Annex II

1 Annex II: Useful relationships between survivor ratios and person-years
- 1.1 Full exposure to the risk of dying
- 1.2 Partial exposure to the risk of dying
2 Bibliography

1 Annex II: Useful relationships between survivor ratios and person-years

The period life table forms the basis for developing the mathematics of population projections in this documentation, including the calculation of associated events (deaths, migrants). It is helpful to recall that the L_x column of the life table represents the person-years lived of a life table population and therefore represents a (stationary) population. Consequently, using the L_x column of the life table and associated statistics (like the survivor proportion), the cohort component projection method was developed. There are other approaches to formulate the cohort-component method, but they are not covered here¹.

In this annex, we explore the relationship between survivor ratios and person years by developing useful expressions for full or partial exposure to the risks of surviving or withdrawal (outmigration, dying).

1.1 Full exposure to the risk of dying

Closed age groups:

${}_n{S_x} = {{{}_n{L_{x + n}}} \over {{}_n{L_x}}}$

First Lexis triangle (survival from birth to age group 0 to n):

${}_n^u{S_0} = {{{}_n{L_0}} \over {n*{l_0}}}$

To distinguish the survival ratio for the first Lexis triangle from the other, some demographers index this one with -1:

${}_n^u{S_{ - 1}} = {{{}_n{L_0}} \over {n*{l_0}}}$

The survivor ratio for the last open-ended age group is calculated differently depending on the availability of an extended age categories for the underlying life table.

If last age of population and life table are the same $\left( {\omega = z} \right)$ , then

${}_\infty {S_{z - n}} = {{{T_z}} \over {{T_{z - n}}}}$

If the life table has more age groups than the population, then

${}_\infty {S_z} = {{{T_{z + n}}} \over {{T_z}}}$

The complement of the survivor ratio calculates the deaths/withdrawals ${}_n{W_x}$ associated with it. Note that these deaths are in a period-cohort age format.

$\matrix{ {{}_n{W_x}} \hfill & = \hfill & {1 - {}_n{S_x}} \hfill \cr {} \hfill & = \hfill & {1 - {{{}_n{L_{x + n}}} \over {{}_n{L_x}}}} \hfill \cr {} \hfill & = \hfill & {{{{}_n{L_x} - {}_n{L_{x + n}}} \over {{}_n{L_x}}}} \hfill \cr }$

The last line shows that the withdrawal proportion ${}_n{W_x}$ is the ratio of the cohort deaths of the stationary population to the stationary population at the beginning of the age interval.

Other useful relationships are:

$1 + {}_n{S_x} = 1 + {{{}_n{L_{x + n}}} \over {{}_n{L_x}}} = {{{}_n{L_x} + {}_n{L_{x + n}}} \over {{}_n{L_x}}}$

${{1 + {}_n{S_x}} \over {{}_n{S_x}}} = {{{}_n{L_x} + {}_n{L_{x + n}}} \over {{}_n{L_{x + n}}}}$

${{1 - {}_n{S_x}} \over {1 + {}_n{S_x}}} = {{{}_n{L_x} - {}_n{L_{x + n}}} \over {{}_n{L_x} + {}_n{L_{x + n}}}}$

1.2 Partial exposure to the risk of dying

We want to find an expression for the case when the population is not fully exposed to risk of dying or withdrawal, as often is applied to (international) migration. We will develop the mathematics using ordinary life table variables, and then translate these into the indicators used in the cohort-component framework.

We start by recalling that the risk of dying in a life table is represented by _nq_x, the probability of dying between age x and x+n, and where _np_x is the corresponding probability of surviving from age age x to age x+n.

$\eqalign{ & {}_n{q_x} = 1 - {}_n{p_x} \cr & {}_n{p_x} = 1 - {}_n{q_x} \cr}$

After the formulas have been developed by using the usual life table analogy, the probability of surviving _np_x can be replaced by the survivor ratio needed for the cohort-component model.

1.2.1 Additive assumption

In the additive assumption, partial exposure to the risk of dying is assumed to be linearly related to survival.

1.2.1.1 Survival associated with half the risk of dying

The probability of survival associated with half the probability of dying is

${}_{n/2}{p_x} = 1 - {{{}_n{q_x}} \over 2}$

This relationship can be expressed using the probabilities of surviving only

${}_{n/2}{p_x} = 1 - {{\left( {1 - {}_n{p_x}} \right)} \over 2}$

This formula can further be simplified to:

${}_{n/2}{p_x} = {1 \over 2}\left( {1 + {}_n{p_x}} \right)$

We now have an expression for the survival associated with half the risk of dying, and fully expressed in the (known) quantity of the probability of surviving from age x to x+n, that is _np_x.

