SISA Research Paper 
Discounting and mortality adjusting Years of Potential Life Lost (YPLL)
Years of Potential Life Lost (YPLL) or Potential Years of Life Lost (PYLL) is an often used statistic in practical epidemiology and demography. The measure is easy to calculate, easy to understand and has a strong intuitive appeal. The YPLL measures for a group of individuals the total number of years these people would have additionally lived up to some point in the future, would they not have died from a particular cause of death. Mostly, as the age where life is "lost" by a premature death, the life expectancy for a population or the age of productivity, from 15 to 65 or 70, are chosen. Besides the obvious advantages of the YPLL there are a number of problems. Two of these problems are discussed in this paper, one the practical problem of correction of the YPLL for people surviving but then dying from an other cause of death, second the theoretical and philosophical problem of discounting of the value of life lived in the far away future.
Traditional mortality statistics are strongly dominated by the high quantity of deaths among older age groups. This denies the fact that death at a young age is generally considered to be a greater loss to the individual and society compared with death at a high age. The YPLL weighs death at a young age more heavily compared with death at a higher age. One of the problems is that death at a young age seems sometimes to be too heavily weighted in calculating the YPLL (1). All future years of life are weighed equally; for example, no difference is made between the loss of the life years 2529 compared with the life years 6064, for an individual who dies at 24. Particularly in the case of postnatal and infant death this has the consequence that a large number of years of lost life are accumulated, many in the very far away future. In economic "discounting" theory the value of events in the far away future are considered to be less compared with similar events in the near future. Thus, for an individual aged 24, the ages 2529 are considered more "valuable" than the ages 6064. There is much discussion if life should be discounted, like other commodities (2). However, an opinion is developing that gains in health should be discounted at an annual rate of at least 1.5%. Health economists in the U.S. advice to discount at 3% (3). Here not much attention is further given to this philosophical discussion. In this paper some of the mathematics required to discount the YPLL are discussed. After this the effect of discounting on the YPLL is studied on the basis of mortality statistics from the Netherlands.
An important methodological problem with the YPLL is the fact that this measure does not consider competing causes of death (4). If the death of a group of people is prevented the fact that these people can die from other causes before the upper cutoff point in calculating the YPLL is reached, is not considered. Particularly in the case of populations with high mortality this can lead to an overestimation of the importance of preventing a particular cause of death. In this paper a number of formulae, mostly from the life table literature, will be discussed and applied to the YPLL. After this the effect of mortality adjustment on the YPLL is studied using mortality data.
METHODS
Calculating the YPLL. For an annotated version of this section go here.
The basic equation for calculating the YPLL after removal of a particular cause of death k for a particular age group i equals (1):
1) YPLL_{k.i}=((number of death at a given age)*(weight for that age))=D_{k.i}W_{i}
It is customary to summarize the individual age YPLL(i) , to represent loss of life in a range of ages between the start of the index i and the endpoint x:
2) YPLL_{k.i..x}= å _{(i..x) }YPLL_{k.i} = å _{(i..x) }D_{k.i}W_{i}
Often the age range of productivity, 15 to 65 (or 70) years of age is used
For the following discussion it is considered that:
3) Weight for that age W_{i}=Sum(weight for each year of life remaining) =å w_{j}
whereby j is in the range i..x
For example, in the case of calculating the YPLL for the productive age groups 15 to 64, w(j) equals zero for years of potential life below the age of 15, zero for years of potential life above the age 64, and w(j) is one for the age groups 15 up to and including age 64. W(i) in these age groups equals å w_{j} whereby j=i..64; i>14.
In the case of discounting w(j) will be a proportion which will be smaller for larger differences between i and j: representing that more time has elapsed between observed death and a potential year of life lost and the decreasing value of future life. In the case of mortality adjusting w(j) will be a proportion representing the probability to die from causes of death which "compete" with the cause of death k in a certain age category.
Discounting the YPLL. For an annotated version of this section go here.
Considering that the weight for the first year of potential life saved equals 1, the value of the weight for the second year at a discount rate of r equals approximately 1/(1+r) and the value of the jth year saved equals approximately (5):
4) w_{j}=(1+r)^{(}j1) Please read the note on this formula
and j is in the range 1..xi1 (whereby i is the age at which the deaths at interest took place and j is the jth year of potential life saved for a death at age i)
For example, the weight w_{10} for the 10th year saved at a discount rate of 1.5% (r=0.015) equals 0.875.
