## Standardized mortality ratio and loss of life expectancy

### Michael Væth, Section for Biostatistics, Aarhus University

Standardized mortality ratio and loss of life expectancy are both used as
mortality indices when comparing the mortality in different populations.
The relationship between these mortality indices is investigated. An exact
expression for the first derivative of the life expectancy with respect to
a proportional change the age-specific mortality ratio is obtained when the
age-specific mortality rate is log-linear in age. This relationship is used
to show that the change in life expectancy is essentially linear in the
logarithm of the proportional change in the mortality rate and that the
slope of this linear relationship is approximately equal to minus the inverse
of the slope of log-linear age dependence of the mortality rate. Mortality
data from Denmark, Finland, Norway and Sweden in the period 1950-2010 are
used to study the robustness of the linear approximation to deviations from
proportional changes of the mortality rate. The results are related to the
concept of effective age recently proposed by Spiegelhalter. This study was
carried out in collaboration with Henrik StÃ¸vring and Mette Vinther Skriver.