{"id":602,"date":"2017-08-30T17:03:42","date_gmt":"2017-08-30T15:03:42","guid":{"rendered":"http:\/\/abcabacus.org\/?page_id=602"},"modified":"2021-02-08T15:53:31","modified_gmt":"2021-02-08T14:53:31","slug":"annex-ii-transformation-of-cohort-death-into-deaths-by-age","status":"publish","type":"page","link":"http:\/\/abcabacus.org\/?page_id=602","title":{"rendered":"Annex II: Useful relationships between survivor ratios and person-years"},"content":{"rendered":"


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\nAnnex II<\/title><\/p>\n<style type=\"text\/css\">code{white-space: pre;}<\/style>\n<div id=\"header\">\n<h1 class=\"title\">Annex II<\/h1>\n<\/div>\n<div id=\"TOC\">\n<ul>\n<li><a href=\"#annex-ii-useful-relationships-between-survivor-ratios-and-person-years\"><span class=\"toc-section-number\">1<\/span> Annex II: Useful relationships between survivor ratios and person-years<\/a>\n<ul>\n<li><a href=\"#full-exposure-to-the-risk-of-dying\"><span class=\"toc-section-number\">1.1<\/span> Full exposure to the risk of dying<\/a><\/li>\n<li><a href=\"#partial-exposure-to-the-risk-of-dying\"><span class=\"toc-section-number\">1.2<\/span> Partial exposure to the risk of dying<\/a>\n<ul>\n<li><a href=\"#additive-assumption\"><span class=\"toc-section-number\">1.2.1<\/span> Additive assumption<\/a><\/li>\n<li><a href=\"#multiplicative-assumption\"><span class=\"toc-section-number\">1.2.2<\/span> Multiplicative assumption<\/a><\/li>\n<li><a href=\"#comparison-of-the-additive-and-multiplicative-approach\"><span class=\"toc-section-number\">1.2.3<\/span> Comparison of the additive and multiplicative approach<\/a><\/li>\n<li><a href=\"#summary\"><span class=\"toc-section-number\">1.2.4<\/span> Summary<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li><a href=\"#bibliography\"><span class=\"toc-section-number\">2<\/span> Bibliography<\/a><\/li>\n<\/ul>\n<\/div>\n<h1 id=\"annex-ii-useful-relationships-between-survivor-ratios-and-person-years\"><span class=\"header-section-number\">1<\/span> Annex II: Useful relationships between survivor ratios and person-years<\/h1>\n<p>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<sub>x<\/sub> 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<sub>x<\/sub> 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 here<a href=\"#fn1\" class=\"footnoteRef\" id=\"fnref1\"><sup>1<\/sup><\/a>.<\/p>\n<p>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).<\/p>\n<h2 id=\"full-exposure-to-the-risk-of-dying\"><span class=\"header-section-number\">1.1<\/span> Full exposure to the risk of dying<\/h2>\n<p>Closed age groups:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-31ac4ac9e9f01b0a82f7d6026e4fdd33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n{S_x} = {{{}_n{L_{x + n}}} \over {{}_n{L_x}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"96\" style=\"vertical-align: -8px;\"\/><\/p>\n<p>First Lexis triangle (survival from birth to age group 0 to n):<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-e9d453f54b6f440e82e628276a9f8406_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n^u{S_0} = {{{}_n{L_0}} \over {n*{l_0}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"80\" style=\"vertical-align: -8px;\"\/><\/p>\n<p>To distinguish the survival ratio for the first Lexis triangle from the other, some demographers index this one with -1:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-0fc54e6335b2c5b7c5cff2254438d2c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n^u{S_{ - 1}} = {{{}_n{L_0}} \over {n*{l_0}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"91\" style=\"vertical-align: -8px;\"\/><\/p>\n<p>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.<\/p>\n<p>If last age of population and life table are the same <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-bf2ef0d576bc2d80397c2c8e9978feed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\left( {\omega = z} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\"\/>, then<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-5585956ee3c9975e799738cb09073faf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_\infty {S_{z - n}} = {{{T_z}} \over {{T_{z - n}}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"110\" style=\"vertical-align: -8px;\"\/><\/p>\n<p>If the life table has more age groups than the population, then<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-632f377d83b3c8fc0c9a11511640bf88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_\infty {S_z} = {{{T_{z + n}}} \over {{T_z}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"91\" style=\"vertical-align: -8px;\"\/><\/p>\n<p>The complement of the survivor ratio calculates the deaths\/withdrawals <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-380f50493d458b4f35acf0a75cabda48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n{W_x}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"34\" style=\"vertical-align: -3px;\"\/> associated with it. Note that these deaths are in a period-cohort age format.