Considering that death does not take place at a single moment in time at the beginning of the year, in each age period there is a fraction a_{i} of life lived by the death in that period. In life table analysis this fraction is often set at 0.5, considering that death takes place more or less equally spaced throughout an age period. For i=0 this fraction should be set at a_{0}=0.1, considering that in the first year of life deaths occur disproportionally often in the beginning of the year (6).
The calculation of the weight W_{i} for the discounted YPLL_{i} for age i up to x would be:
5) W(i)= å _{(j=i...x) }w_{j }=1+å _{(}j=2...xi1) (1+r)^{(}j1) + a_{i} * (1+r)^{(xi1) }
In formulae five there is first a not discounted year of benefit, followed by a number of years of benefits of decreasing importance and, lastly, a partial discounted year of relatively low benefit.
In the case of abridged tables, if ages are grouped in, for example, 5 or 10 year age categories, the weight of an age category is the sum of years contained in the individual age categories in that group. For an abridged table category c with width n and considering the age range i..i+n
6) w_{c} =å _{(j=i...} _{i+n ) }w_{j}
Mortality adjusting the YPLL.
Mortality adjusting the YPLL is done easiest by developing a series. At the start of the first year i the number of survivors of other causes of death is 1, or 100%. The weight for the YPLL at the beginning of each subsequent age year of life lost is related to the proportion of individuals who survived the previous year:
7) pr_{j}=pr_{(j1) } (pr_{(j1)}*d_{(j1)})
whereby d is the death rate for a single age year in a single year of observation, and pr the proportion survivors at the beginning of each age period.
For an abridged table category c with width n the proportion of individuals who survived the previous period equals:
8) pr_{c}=pr_{(c1) } (pr_{(c1)}*n _{c}*d_{(c1)})
whereby d is the death rate in an age category in a single year, and pr the proportion survivors at the beginning of each age category.
To correct for mortality, which takes place during a time period the following method can be applied to the outcome of the series, and the weight, w(j) is derived. For the first year:
9) w_{j }= (1a_{j}) * {1d_{j} + a_{j}*d_{j}}; j=1
and for all subsequent years:
10) w_{j }= pr_{j}d_{j} + a_{j}*d_{j}; j>1
Two different formulas are required in each case primarily because the prevented deaths will gain 1a(j) period in the first instance, and full periods in the years thereafter.
These formulae directly based on the death rates will usually give quite a satisfactory mortality adjustment for the YPLL. However, by replacing the death rate d with the proportion dying, q, the procedure will become more precise. Death rates are based on the number of deaths at age n relative to the mean population aged n, the proportion dying considers he number of deaths aged n relative to the number who celebrated their nth birthday. q(j) is related to d(j) as follows (6):
11) q_{j}=d_{j }/ {1+(1a_{j})*d_{j}}
and q(c) for the abridged table, i.e., the probability for a person who celebrated the nth birthday to die in the age interval n..n+n , equals,
12) q_{c}= n _{c}*d_{c }/ {1+(1a_{c})*n _{c}*d_{c}}
A further improvement is to take the prevented cause of death k out of the proportion dying (4,7):
15) q_{j}'=q_{j}q_{k.j}=1(1q_{j})^{(qjqk.j)/qj}
Use this formula for both abridged and not abridged tables.
RESULTS
Cancer in the Netherlands.
Table one. Total and discounted YPLL after preventing cancer as cause of death, age groups 4564 years, The Netherlands 19962000.
