Survival and Life Expectancy (LE) Let S(t) = number alive at start of year t, s(t) = survival rate from year t to t+1, so S(t+1) = s(t)S(t). What is S(t) - S(t+1)? Life Expectancy = Average years to be lived = (∑ S(t)) / S(0). “age” at death for those dying S(t) S(t+1) Number surviving, S(t) Years from start Start t t+1 S(0) another year for survivors 1000? area under survival curve
Life Expectancy (LE) with continuous death Let S(t) = number alive at time t, and dS/dt = -h(t)S(t). h(t) is called the hazard of dying. S(t) = S(0) exp (∫-h(x)dx) Life Expectancy = Average years lived = (∫S(t)) / S(0). If h(t) = h, LE = 1/h S(t) Number surviving, S(t) Years from start Start t S(0) more years for survivors
Life Tables for males, US 2004 Age death rate in interval number living at start deaths in interval years lived in interval years lived in this and all future intervals life expect at start 0-1.0071000799475,17575.2 1-2.001 9931992.574,17174.7 100+1.0012.5 30 2.4 ??
Prob of Survival 1.0 Time from Beginning of intervention 0 1 2 3 Area under curve = Life Expectancy after intervention If we divide S(t) by the number treated then S(0) = 1, S(t) is a probability, and LE = area under survival curve/1. Cohort Life Expectancy after treatment
DEALE: If death rate is constant d, Life Expectancy = 1/d Here death rate =. 25, so LE = 4 years. For proof see next slide. Note L=1/d d= 1/L