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Life tables, cohort-component projections Pia Wohland Basics of demographic analysis (2):

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Why do we need life tables to measure mortality experience? Life expectancy at birth, UK, from period life tables, 1980-82 to 2005-07 Mortality rates, by age, England and Wales

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Life table observed data:D x and P x mortality rates:M x = D x /P x mortality probabilities:q x = M x /(1 + 0.5 M x ) survival probabilities:p x = 1 - q x number surviving:l x = l x-1 p x-1 number dying:d x = l x – l x+1 life years lived:L x = l x+1 + 0.5l x total life years lived:T x = T x+1 + L x life expectancy:e x = T x /l x

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Age-time graph (aka Lexis diagram) Age-time spaces: Period: e.g. 2003, vertical column Age: e.g. 2, horizontal row Cohort: e.g. born 2003, diagonal Period-age: e.g. 2 in 2005, square Period-cohort: e.g. 1 on 1.1.04 to 2 on 1.1.05, parallelogram Age-cohort: e.g. birthday 1 in 2004 to birthday 2 in 2005, parallelogram Age-period-cohort: e.g. age1, 2004, born 2003, triangle Point and interval measures Referring to exact age: l x, T x, e x Functions referring to the interval between exact ages x to x+n: n q x, n p x, n d x, n L x

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AgeDeathsPopulation xDxDx PxPx 0D0D0 P0P0 1D1D1 P1P1 2D2D2 P2P2 ::: 99D 99 P 99 100 (=100+)D 100 P 100 Given data

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Life table, step 1: UK males 1998, numbers AgeDeathsPopulation xDxDx PxPx 02327364800 1190375100 2117369000 ::: 99215561 100319678

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Life table, step 2: compute observed mortality rates by age, M x Mortality rate at age x = Deaths at age x/Population at risk at age x Deaths are recorded in period-age age-time spaces Population at risk either mid-year population aged x or average of start and end of year population aged x

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Life table, step 2: compute observed mortality rates by age, M x, variables AgeDeathsPopulatio n Mortality rates xDxDx PxPx M x =D x /P x 0D0D0 P0P0 M 0 =D 0 /P 0 1D1D1 P1P1 M 1 =D 1 /P 1 2D2D2 P2P2 M 2 =D 2 /P 2 :::: 99D 99 P 99 M 99 =D 99 /P 99 100 (=100+)D 100 P 100 M 100 =D 100 /P 10 0

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Life table, step 2: compute observed mortality rates by age, M x, UK males 1998, numbers AgeDeathsPopulationMortality rate xDxDx PxPx MxMx 023273648000.006379 11903751000.000507 21173690000.000317 :::: 992155610.382851 1003196780.469674

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Life table, step 3: compute the probabilities of dying between age x and x+1, q x The equation q x = M x /(1 + 0.5M x ) sometimes given as q x = 2M x /(2 + M x ) Derivation Persons with x th birthday in year = P x + 0.5 D x Probability of dying between ages x and x+1 = D x /(P x + 0.5 D x ) Divide top and bottom by P x q x = (D x /P x ) / ([P x /P x ] + 0.5 [D x /P x ]) Substitute definition of M x q x = M x / (1 + 0.5 M x )

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x+1 x period MxMx Assume MxMx P x + 0.5 D x PxPx DxDx Step 3: graphical illustration using age-time diagram

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Life table, step 3: compute the probabilities of dying between age x and x+1, q x, variables AgeMortality rate Mortality probability x M x q x 0M0M0 q 0 =M 0 /(1+0.5M 0 ) 1M1M1 q 1 =M 1 /(1+0.5M 1 ) 2M2M2 q 2 =M 2 /(1+0.5M 2 ) :: 99M 99 q 99 =M 99 /(1+0.5M 99 ) 100 (=100+)M 100 q 100 = 1.0000

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Life table, step 3: compute the probabilities of dying between age x and x+1, q x, UK males 1998 AgeMortality rateMortality probability x M x q x 00.0063790.006342 10.0005070.000506 20.000317 ::: 990.3828510.321338 100 (=100+)0.4696741.000000

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Life table, step 4: compute the probabilities of surviving from age x to x+1, p x Probability of survival from age x to x+1, p x p x = (1 – q x ) x x+ 1 time Age t t+1

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Life table, step 4: compute the probabilities of surviving from age x to x+1, p x, UK males 1998 AgeMortality rate Mortality probability Survival probability xMxMx q x p x 00.0063790.0063420.993658 10.0005070.0005060.999494 20.000317 0.999683 :::: 990.3828510.3213380.678662 1000.4696741.0000000.000000

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Life table, step 5: compute the numbers surviving to age x, l x - concept Life tables have a radix (number base) = hypothetical constant number born each year into a stationary population Usually radix = 100000 but you can use 10000 or 1000000 or 1 (in this case the survivors variable has a probability interpretation) l x = number of survivors of birth cohort who have attained age x (exact age or birthday) The number surviving to age x is the number surviving to age x-1 times the probability of surviving from age x- 1 to age x