Age 
Total Death Dc 
Cancer Dk.c 
Population Pc 
Not discounted 
Discounted at 1.5% 

Wi 
YPLLi 
YPLLi..x 
Wi 
YPLLi 
YPLLi..x 

4549 
14773 
6101 
5746208 
17.5 
106767.5 
371460.0 
15.52 
94684.2 
344242.4 
5054 
21586 
10051 
5357130 
12.5 
125637.5 
264692.5 
11.50 
115479.1 
249558.2 
5559 
26541 
12646 
4106182 
7.5 
94845.0 
139055.0 
7.15 
90389.8 
134079.1 
6064 
38358 
17684 
3541569 
2.5 
44210.0 
44210.0 
2.47 
43689.3 
43689.3 
In table one, Dutch data regarding the total number of deaths, the number of cancer deaths and the mean population are shown for the years 19962000 and for the four five year age groups 45 to 64. The not discounted "crude" YPLL is compared with the discounted YPLL. The death in the age group 4549 would have lived an additional 17.5 years would they have made the age of 65. This is reduced to 15.5 years would each additional year of gained life have been discounted at 1.5% compared with the previous year. The total weight (Wi) 7.15 for the age group 5559 is produced by seven years with weights (wj) 1,000; 0,985; 0,971; 0,956; 0,942; 0,928 and 0,915 respectively, and halve a year with weight 0.901. The resulting 7.3% reduction in total YPLL for the age groups 4564 from 371460 to 344242.4 as a consequence of discounting does not seem to be particularly large.
Table two. Mortality adjusted YPLL after preventing cancer as a cause of death, age groups 4564 years, The Netherlands 19962000.
Age 
Death rate for all causes: d_{c} 
Death rate for cancer d_{k.c} 
Wi 
YPLLi 
YPLLi..x 
4549 
0,0026 
0,0011 
17.172 
104710.0 
364574.9 
5054 
0,0040 
0,0019 
12.289 
123377.8 
259864.9 
5559 
0,0065 
0,0031 
7.403 
93349.2 
136487.1 
6064 
0,0108 
0,0050 
2.475 
43137.9 
43137.9 
Table two is based on the same data and shows the effect of mortality adjusting the data. The adjusted YPLL for the age group 4549 exclusively is based on the probability of 99.3% of the prevented deaths surviving up to age 50, 98.2% surviving up to 55, and 96.6% up to 60 years of age. For the first 2.5 years of potential life lost in the ages 4549 by deaths in the age group 4549 the weight (wj) is 2.49, for the five years lost in the ages 5054 by deaths in the age group 4549 the weight is 4.96 for the following five years wj is 4.90 and for the years 6064 it is 4.81 (numbers not in table). This results in a total weight (Wi) for the age group 4549 for years of potential life lost in the ages 4564 of 17.163. Overall the effect of mortality adjusting the YPLL in the age groups 4564 is limited, consisting of a decrease in the total YPLL for this age group of 1.9%, from 371460 to 364574.9.
Table three. Statistics of mortality in the Netherlands, 19962000. Age groups 1564; various causes; numbers per 100,000.

All causes 
Cancers 
Stroke 
Respiratory 
Trauma 
Congenital 
Total Death_{15..65} 
239.91 
100.19 
10.39 
8.39 
14.34 
1.60 
YPLL_{15..65} 






Crude 
3071.43 
1061.31 
115.81 
78.25 
210.32 
32.32 
Discounted (1.5%) 
2652.01 
943.79 
102.06 
69.28 
179.15 
26.39 
Discounted (3.0%) 
2333.52 
850.28 
91.25 
62.31 
155.79 
22.14 
Mortality Adjusted 
2974.96 
1041.55 
112.12 
75.74 
204.00 
31.34 
In table 3 total YPLL, discounted YPLL at 1.5% and 3% and mortality adjusted YPLL is given for the age groups 15 to 64 years. The figures relate to the total number of death per 100000 and the number of YPLL per 100000 of the population in the age groups 15 to 64 years. The data is given for all causes of death, all cancer death and death caused by stroke, respiratory diseases, trauma and congenital anomalies respectively. The last four causes have been selected because they give a good view of the effect of age and number dying on the different ways of calculating YPLL. As can be seen in the table, the effect of discounting at 1.5% seems modest. In the case of all causes the YPLL decreases with 13.7%, for the other diseases the effect is between 11% for cancers and 18% for congenital anomalies. Discounting at the 3% level has a more powerful effect, a decrease in the YPLL of 24.0% for all causes, while for the other causes it ranges from 20% for cancers up to 32% for congenital anomalies. The effect of mortality adjustment is small, for all causes the YPLL decreases with 3.0%, for all cancers with 1.9% for congenital anomalies with 3%. The rank order of the causes of death as expressed in the YPLL are the same before and after discounting and adjusting and also if compared with the numbers per 100000 of the population.