The associated probability of dying is then:

$\matrix{ {{}_{n/2}{q_x}} \hfill & = \hfill & {1 - {1 \over 2}\left( {1 + {}_n{p_x}} \right)} \hfill \cr {} \hfill & = \hfill & {1 - \left( {{1 \over 2} + {{{}_n{p_x}} \over 2}} \right)} \hfill \cr {} \hfill & = \hfill & {{{2 - 1 - {}_n{p_x}} \over 2}} \hfill \cr {} \hfill & = \hfill & {{1 \over 2}\left( {1 - {}_n{p_x}} \right)} \hfill \cr }$

1.2.1.2 Survival associated with two-thirds the risk of dying

Pollard suggested exposing migrants in the first age group to one third the risk of dying. The probability of survival associated with half the probability of dying can be written as:

${}_{2n/3}{p_x} = 1 - {2 \over 3}{}_n{q_x}$

This relationship can be expressed using the probabilities of surviving only

${}_{2n/3}{p_x} = 1 - {2 \over 3}\left( {1 - {}_n{p_x}} \right)$

This formula can further be simplified

$\matrix{ {{}_{2n/3}{p_x}} \hfill & = \hfill & {{3 \over 3} - {{2\left( {1 - {}_n{p_x}} \right)} \over 3} = {{3 - 2\left( {1 - {}_n{p_x}} \right)} \over 3} = {{3 - 2 + 2{}_n{p_x}} \over 3} = {{1 + 2{}_n{p_x}} \over 3}} \hfill \cr {} \hfill & = \hfill & {{1 \over 3}\left( {1 + 2{}_n{p_x}} \right)} \hfill \cr }$

As for the case of half the risk of dying before, we now have an expression for the survival associated with two-thirds the risk of dying, and fully expressed in the (known) quantity of the probability of surviving from age x to x+n (_np_x).

The associated probability of dying is:

$\matrix{ {{}_{2n/3}{q_x}} \hfill & = \hfill & {1 - \left[ {{1 \over 3}\left( {1 + 2{}_n{p_x}} \right)} \right]} \hfill \cr {} \hfill & = \hfill & {{2 \over 3}\left( {1 - {}_n{p_x}} \right)} \hfill \cr }$

1.2.2 Multiplicative assumption

Under the assumption of a constant force of mortality through an age interval, partial probabilities are linked to each other in a multiplicative way (Hill, 1990).

1.2.2.1 Survival associated with half the risk of dying

Assuming the force of mortality is constant throughout the interval, the probability of surviving associated with half the risk of dying is simply the square root of the survival probability.

${}_{n/2}{p_x} = \root 2 \of {{}_n{p_x}}$

Similarly, the probability of dying expressed with the survival probability is then

${}_{n/2}{q_x} = 1 - \root 2 \of {{}_n{p_x}}$

1.2.2.2 Survival associated with two-thirds the risk of dying

Assuming the force of mortality is constant throughout the interval; partial probabilities are linked to each other in a multiplicative way. Under such assumption, the probability of surviving associated with two-thirds of the risk of dying is

$\matrix{ {{}_{2n/3}{p_x}} \hfill & = \hfill & {\root 3 \of {{{\left( {{}_n{p_x}} \right)}^2}} } \hfill \cr {} \hfill & = \hfill & {{{\left( {{}_n{p_x}} \right)}^{2/3}}} \hfill \cr }$

Similarly, the probability of dying expressed with the given survival probability is

$\matrix{ {{}_{2n/3}{q_x}} \hfill & = \hfill & {1 - \root 3 \of {{}_n{p_x}^2} } \hfill \cr {} \hfill & = \hfill & {1 - {{\left( {{}_n{p_x}} \right)}^{2/3}}} \hfill \cr }$

1.2.3 Comparison of the additive and multiplicative approach

We want now to compare the additive approach (most commonly used) with the multiplicative approach. It should be noted that the additive approach is a convenient approximation for exact multiplicative approach. Nevertheless, a comparison, with numerical examples, might be interesting.

Assume a population at time t of 100,000, exposed to a survivor ratio of 0.90. The population one projection interval ahead would then be

$\matrix{ {{P^{t + n}}} \hfill & = \hfill & {{P^t}*{S^{t,t + n}}} \hfill \cr {} \hfill & = \hfill & {100,000*0.9} \hfill \cr {} \hfill & = \hfill & {90,000} \hfill \cr }$

The associated number of deaths during the period is

$\matrix{ {{D^{t,t + n}}} \hfill & = \hfill & {{P^t}*\left( {1 - {S^{t,t + n}}} \right)} \hfill \cr {} \hfill & = \hfill & {100,000*(1 - 0.9)} \hfill \cr {} \hfill & = \hfill & {10,000} \hfill \cr }$

We now calculate the survivors for half the period, e.g. [t, t+n/2]

$\matrix{ {{P^{t + n/2}}} \hfill & = \hfill & {{P^t}*{1 \over 2}\left( {1 + {S^{t,t + n}}} \right)} \hfill \cr {} \hfill & = \hfill & {100,000*\left( {{1 \over 2}*\left( {1 + 0.9} \right)} \right)} \hfill \cr {} \hfill & = \hfill & {100,000*0.95} \hfill \cr {} \hfill & = \hfill & {95,000} \hfill \cr }$

A problem arises if we continue to survive the 95,000 survivors to the end of the interval with the same adjusted by half survivor ratio of 0.95.