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-510c0b542228dc15d0e7ba8276263377_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"321\" style=\"vertical-align: -8px;\"\/><\/p>\n<p>The last line shows that the withdrawal proportion <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-380f50493d458b4f35acf0a75cabda48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n{W_x}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"34\" style=\"vertical-align: -3px;\"\/> is the ratio of the cohort deaths of the stationary population to the stationary population at the beginning of the age interval.<\/p>\n<p>Other useful relationships are:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-d93934edea43b0b93015d118e68ac16c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1 + {}_n{S_x} = 1 + {{{}_n{L_{x + n}}} \over {{}_n{L_x}}} = {{{}_n{L_x} + {}_n{L_{x + n}}} \over {{}_n{L_x}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"262\" style=\"vertical-align: -8px;\"\/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-192208620fc7904eecd061854ecc934b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{{1 + {}_n{S_x}} \over {{}_n{S_x}}} = {{{}_n{L_x} + {}_n{L_{x + n}}} \over {{}_n{L_{x + n}}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"147\" style=\"vertical-align: -10px;\"\/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-6de2131b9e88a09680434eaa4d6579b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{{1 - {}_n{S_x}} \over {1 + {}_n{S_x}}} = {{{}_n{L_x} - {}_n{L_{x + n}}} \over {{}_n{L_x} + {}_n{L_{x + n}}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"147\" style=\"vertical-align: -10px;\"\/><\/p>\n<h2 id=\"partial-exposure-to-the-risk-of-dying\"><span class=\"header-section-number\">1.2<\/span> Partial exposure to the risk of dying<\/h2>\n<p>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.<\/p>\n<p>We start by recalling that the risk of dying in a life table is represented by <sub>n<\/sub>q<sub>x<\/sub>, the probability of dying between age <em>x<\/em> and <em>x+n<\/em>, and where <sub>n<\/sub>p<sub>x<\/sub> is the corresponding probability of surviving from age age <em>x<\/em> to age <em>x+n<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-ecb2909ac4f4b319e7c656a0b88e513d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\eqalign{ & {}_n{q_x} = 1 - {}_n{p_x} \cr & {}_n{p_x} = 1 - {}_n{q_x} \cr}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"212\" style=\"vertical-align: -4px;\"\/><\/p>\n<p>After the formulas have been developed by using the usual life table analogy, the probability of surviving <sub>n<\/sub>p<sub>x<\/sub> can be replaced by the survivor ratio needed for the cohort-component model.<\/p>\n<h3 id=\"additive-assumption\"><span class=\"header-section-number\">1.2.1<\/span> Additive assumption<\/h3>\n<p>In the additive assumption, partial exposure to the risk of dying is assumed to be linearly related to survival.<\/p>\n<h4 id=\"survival-associated-with-half-the-risk-of-dying\"><span class=\"header-section-number\">1.2.1.1<\/span> Survival associated with half the risk of dying<\/h4>\n<p>The probability of survival associated with half the probability of dying is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-6315ba568a4f286db544509a19dbe413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{p_x} = 1 - {{{}_n{q_x}} \over 2}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"120\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>This relationship can be expressed using the probabilities of surviving only<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-626c6dd1a148debbe2f087b4e4f8d904_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{p_x} = 1 - {{\left( {1 - {}_n{p_x}} \right)} \over 2}\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"149\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>This formula can further be simplified to:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-efd74e4c52c16f6dd6b56357785c79da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{p_x} = {1 \over 2}\left( {1 + {}_n{p_x}} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"148\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>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 <em>x<\/em> to <em>x+n<\/em>, that is <sub>n<\/sub>p<sub>x<\/sub>.