earnings, loss of pension income and lifetime losses (costs the claimant is expected to incur until death, such as care costs). Based on the Damages Act 1996, UK courts calculate the claimant’s annual loss (the multiplicand) and multiply this by a multiplier from the Ogden tables, which are cohort tables that allow for future improvements in mortality. The multiplier is the number of years the payments are assumed to continue for, subject to the current discount rate (currently -0.25% in England and Wales). For loss of earnings the multiplier is thus the number of years until retirement subject to the current discount rate, while for loss of pension income the multiplier is the number of years from retirement until expected age of death – both allowing for the fact the claimant might not survive to retirement age, and subject to the current discount rate. For lifetime losses, assumed to begin at the time of the accident and continue until death, the multiplier is the LE (again, subject to the current discount rate).

Typically, a claimant’s LE is assumed to be normal and is derived directly from the Ogden tables. However, if there is clear evidence suggesting an individual is ‘atypical’ (they have a shorter or longer life expectancy), the Ogden tables may be departed from. In such cases, the court will primarily seek advice from clinical experts, who consider all the negative and positive pre and post-injury factors that might affect LE. However, if the predicted remaining years of life cannot be agreed upon, a statistician or actuary might be consulted. What methods might be used to estimate the LE of a claimant with one or more additional mortality risks (impairments)?

**Accounting for risk: numerical rating method **

One method typically employed is the numerical rating method. This uses a mortality ratio (MR), defined as the ratio of the age-specific mortality rates of an impaired life to those of a standard life. The notation R = 100 . MR is often used in practice. Standard lives thus have MR = 1.0 and any additional mortality risks are added to this baseline, for example if MR = 2.0, the risk of dying within a year is twice that of standard lives. To account for the impairment, age-specific mortality rates (*q** _{x}*) of standard lives are adjusted using the formula q*

*= q*

_{x}*. MR.*

_{x}A life table is then constructed by setting l*x = (say) 100,000 and computing l** _{x+t}* recursively, using the formula l*

_{x}_{+}

_{t}_{+1}= l*

_{x}_{+}

_{t}_{*}(1 –

*q**

_{x}_{+}

*) for t = 0, 1, 2, ...*

_{t}Finally LE*, the adjusted LE, is calculated using the formula

**What happens when a life has several impairments? **

If a claimant has more than one impairment, the reduction in LE for each impairment is computed separately. A common practice is for the reductions in LE to be summed and the total reduction subtracted from the LE of a standard life aged x. However, this method may underestimate LE*, as shown in the following example.

Consider a male aged 45 in 2020, with traumatic brain injury (TBI) and type 1 diabetes. For TBI, MR varies with post-injury mobility level. We use MR = 1.8, the average for all persons that can ‘walk well alone, at least 20 feet’, based on a large long-term cohort study. For diabetes, we use MR = 2.0, based on *Brackenridge’s Medical Selection of Life Risks* and diagnosis at age 13 (the average for type 1 diagnosis in the UK). Using the mortality assumptions from the ONS 2018-based National Population Projections we compute a life table for a standard male life aged 45 in 2020, allowing for future improvements in mortality. From this, life expectancy when MR = 1.0, 1.8, 2.0 and 2.8 is calculated, giving 38.943, 33.205, 32.173 and 28.881 respectively. Next, we compute the reduction in LE for TBI only, diabetes only, and both conditions (ie for MR = 1.8, 2.0, 2.8). Summing the values for MR = 1.8 and MR = 2.0, we obtain a reduction in LE of 12.508 years. By comparison with the value obtained for MR = 2.8, a reduction of 10.061 years, we see that the numerical rating method gives a considerably smaller reduction in LE than that obtained by summing the two separate reductions.

**An alternative method: rating up **

‘Rating up’ is an alternative method that treats impaired lives as though they experience the mortality of a standard life but are *k* years older. This allows all calculations to be based on a single set of tables, assuming the following:

- The standard cohort table for a life currently aged x follows Gompertz’s law
- The proportional hazards assumption holds for the impairments – that is, the hazard rate at each future time years is equal to the standard hazard rate multiplied by a constant (λ).

It follows that the mortality of an impaired life currently aged x is the same as the mortality of a standard life in *k* years’ time. If there are two independent impairments, their combined λ is the product of the two separate values of λ. Thus, *k *is the sum of the corresponding values for the separate impairments.

If, as is likely, Gompertz’s law does not hold exactly, we may estimate *k* for a given impairment from tables of life expectancies. LE* is found as the life expectancy of a standard life *k* years older. These calculations are illustrated by the following example, which makes use of the values of l*x+t for ‘MR = 1.0’ illustrated in *Figure 1* and the reductions in life expectancies due to individual impairments from the previous example.

A standard male life aged 45 has an LE of 38.943. For TBI, LE* = 33.205, equivalent to the LE of a standard male life aged 51.338, giving *k*_{1} = 6.338. For diabetes, LE* = 32.173, equivalent to the LE of a standard male life aged 52.492, giving *k*_{2} = 7.492. Thus, the rating up method for a man with both impairments regards the life as *k*_{1} + *k*_{2} = 13.830 years older, or as having the LE of a standard male aged 58.830. Interpolating from *Table 1*, a standard life aged 58.830 has a life expectancy of 26.619, so the total reduction to LE using the rating up method is 12.323 years.

*Table 1* gives the life expectancy of a male aged 45 at his current and later ages, using the values of *k* given above (rounded to the nearest integer). For many conditions it is unlikely that *k* would be constant for the remainder of one’s life, but this table illustrates the effect the rating up method would have if *k* were to take these values at the given age. By comparing 12.323, the total reduction to LE obtained using the rating up method, with 12.508, the reduction in LE for TBI and diabetes computed separately using the numerical rating method, we see that when using the rating up method the reduction in LE is similar to that obtained by summing the two separate reductions.

**The problem of assuming a constant relative mortality risk **

Both the numerical rating method and rating up method underestimate life expectancy due to their reliance on mortality ratios (MR) or hazard ratios (λ), which compare the mortality of impaired lives to the mortality experienced by standard lives. These ratios are assumed to be constant but, in reality, this is unlikely. As the standard population ages, the proportion of deaths from natural causes increases, which implies that the MR/ for a given impairment decreases with age. If these ratios are based on data for the early durations since the onset of the impairment, they may then be too high at later durations. Put simply, neither method accounts for the fact that relative risk of mortality generally declines with age. When multiple impairments are considered, underestimations of LE may be amplified.

It is clear from the examples above that the usual method of summing the reductions in life expectancy caused by separate impairments may overestimate the total reduction. It seems that if the rating up method is used the difference is small, but if the numerical rating method is employed the difference may be considerable. The rating up method is largely based on the proportional hazards assumption, while the numerical rating method is not. In practice, however, the numerical rating method may be more suitable, since medical care for one impairment might relieve the other; if this is the case, the two impairments are not completely independent, as the rating up method assumes.

*The authors gratefully acknowledge the assistance of Dr William Scott, with thanks to Mac-Migs for a summer vacation scholarship.*

**Madeleine Reid** is an undergraduate studying mathematics and its applications at the University of Stirling

**Nicole Gray** is a graduate of Oxford University, where she studied human sciences