$\matrix{ {{P^{t + n}}} \hfill & = \hfill & {{P^{t + n/2}}*{1 \over 2}\left( {1 + {S^{t,t + n}}} \right)} \hfill \cr {} \hfill & = \hfill & {95,000*0.95} \hfill \cr {} \hfill & = \hfill & {90,250} \hfill \cr }$

Surprisingly, the additive approach produces 250 additional survivors when compared with the expected 90,000 survivors using the original survivor ratio for the full interval.

We now show that the multiplicative approach is consistent between full and half period survival. First, the base population is forward survived to the middle of the interval:

$\matrix{ {{P^{t + n/2}}} \hfill & = \hfill & {{P^t}*\root 2 \of {{S^{t,t + n}}} } \hfill \cr {} \hfill & = \hfill & {100,000*0.94868} \hfill \cr {} \hfill & = \hfill & {94,868} \hfill \cr }$

Second, the mid-interval population ${P^{t + n/2}}$ is forward survived to the end of the interval:

$\matrix{ {{P^{t + n}}} \hfill & = \hfill & {{P^{t + n/2}}*\root 2 \of {{S^{t,t + n}}} } \hfill \cr {} \hfill & = \hfill & {94,868*0.94868} \hfill \cr {} \hfill & = \hfill & {90,000} \hfill \cr }$

Note that the relative error incurred by applying the short-hand approach of an additive model is smallest when survival ratios are close to 1.0, and grow when survival ratios get smaller.

1.2.4 Summary

Survivor ratios and withdrawal ratios for fractional exposure in terms of survivor ratios are summarized in the following tables.

Table 1: Survival and withdrawal ratios, additive option

Exposure	Surviving	Withdrawal
Full	${}_n{S_x}$	${}_n{W_x} = 1 - {}_n{S_x}$
1/3	${}_{n/3}{S_x} = {1 \over 3}\left( {2 + {}_n{S_x}} \right)$	${}_{n/3}{W_x} = {1 \over 3}\left( {1 - {}_n{S_x}} \right)$
1/2	${}_{n/2}{S_x} = {1 \over 2}\left( {1 + {}_n{S_x}} \right)$	${}_{n/2}{W_x} = {1 \over 2}\left( {1 - {}_n{S_x}} \right)$
2/3	${}_{2n/3}{S_x} = {1 \over 3}\left( {1 + 2*{}_n{S_x}} \right)$	${}_{2n/3}{W_x} = {1 \over 3}\left( {1 - {}_n{S_x}} \right)$

Table 2: Survival and withdrawal ratios, multiplicative option

Exposure	Surviving	Withdrawal
Full	${}_n{S_x}$	${}_n{W_x} = 1 - {}_n{S_x}$
1/3	${}_{n/3}{S_x} = \root 3 \of {{}_n{S_x}}$	${}_{n/3}{W_x} = 1 - \root 3 \of {{}_n{S_x}}$
1/2	${}_{n/2}{S_x} = \root 2 \of {{}_n{S_x}}$	${}_{n/2}{W_x} = 1 - \root 2 \of {{}_n{S_x}}$
2/3	${}_{2n/3}{S_x} = \root 3 \of {{{\left( {{S_x}} \right)}^2}}$	${}_{2n/3}{W_x} = 1 - \root 3 \of {{{\left( {{S_x}} \right)}^2}}$

2 Bibliography

Arriaga, Eduardo E. 1994. Population Analysis with Micromputers. Volume II: Software and Documentation. https://www.census.gov/population/international/software/pas/pasdocs.html.

Bohk, C. (2011). Ein probabilistisches Bevölkerungsprognosemodel. Springer VS, Wiesbaden. (A probabilistic population projection model)

Bretz, M. (2000). Methoden der Bevölkerungsvorausberechnung in: Müller U., Nauck B., Dieckmann A. “Handbuch der Demographie” Band 1, S. 643-681.

Hill, K. (1990). PROJ3S – A Computer Program for Population Projections: Diskettes and Reference Guide.

Hinde, Andrew. (1998). Demographic Methods. London: Arnold.

Statistisches Bundesamt (DESTATIS). 2010. “Model der Bevölkerungsvorausberechnungen.” Wiesbaden, Germany.

Keilman, Nico. 2000. “Review: Hinde (1998). Demographic Methods.” European Journal of Population/ Revue Europenne de Dmographie 16 (2): 187–88. doi:10.1023/A:1006353422154.

Australian Bureau of Statistics. 1999. “Demographic Estimates and Projections: Concepts, Sources and Methods, 1999.” Canberra. http://www.abs.gov.au/ausstats/abs@.nsf/mf/3228.0.

See, for instance, the approach used by the US Census Bureau that is using exposure-occurrence rates centered around mid-period and explicitly including half-year intervals (Arriaga, 1994, p. 371-374). Bokh (2011) used a similar approach. Other authors (Hinde 1999, Australian Bureau of Statistics 1999, Statistics UK, DESTATIS 2011, Breetz, 2000.) formulate the mathematics of the projection method in terms of true probabilities instead of survival ratios, which is a crude approximation (see Keilman 2000 for a critique). It seems that there is no authoritative, consistent and comprehensive mathematical description of the cohort-component available.↩