<\/p>\n<p>The associated probability of dying is then:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-30ba4f49debda3eece2022327216118c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"501\" style=\"vertical-align: -7px;\"\/><\/p>\n<h4 id=\"survival-associated-with-two-thirds-the-risk-of-dying\"><span class=\"header-section-number\">1.2.1.2<\/span> Survival associated with two-thirds the risk of dying<\/h4>\n<p>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:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-f93182eda168f27c544f636aa6494574_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{2n/3}{p_x} = 1 - {2 \over 3}{}_n{q_x}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"137\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>This relationship can be expressed using the probabilities of surviving only<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-b095e7a763394067f52717192ccb11ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{2n/3}{p_x} = 1 - {2 \over 3}\left( {1 - {}_n{p_x}} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"185\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>This formula can further be simplified<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-67b2d11b158e1fe1b59c0cc55e1bb7ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"552\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>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 <em>x<\/em> to <em>x+n<\/em> (<sub>n<\/sub>p<sub>x).<\/sub><\/p>\n<p>The associated probability of dying is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-d6ce1e0648dc8b2d827f2f71711cc233_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"316\" style=\"vertical-align: -7px;\"\/><\/p>\n<h3 id=\"multiplicative-assumption\"><span class=\"header-section-number\">1.2.2<\/span> Multiplicative assumption<\/h3>\n<p>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).<\/p>\n<h4 id=\"survival-associated-with-half-the-risk-of-dying-1\"><span class=\"header-section-number\">1.2.2.1<\/span> Survival associated with half the risk of dying<\/h4>\n<p>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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-bc790b389ffbbeff67b41886f37480e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{p_x} = \root 2 \of {{}_n{p_x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>Similarly, the probability of dying expressed with the survival probability is then<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-f78e5653195bb39a5708a1c3d72cf0f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{q_x} = 1 - \root 2 \of {{}_n{p_x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -7px;\"\/><\/p>\n<h4 id=\"survival-associated-with-two-thirds-the-risk-of-dying-1\"><span class=\"header-section-number\">1.2.2.2<\/span> Survival associated with two-thirds the risk of dying<\/h4>\n<p>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<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-6cee833041d02112588800e41982bf49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"223\" style=\"vertical-align: -10px;\"\/><\/p>\n<p>Similarly, the probability of dying expressed with the given survival probability is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-a73c68199ddbfe039dcfe3567262f4f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"269\" style=\"vertical-align: -7px;\"\/><\/p>\n<h3 id=\"comparison-of-the-additive-and-multiplicative-approach\"><span class=\"header-section-number\">1.2.3<\/span> Comparison of the additive and multiplicative approach<\/h3>\n<p>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.<\/p>\n<p>Assume a population at time <em>t<\/em> of 100,000, exposed to a survivor ratio of 0.90. The population one projection interval ahead would then be<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-2c402721a6636fa7f351e1c0e646f428_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"345\" style=\"vertical-align: -4px;\"\/><\/p>\n<p>The associated number of deaths during the period is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-75290c211b2ed4b4836c2062cfc48119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"446\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>We now calculate the survivors for half the period, e.g. [t, t+n\/2]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-40ff81e6b2ff05bfa159346a172dd12e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"640\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-cb7885a5ac5cabb915bf2daee80ee84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"438\" style=\"vertical-align: -7px;\"\/><\/p>\n<p>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.