Life table

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2003 US mortality table, Table 1, Page 1

In actuarial science, a life table (also called a mortality table or actuarial table) is a table which shows, for a person at each age, what the probability is that they die before their next birthday. From this starting point, a number of statistics can be derived and thus also included in the table:

  • the probability of surviving any particular year of age
  • remaining life expectancy for people at different ages
  • the proportion of the original birth cohort still alive
  • estimates of a cohort's longevity characteristics.

Life tables are usually constructed separately for men and for women because of their substantially different mortality rates. Other characteristics can also be used to distinguish different risks, such as smoking-status, occupation, socio-economic class, and others.

Life tables can be easily extended to include other information in addition to mortality, for instance health information to calculate health expectancy. Health expectancies, of which disability-free life expectancy (DFLE) and Healthy Life Years (HLY) are the best- known examples, are the remaining life expectancy a person can expect to live in a specific health state, such as free of disability. Two types of life table techniques are used to divide the life expectancy into life spent in various states: 1) multi-state life tables (also known as increment-decrement life tables) based on transition rates in and out the different states and to death, and 2) prevelence-based life tables (also known as the Sullivan method) based on external information on the proportion in each state. Apart from life tables for health expectancy applications, these extensions include life expectancies in different labor force states or marital status states.

Life tables are also used extensively in biology and epidemiology. The concept is of importance, as well, to product life cycle management.

Contents

[edit] Insurance applications

In order to price insurance products, and ensure the solvency of insurance companies through adequate reserves, actuaries must develop projections of future insured events (such as death, sickness, disability, etc.). To do this, actuaries develop mathematical models of the causes of these events, as well as the amount and timing of the events. They do this by studying the incidence and severity of these events in the recent past, developing expectations about how the drivers of these past events will change over time (for example, whether the increase in life expectancy that has been experienced by most generations over prior generations will continue) and, accordingly, develop an expectation for what the timing and amount of such events will be into the future. These expectations usually take the form of tables of percentages indicating the number of such events that will occur in a population, usually based on the age or other relevant characteristics of the population. More specifically, they may be referred to as mortality tables (if they provide rates of mortality, or death), morbidity tables (if they provide rates of disability and recovery), or by other names if they cover other decrements.

The invention of computers and the proliferation of data gathering about individuals has led to fundamental changes in the way actuarial tables are computed for different uses, and a variety of emerging methods factor a range of non-traditional behaviors (e.g. gambling, debt load) into specialized calculations utilized by some institutions for evaluating risk.

[edit] The mathematics

tPx Chart from Table 1. Life table for the total population: United States, 2003, Page 8

To give an indication of how life tables are used, here are a few sample calculations. These samples may not be obvious to someone who has never studied probability theory, but are intended to introduce new ideas to people who have some understanding of discrete probability theory.

  • \,q_x: the probability that someone aged exactly \,x will die before reaching age \,(x+1).
  • \,p_x: the probability that someone aged exactly \,x will survive to age \,(x+1).
\,p_x = 1-q_x
  • \,l_x: the number of people who survive to age \,x
note that this is based on a starting point of \,l_0 lives, typically 100,000
\,l_{x + 1} = l_x \cdot (1-q_x) = l_x \cdot p_x
\,{l_{x + 1} \over l_x} = p_x
  • \,d_x: the number of people who die aged \,x
\,d_x = l_x-l_{x+1} = l_x \cdot (1-p_x) = l_x \cdot q_x
  • \,{}_tp_x: the probability that someone aged exactly \,x will survive for \,t more years, i.e. live up to at least age \,x+t years
\,{}_tp_x = {l_{x+t} \over l_x}
  • \,{}_{t|k}q_x: the probability that someone aged exactly \,x will survive for \,t more years, then die within the next \,k years
\,{}_{t|k}q_x = {}_t p_x \cdot {}_k q_{x+t} = {l_{x+t} - l_{x+t+k} \over l_x}
  • \,{}_tm_x: the mortality rate between exact age \,x and exact age \,x+t
\,{}_tm_x = {ln(1-_tq_x) \cdot }{-1 \over t}

[edit] Biology

When biologists use life tables, they will normally also include fertility for each age. The extra parameter used is

  • \,m_x: expected number of progeny for an individual aged \,x

[edit] Epidemiology

In epidemiology and public health, both standard life tables to calculate life expectancy and Sullivan and multistate life tables to calculate health expectancy are commonly used. The latter include information on health in addition to mortality.

[edit] See also

[edit] Citation

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Life_table"
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