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Life table, step 5: compute the numbers surviving to age x, l x - formulae l x = l x-1 p x-1 e.g. l 2 = l 1 p 1 We can include prior equations to obtain an expression for l x which is linked to the radix l x = l 0 p 0 p 1 … p x-1 = l 0 y=0,x-1 p y A picture can clarify this

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Age Time 0 1 2 3 4 tt+1t+2t+3 l0l0 l1l1 p0p0 p1p1 l2l2 p2p2 l3l3 Surviving a birth cohort from one birthday to the next

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Life table, step 5: compute the numbers surviving to age x, l x, UK males 1998 AgeSurvival probability Cohort survivors x p x l x 00.993658100000 10.99949499366 20.99968399315 ::: 990.678662619 1000.000000420

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Life table, step 6: compute the numbers dying between ages x and x+1, d x Some of the stationary birth cohort will die between exact ages The number, d x, is computed from Successive cohort survivors d x = l x – l x+1 Multiplication of cohort survivors by mortality probability d x = l x q x

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Life table, step 6: compute the numbers dying between ages x and x+1, d x AgeMortality probability Cohort survivors Cohort non- survivors x q x l x d x 00.006342100000634 10.0005069936650 20.0003179931531 :::: 990.321338619199 1001.000000420

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Life table, step 8: compute the life years lived between ages x and x+1, L x, concepts The next step is to work out how many years people in the birth cohort live between exact ages This quantity is called L x and is made up of two parts, the life years lived by persons surviving through the interval and the life years lived by those who die in the interval We make an assumption about the fraction of the interval, called a x that non-survivors are alive for e.g. a x = 0.5 for most ages (except the very young and oldest)

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Life table, step 8: compute the life years lived between ages x and x+1, L x, formulae Life years lived are given by L x = 1 l x+1 + a x d x = l x+1 + a x d x e.g. L 40 = l 41 + 0.5 d 40 = 96486 + 0.5 155 = 96563 An alternative formula averages successive numbers of survivors (see Rowland 2003, p.280) L x = 0.5 (l x + l x+1 ) We need a different a x for age 0 a 0 = 0.1 (Rees using ONS 2003 method) a 0 = 0.3 (Rowland using earlier authors) We need a different a x for age 0 and age z (the last) a z = 1/M z i.e. if the mortality rate is 0.5, we expect people to live for a further 2 years on average Detailed empirical study of deaths below age 1 by week and month is needed to improve on these approximations Detailed study of deaths beyond age 100 is also needed

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Age Time x x+1 x+2 tt+1 lxlx l x+1 LxLx Illustration of the two meanings of L x (1)Life years lived between ages (2)Stationary population in age group LxLx L x+1 SxSx

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Life table, step 8: compute the life years lived between ages x and x+1, L x, UK males 1998 AgeCohort survivors Fraction live Cohort non- survivors Life years xlxlx axax DxDx LxLx 01000000.163499429 1993660.55099341 2993150.53199300 ::::: 996190.5199520 1004202.1420894

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Life table, step 9: compute the total life years lived beyond age y, T y The next step is to add up the L x variables beyond each age. This variable is called the Total Life Years lived beyond age y, T y We use y rather than x because we need to sum over x The formula is T y = x=y to x=z L x i.e. we sum L x from the current y value to the last age z To do the arithmetic is easier to start with the last age and then work backwards to younger ages T z = L z then T z-1 = T z + L z-1 etc In the next slide T 99 = T 100 + L 99 = 894 + 520 = 1414

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Life table, step 9: compute the total life years lived beyond age x, T x, UK males 1998 AgeLife yearsTotal life years xLxLx TxTx 0994297472831 1993417373402 2993007274062 ::: 995201414 100894

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Life table, step 10: compute life expectancy beyond age x, e x The life expectancy at age x is the total number of life years lived beyond age x divided by the numbers in the birth cohort surviving to age x e x = T x / l x e.g. e 0 = T 0 / l 0 = life expectancy at birth Also of interest are e 65,e 70,e 75,e 80 for pension, health care and personal care planning for the older population

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Life table, step 10: compute life expectancy beyond age x, e x, UK males 1998 AgeCohort survivors Total life years Life expectanc y x l x TxTx e x 0100000747283174.73 199366737340274.20 299315727406273.24 :::: 9961914142.28 1004208942.13

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Now you will understand how the statistics in these two maps were compute to give a picture of the spatial pattern of life expectancies for males in Leeds in (a) 1990-92 and (b) 2000-02 Northern and Eastern suburbs favoured Spatial pattern very stable over 10 years e 0 improves 2.68 years for men, 2.50 for women over 10 years For England & Wales, over the 10 years mens e 0 rose by 2.57 years and womens by 1.67 years Where Phil boards the bus in the morning (Cookridge ward), male life expectancy is 79 years (2000-02) Where he alights from the bus (University ward), men can expect to live only 71 years. Source: Figure 2.12 in Rees, Stillwell & Tyler- Jones (2004)