<\/p>\n<p>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:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-c3342d9042563e024c5f4da1c2394c39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"410\" style=\"vertical-align: -4px;\"\/><\/p>\n<p>Second, the mid-interval population <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-bcec46ce2a4a9b9bf6d73a872b54c33c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{P^{t + n/2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: 0px;\"\/> is forward survived to the end of the interval:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-fa6e6a986f56392875ec6ae5db5efc13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\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 }\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"420\" style=\"vertical-align: -4px;\"\/><\/p>\n<p>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.<\/p>\n<h3 id=\"summary\"><span class=\"header-section-number\">1.2.4<\/span> Summary<\/h3>\n<p>Survivor ratios and withdrawal ratios for fractional exposure in terms of survivor ratios are summarized in the following tables.<\/p>\n<p>Table 1: Survival and withdrawal ratios, additive option<\/p>\n<table>\n<thead>\n<tr class=\"header\">\n<th><strong>Exposure<\/strong><\/th>\n<th><strong>Surviving<\/strong><\/th>\n<th><strong>Withdrawal<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"odd\">\n<td>Full<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-a958a3c4c234ca7aa048253e5f29c42a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n{S_x}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"28\" style=\"vertical-align: -3px;\"\/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-78b6fa14b1661fa6ba276787ed3f0806_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n{W_x} = 1 - {}_n{S_x}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"117\" style=\"vertical-align: -3px;\"\/><\/td>\n<\/tr>\n<tr class=\"even\">\n<td>1\/3<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-b9dd135ee0d09ab6832593eedb7f74ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/3}{S_x} = {1 \over 3}\left( {2 + {}_n{S_x}} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"152\" style=\"vertical-align: -7px;\"\/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-09385d4a96d6882d14d86e21c7e1c749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/3}{W_x} = {1 \over 3}\left( {1 - {}_n{S_x}} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"158\" style=\"vertical-align: -7px;\"\/><\/td>\n<\/tr>\n<tr class=\"odd\">\n<td>1\/2<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-5c66d8cf7a2911ab3628a9f2209e140c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{S_x} = {1 \over 2}\left( {1 + {}_n{S_x}} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"152\" style=\"vertical-align: -7px;\"\/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-f45ed10091cd0d587e83521eb82aa3ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{W_x} = {1 \over 2}\left( {1 - {}_n{S_x}} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"158\" style=\"vertical-align: -7px;\"\/><\/td>\n<\/tr>\n<tr class=\"even\">\n<td>2\/3<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-1efa99b70ea99627b5a0064b1acb002f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{2n/3}{S_x} = {1 \over 3}\left( {1 + 2*{}_n{S_x}} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"184\" style=\"vertical-align: -8px;\"\/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-41f4f53954674926278ed9ae57a07f47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{2n/3}{W_x} = {1 \over 3}\left( {1 - {}_n{S_x}} \right)\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"164\" style=\"vertical-align: -8px;\"\/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Table 2: Survival and withdrawal ratios, multiplicative option<\/p>\n<table>\n<thead>\n<tr class=\"header\">\n<th><strong>Exposure<\/strong><\/th>\n<th><strong>Surviving<\/strong><\/th>\n<th><strong>Withdrawal<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"odd\">\n<td>Full<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-a958a3c4c234ca7aa048253e5f29c42a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n{S_x}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"28\" style=\"vertical-align: -3px;\"\/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-78b6fa14b1661fa6ba276787ed3f0806_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_n{W_x} = 1 - {}_n{S_x}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"117\" style=\"vertical-align: -3px;\"\/><\/td>\n<\/tr>\n<tr class=\"even\">\n<td>1\/3<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-7f23161bea4f51d0e80842fd5a4d6e3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/3}{S_x} = \root 3 \of {{}_n{S_x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"111\" style=\"vertical-align: -8px;\"\/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-f11f444210d829d1028c76b86f441c47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/3}{W_x} = 1 - \root 3 \of {{}_n{S_x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"147\" style=\"vertical-align: -8px;\"\/><\/td>\n<\/tr>\n<tr class=\"odd\">\n<td>1\/2<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-4bf5f80c2a5226dd58aa6e3b80ebbdd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{S_x} = \root 2 \of {{}_n{S_x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"111\" style=\"vertical-align: -8px;\"\/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-aabec406aeff257f871d3ee0853319bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{n/2}{W_x} = 1 - \root 2 \of {{}_n{S_x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"147\" style=\"vertical-align: -8px;\"\/><\/td>\n<\/tr>\n<tr class=\"even\">\n<td>2\/3<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-1992db40d0122bc8deb61d626694e3de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{2n/3}{S_x} = \root 3 \of {{{\left( {{S_x}} \right)}^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"133\" style=\"vertical-align: -9px;\"\/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/abcabacus.org\/wp-content\/ql-cache\/quicklatex.com-683f2ba387653eaa9b990fb7e2993660_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"{}_{2n/3}{W_x} = 1 - \root 3 \of {{{\left( {{S_x}} \right)}^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"169\" style=\"vertical-align: -9px;\"\/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1 id=\"bibliography\"><span class=\"header-section-number\">2<\/span> Bibliography<\/h1>\n<p>Arriaga, Eduardo E. 1994. <em>Population Analysis with Micromputers. Volume II: Software and Documentation<\/em>. <a href=\"https:\/\/www.census.gov\/population\/international\/software\/pas\/pasdocs.html\" class=\"uri\">https:\/\/www.census.gov\/population\/international\/software\/pas\/pasdocs.html<\/a>.<\/p>\n<p>Bohk, C. (2011). Ein probabilistisches Bev\u00f6lkerungsprognosemodel. Springer VS, Wiesbaden. (A probabilistic population projection model)<\/p>\n<p>Bretz, M. (2000). Methoden der Bev\u00f6lkerungsvorausberechnung in: M\u00fcller U., Nauck B., Dieckmann A. “Handbuch der Demographie” Band 1, S. 643-681.<\/p>\n<p>Hill, K. (1990). PROJ3S \u2013 A Computer Program for Population Projections: Diskettes and Reference Guide.<\/p>\n<p>Hinde, Andrew. (1998). <em>Demographic Methods<\/em>. London: Arnold.<\/p>\n<p>Statistisches Bundesamt (DESTATIS). 2010. \u201cModel der Bev\u00f6lkerungsvorausberechnungen.\u201d Wiesbaden, Germany.<\/p>\n<p>Keilman, Nico. 2000. \u201cReview: Hinde (1998). Demographic Methods.\u201d <em>European Journal of Population\/ Revue Europenne de Dmographie<\/em> 16 (2): 187\u201388. doi:10.1023\/A:1006353422154.<\/p>\n<p>Australian Bureau of Statistics. 1999. \u201cDemographic Estimates and Projections: Concepts, Sources and Methods, 1999.\u201d Canberra. http:\/\/www.abs.gov.au\/ausstats\/abs@.nsf\/mf\/3228.0.<\/p>\n<div class=\"footnotes\">\n<hr>\n<ol>\n<li id=\"fn1\">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.<a href=\"#fnref1\">\u21a9<\/a><\/li>\n<\/ol>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Annex II 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 1.2.1 Additive assumption 1.2.2 Multiplicative assumption 1.2.3 Comparison of the additive and multiplicative approach 1.2.4 Summary 2 Bibliography 1 Annex II: Useful relationships between survivor … <a href=\"http:\/\/abcabacus.org\/?page_id=602\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> “Annex II: Useful relationships between survivor ratios and person-years”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":374,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-602","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/abcabacus.org\/index.php?rest_route=\/wp\/v2\/pages\/602","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/abcabacus.org\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/abcabacus.org\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/abcabacus.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/abcabacus.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=602"}],"version-history":[{"count":6,"href":"http:\/\/abcabacus.org\/index.php?rest_route=\/wp\/v2\/pages\/602\/revisions"}],"predecessor-version":[{"id":743,"href":"http:\/\/abcabacus.org\/index.php?rest_route=\/wp\/v2\/pages\/602\/revisions\/743"}],"up":[{"embeddable":true,"href":"http:\/\/abcabacus.org\/index.php?rest_route=\/wp\/v2\/pages\/374"}],"wp:attachment":[{"href":"http:\/\/abcabacus.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}