Life Expectancy

An $l_x$ table is a tabulation of numbers which is used to calculate life expectancies.

$x$	$n_x$	$d_x$	$l_x$	$q_x$	$L_x$	$T_x$	$e_x$
0	1000	200	1.00	0.20	0.90	2.70	2.70
1	800	100	0.80	0.12	0.75	1.80	2.25
2	700	200	0.70	0.29	0.60	1.05	1.50
3	500	300	0.50	0.60	0.35	0.45	0.90
4	200	200	0.20	1.00	0.10	0.10	0.50
5	0	0	0.00	--	0.00	0.00	--
$\sum$		1000	2.70

$x$: Age category ($x=0$, 1, ..., $k$). These values can be in any convenient units, but must be chosen so that no observed lifespan extends past category $k-1$.

$n_x$: Census size, defined as the number of individuals in the study population who survive to the beginning of age category $x$. Therefore, $n_0=N$ (the total population size) and $n_k=0$.

$d_x$: $=n_x-n_{x+1}$; $\sum_{i=0}^k d_i=n_0$. Crude death rate, which measures the number of individuals who die within age category $x$.

$l_x$: $=n_x/n_0$. Survivorship, which measures the proportion of individuals who survive to the beginning of age category $x$.

$q_x$: $=d_x/n_x$; $q_{k-1}=1$. Proportional death rate, or ``risk,'' which measures the proportion of individuals surviving to the beginning of age category $x$ who die within that category.

$L_x$: $=(l_x+l_{x+1})/2$. Midpoint survivorship, which measures the proportion of individuals surviving to the midpoint of age category $x$. Note that the simple averaging formula must be replaced by a more complicated expression if survivorship is nonlinear within age categories. The sum $\sum_{i=0}^k L_x$ gives the total number of age categories lived by the entire study population.

$T_x$: $=T_{x-1}-L_{x-1}$; $T_0=\sum_{i=0}^k L_x$. Measures the total number of age categories left to be lived by all individuals who survive to the beginning of age category $x$.

$e_x$: $=T_x/l_x$; $e_{k-1}=1/2$. Life expectancy, which is the mean number of age categories remaining until death for individuals surviving to the beginning of age category $x$.

For all $x$, $e_{x+1}+1>e_x$. This means that the total expected lifespan increases monotonically. For instance, in the table above, the one-year-olds have an average age at death of $2.25+1=3.25$, compared to 2.70 for newborns. In effect, the age of death of older individuals is a distribution conditioned on the fact that they have survived to their present age.

It is common to study survivorship as a semilog plot of $l_x$ vs. $x$, known as a Survivorship Curve. A so-called $l_x m_x$ table can be used to calculate the mean generation time of a population. Two $l_x m_x$ tables are illustrated below.

$x$	$l_x$	$m_x$	$l_x m_x$	$xl_xm_x$
0	1.00	0.00	0.00	0.00
1	0.70	0.50	0.35	0.35
2	0.50	1.50	0.75	1.50
3	0.20	0.00	0.00	0.00
4	0.00	0.00	0.00	0.00
			$R_0=1.10$	$\sum=1.85$

$$T = {\sum xl_xm_x\over\sum l_xm_x}={1.85\over 1.10} = 1.68$$

$$r = {\ln R_0\over T}={\ln 1.10\over 1.68} = 0.057.$$

$x$	$l_x$	$m_x$	$l_x m_x$	$xl_xm_x$
0	1.00	0.00	0.00	0.00
1	0.70	0.00	0.00	0.00
2	0.50	2.00	1.00	2.00
3	0.20	0.50	0.10	0.30
4	0.00	0.00	0.00	0.00
			$R_0=1.10$	$\sum=2.30$

$$T = {\sum xl_xm_x\over\sum l_xm_x}={2.30\over 1.10} = 2.09$$

$$r = {\ln R_0\over T}={\ln 1.10\over 2.09} = 0.046.$$

$x$: Age category ($x=0$, 1, ..., $k$). These values can be in any convenient units, but must be chosen so that no observed lifespan extends past category $k-1$ (as in an $l_x$ table).

$l_x$: $=n_x/n_0$. Survivorship, which measures the proportion of individuals who survive to the beginning of age category $x$ (as in an $l_x$ table).

$m_x$: The average number of offspring produced by an individual in age category $x$ while in that age category. $\sum_{i=0}^k m_x$ therefore represents the average lifetime number of offspring produced by an individual of maximum lifespan.

$l_x m_x$: The average number of offspring produced by an individual within age category $x$ weighted by the probability of surviving to the beginning of that age category. $\sum_{i=0}^k l_xm_x$ therefore represents the average lifetime number of offspring produced by a member of the study population. It is called the net reproductive rate per generation and is often denoted $R_0$.

$xl_xm_x$: A column weighting the offspring counted in the previous column by their parents' age when they were born. Therefore, the ratio $T=\sum(xl_xm_x)/\sum(l_xm_x)$ is the mean generation time of the population.

The Malthusian Parameter $r$ measures the reproductive rate per unit time and can be calculated as $r=(\ln R_0)/T$. For an exponentially increasing population, the population size $N(t)$ at time $t$ is then given by

$$N(t)=N_0 e^{rt}.$$

In the above two tables, the populations have identical reproductive rates of $R_0=1.10$. However, the shift toward later reproduction in population 2 increases the generation time, thus slowing the rate of Population Growth. Often, a slight delay of reproduction decreases Population Growth more strongly than does even a fairly large reduction in reproductive rate.

See also Gompertz Curve, Logistic Growth Curve, Makeham Curve, Malthusian Parameter, Population Growth, Survivorship Curve