Counting and discounting gained life-years.

The life expectancy gain produced by a reduction in mortality can be determined by three different methods with respect to the timing of the gained life-years. One method adds the life expectancy gain to the expected end of life. Another method places the gain at the time of occurrence of the mortality reduction. A third method distributes the gained life-years over the maximum lifespan according to the differences in survival probabilities after and before the reduction in mortality. The three methods are all used in the literature together with a quasi-deterministic and a probabilistic approach to the notion of life expectancy. The counted numbers of gained life-years are the same, but due to different timing of life expectancy gains the discounted numbers are different. Several discounting models are identified when combining the three methods of counting with the deterministic and the probabilistic approaches to life expectancy. Some are symmetrical, some are not. However, most importantly, they come out with potentially very large differences in the discounted number of gained life-years. They differ by a factor of approximately (1 + r)e(a)-1, where r is a constant discount rate and e(a) is remaining life expectancy at age a, when the reduction of mortality occurs. For a new-born, discounting at 7% p.a., one discounting model provides a present value that is 150 times larger than another discounting model, the other models being in between. The various counting and discounting models for life expectancy gains are presented formally, graphically, and with numerical examples using Danish male mortality data. We show how three different discounting models provide large differences in discounted life expectancy gains and hence cost-effectiveness ratios in an economic evaluation of a colorectal cancer screening programme in Denmark. These different discounting models co-exist in the evaluation literature. Choice of method is rarely made explicit. Sensitivity analysis with respect to this choice is even rarer. We argue that one counting-discounting model is sufficient and that this should be to discount the differences between the two survival probability curves.