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Life table, summary of formulae Step 1, observed data:D x and P x Step 2, mortality rates:M x = D x /P x Step 3, mortality probabilities:q x = M x /(1 + 0.5 M x ) Step 4, survival probabilities:p x = 1 - q x Step 5, number surviving:l x = l x-1 p x-1 Step 7, number dying:d x = l x – l x+1 Step 8, life years lived:L x = l x+1 + 0.5l x first ageL 0 = l 1 + 0.1l 0 last ageL z = l z (1/M z ) Step 9, total life years lived:T x = T x+1 + L x Step 10, life expectancy:e x = T x /l x

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Median life expectancy This is the age at which half the birth cohort has died and half are still alive That is, where l x = 50000 Find ages x and x+1 such that l x > 50000 and l x+1 <50000 Then the median can be computed using the formula (assuming a one year interval) x median = x + (l x – 50000)/(l x – l x+1 )

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Cohort component projection

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Cohort Component Model Type of population projection model Separately accounts for 3 components of population change: Births Deaths Migration +Age, sex, geographic, ethnicity The model is used to account for population change.

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http://home.business.utah.edu/bebrpsp/URPL5020/Demog/CohortComponent.pdf

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Beginning population + Births Fertility Survivors Survival probabilities Non survivors Migration rates and flows Migrants - -+ End population Beginning population + Births

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The cohort-component projection model Components of population change: Population next year = Population last year minus Deaths plus Births plus Net Migration Cohort Each component needs to be decomposed by age Births are introduced into the first age group Deaths cluster in older ages Migration is highest in the young adult ages Fertility is age dependent Cohort-component intensities are used Survival probabilities (the complement of mortality probabilities) Fertility rates (occurrence-exposure) by age of mother Net migration counts by age The C-C model normally uses both sexes

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Survival probabilities for the cohort-component model (for period-cohorts) (aka survivorship rates) First period-cohort (birth to age 0) S -1 = L 0 /l 0 Period-cohorts x to x+1 from x=0 to x=z-1 (where z is the last age) S x = L x+1 /L x Final period cohort S z-1+ = L z+ /(L z+ +L z-1 ) S z-1 = S z-1+ S z = S z-1+ We can only measure the survival of the combined last but one and last (open ended) age and assume this survival probability applies to both ages (any error will be small)

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l0l0 L0L0 time age 0 First period-cohort S -1 t t+1 x x+1 Standard period-cohort x+1 x+2 LxLx L x+1 t t+1 z-1 z L z-1 LzLz LzLz Last but one and last period-cohorts

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The cohort-component model (1) Notation P = population s = survival probability N = net migration B = births x = age (usually period-cohort) z = last age (open ended), z-1 = last but one age, -1 = birth g = gender with values m (males) and f (females) t = time point (start of interval) w = sex probability (at birth) We need three sets of equations for The infant period-cohort, birth to age 0 The standard period-cohorts, from age 0 to age z-1 The last period-cohort, from age z+ to age z+

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The cohort-component model (2) Standard period-cohort projection equation P x+1 g (t+1) = s x g (t) P x g (t) + [s x g (t)] ½ N x g (t) Where P x g (t) = population aged x, gender g at time t N x g (t) = net migration of persons aged x, gender g in time interval t to t+1 s x g (t) = survival probability for period cohort x to x+1 for persons of gender g in time interval t to t+1 P x+1 g (t+1) = population aged x+1 of gender g at time t+1 This applies from x=0 to x=z-1, the period-cohort age 0 to age 1 to the period-cohort age z-1 to age z

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The cohort-component model (3) Last period-cohort projection equation P z g (t+1) = s z-1 g (t) P z-1 g (t) + [s z-1 g (t)] ½ N z-1 g (t) + s z g (t) P z g (t) + [s z g (t)] ½ N z g (t) We survive persons aged z-1 into z and add persons surviving within z, the last open ended age group e.g. 100 + = survivors from 99 to 100 plus from 100+ to 100+

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The cohort-component model (4) For the first period-cohort the projection equation is P 0 g (t+1) = s -1 g (t) B g (t) + [s -1 g (t)] ½ N -1 g (t) Population in age 0 at time t+1 = infant survival probability times projected number of births in interval t to t+1 plus the infant survival probability for net migrants times the projected number of net migrants

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The cohort-component model (5) The equation for projecting the number of births is B g (t) = w g × x=x1,x2 f x ½[P x f (t) + P x g (t+1)] Births of gender g in time interval t to t+1 equals sex probability for gender g times the sum over fertile age x1 to x2 of the age specific fertility rate times average population of women aged x in interval t to t+1

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More to consider: changes over time migration new births/fertility survival

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Some useful definitions Population projection = the computation of a future population size and structure under given assumptions Population forecast = the projection adopted as the most likely from a range of projections Population scenario = a population projection using assumptions known to be untrue but which are useful for assessing key factors (e.g. run a zero migration projection against a projection with migration to assess not just the direct differences due to migration but also additional effects such as natural increase contribution of migrants) Projection drivers = the components of change which determine the outcomes of a projections for which particular trajectories are assumed Projection assumptions = future course of the components of change that drive the population projection

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Part- I {Conic Sections}

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