Table four. Statistics of mortality in the Netherlands, 19962000. Age groups 074; various causes numbers per 100,000.

All causes 
Cancers 
Stroke 
Respiratory 
Trauma 
Congenital 
Total Death_{0..74} 
373.34 
146.13 
20.89 
22.40 
18.18 
4.15 
YPLL_{0..74} 






Crude 
5577.73 
1887.52 
219.60 
200.65 
335.39 
242.82 
Discounted (1.5%) 
4576.39 
1631.65 
191.30 
175.55 
269.44 
154.34 
Discounted (3.0%) 
3901.68 
1438.38 
169.95 
157.41 
225.44 
108.02 
Mortality Adjusted 
5161.83 
1797.04 
202.17 
184.94 
311.67 
230.82 
Table 4 the same analysis is presented for the age groups 074 years. This age range is in keeping with current U.S. National Center for Health Statistics practice (8). Compared with table 3 stroke, and particularly respiratory diseases, increase strongly in terms of mortality per 100000 of the population. This increase is primarily caused by adding the age groups 6574. Trauma increases only slightly, due to additional cases in both the youngest and the oldest age groups, while additional cases in the youngest age group increase the number of people dying from congenital anomalies. The rank order for these four causes of death in number per 100000 is respiratory diseases as the most important cause, followed by stroke and trauma, and, clearly less important in numerical terms, congenital anomalies. In YPLL terms this order is very different with the causes of death which are more common at younger ages, trauma and congenital diseases, becoming more important. The effect on respiratory diseases and congenital anomalies is most marked, with respiratory diseases becoming less important and congenital anomalies considerably more important. Discounting changes these conclusions again, with congenital anomalies decreasing strongly in importance compared with the not discounted YPLL. Trauma continuous to be the most important cause of death with regard to the discounted YPLL, which reflects the fact that trauma is quite a common cause of death in all age groups. Discounting at a the higher level of 3% further emphasizes the results but does not change conclusions. Mortality adjusting has very little effect on the data and does not change the conclusion with regard to the crude YPLL.
DISCUSSION
In this paper methods for discounting the YPLL for future health benefit and mortality adjusting the YPLL are discussed. The mathematical framework is introduced and explained after which the methods are applied to Dutch mortality data. The analysis shows that in the age group of productivity, 1564, neither discounting at the minimal recommended level for health benefits nor mortality adjusting the YPLL has a large effect on interpreting mortality data. The effect of discounting at 3% is more pronounced. In general in the age group 1564, the substantive results with regard to crude mortality, the YPLL, the discounted YPLL at various levels, and the mortality adjusted YPLL, are all the same. If the age groups 014 and 6574 are also considered in the analysis this conclusion of no difference between methods is no longer valid. In terms of number dying, stroke, and particularly respiratory diseases, become more important. This is primarily because of the addition of the age group 6574, were there is a relatively high mortality of these causes. However, the fewer cases that are added in the younger age group have a more profound effect on the YPLL. Trauma is in YPLL terms the most important cause of death, and particularly notable is the importance of congenital anomalies in terms of accrued YPLL, more important as both stroke and respiratory diseases. This effect is mostly undone if the data is discounted and particularly congenital anomalies loose much of their weight in the discounted YPLL.
The discussion about which method to use to describe mortality is complex and philosophical. This observation is probably particularly true if one wants to apply discounting of the value of future life in mortality analysis. However, as the analysis shows, if one does use the YPLL discounting should be considered. Discounting will blunt the problem that small numbers of deaths in younger age categories can profoundly alter conclusions drawn on the basis of the YPLL compared with conclusions based on numbers dying. Even discounting at a relatively modest level will have this effect.
If one decides to use the YPLL mortality adjustment should in principle be applied, as mortality adjustment overcomes a methodological flaw in calculating the YPLL. With regard to the data used in this paper, the effect of mortality adjusting is very small indeed. However, it should be considered that in the data used there is a low mortality. In populations with higher mortality the difference between adjusted and unadjusted YPLL will be larger.
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