Cohort Component Method of Population Projection (CCMPP)
+Cohort Component Method of Population Projection +(CCMPP)
Source:vignettes/ccmpp.Rmd
@@ -91,10 +92,14 @@ Cohort Component Method of Population Projection (CCMPP)
Demographic Balancing Equation
-The demographic balancing equation is one of the core concepts of demography. It says that the population at time \(T\) is equal to the population at time \(0\) plus all the people that entered the population through birth or immigration minus the people that left the population through death or emigration.
+The demographic balancing equation is one of the core concepts of +demography. It says that the population at time \(T\) is equal to the population at time +\(0\) plus all the people that entered +the population through birth or immigration minus the people that left +the population through death or emigration.
\[\begin{aligned}
N(T) =& N(0) - E[0,T] - D[0,T] + B[0,T] + I[0,T] \\
- =& N(0) - D[0,T] + B[0,T] + G[0,T] \\
+=& N(0) - D[0,T] + B[0,T] + G[0,T] \\
\\
G[0,T] =& I[0,T] - E[0,T] \\
\\
@@ -110,14 +115,24 @@ The cohort component method of population projection (CCMPP) is the most common method for projecting populations forward in time. It satisfies the age- and sex-specific version of the demographic balancing equation. CCMPP takes as input age and sex specific estimates of the baseline population, survivorship ratios, net migration proportion, age-specific fertility rates and sex-ratio at birth to produce projected population estimates starting from the baseline. Below are two sections, the first gives a quick introduction to the The cohort component method of population projection (CCMPP) is the
+most common method for projecting populations forward in time. It
+satisfies the age- and sex-specific version of the demographic balancing
+equation. CCMPP takes as input age and sex specific estimates of the baseline
+population, survivorship ratios, net migration proportion, age-specific
+fertility rates and sex-ratio at birth to produce projected population
+estimates starting from the baseline. Below are two sections, the first gives a quick introduction to the
+ The There are two inputs needed for the The There are two inputs needed for the Below is a summary of what the Thailand example inputs look like. The baseline population is for the year 1960 and includes both female and male populations for five year age groups up to 80+. The other four inputs define how the population changes every five calendar years between 1960 and 2000. The survivorship ratios go up to age 85+ while the asfr estimates are defined for the reproductive age groups between age 15 and 50. Below is a summary of what the Thailand example inputs look like. The
+baseline population is for the year 1960 and includes both female and
+male populations for five year age groups up to 80+. The other four
+inputs define how the population changes every five calendar years
+between 1960 and 2000. The survivorship ratios go up to age 85+ while
+the asfr estimates are defined for the reproductive age groups between
+age 15 and 50. Once those inputs are ready Once those inputs are ready Explained below are some of the basics of CCMPP using notation from Preston et al 2001, this is a great resource for more detailed explanations of CCMPP and other demography concepts. Explained below are some of the basics of CCMPP using notation from
+Preston et al 2001, this is a great resource for more detailed
+explanations of CCMPP and other demography concepts. In order to understand CCMPP it is helpful to use lexis diagrams. A lexis diagram is a simple tool often used in demography to clearly identify the ages, calendar years and cohorts included in a given population. The lexis diagrams in the following sections were created using In order to understand CCMPP it is helpful to use lexis diagrams. A
+lexis diagram is a simple tool often used in demography to clearly
+identify the ages, calendar years and cohorts included in a given
+population. The lexis diagrams in the following sections were created
+using CCMPP for all age groups except the youngest and oldest when ignoring migration is fairly simple. The example below (blue parallelogram) shows how to age the cohort of people that are 1 year old in 2017 \({_1}N_{1}(2017)\) to 2018 when the same cohort is now 2 years old \({_1}N_{2}(2018)\). The survivorship ratio \(\frac{{_1}L_{2}[2017, 2018]}{{_1}L_{1}[2017, 2018]}\) is used to calculate the proportion of \({_1}N_{1}(2017)\) alive after one year. This ratio could for example come from a \([2017, 2018]\) period specific life table and actually represents the proportion of people age 1 that survive another year in a stationary population subject to this life table. In the example below the orange square captures person years lived by 1 year olds \({_1}L_{1}[2017, 2018]\) and the green square captures person years lived by 2 year olds \({_1}L_{2}[2017, 2018]\) Example equation: \[{_1}N_{2}(2018) = {_1}N_{1}(2017) \cdot \frac{{_1}L_{2}[2017, 2018]}{{_1}L_{1}[2017, 2018]}\] General equation: \[{_n}N_{x}(t + n) = {_n}N_{x - n}(t) \cdot \frac{{_n}L_{x}[t, t + n]}{{_n}L_{x - n}[t, t + n]}\] where: \[\begin{aligned} {_n}N_{x}(t) = & \text{the population at time } t \text{ between age } x \text{ and } x + n \\ {_n}L_{x}[t, t + n] = & \text{the number of person-years lived between time } t \text{ and } t + n \text{ and between age } x \text{ and } x + n \end{aligned}\] CCMPP for all age groups except the youngest and oldest when ignoring
+migration is fairly simple. The example below (blue parallelogram) shows
+how to age the cohort of people that are 1 year old in 2017 \({_1}N_{1}(2017)\) to 2018 when the same
+cohort is now 2 years old \({_1}N_{2}(2018)\). The survivorship ratio
+\(\frac{{_1}L_{2}[2017, 2018]}{{_1}L_{1}[2017,
+2018]}\) is used to calculate the proportion of \({_1}N_{1}(2017)\) alive after one year.
+This ratio could for example come from a \([2017, 2018]\) period specific life table
+and actually represents the proportion of people age 1 that survive
+another year in a stationary population subject to this life table. In
+the example below the orange square captures person years lived by 1
+year olds \({_1}L_{1}[2017, 2018]\) and
+the green square captures person years lived by 2 year olds \({_1}L_{2}[2017, 2018]\) Example equation: \[{_1}N_{2}(2018) = {_1}N_{1}(2017) \cdot
+\frac{{_1}L_{2}[2017, 2018]}{{_1}L_{1}[2017, 2018]}\] General equation: \[{_n}N_{x}(t + n) = {_n}N_{x - n}(t) \cdot
+\frac{{_n}L_{x}[t, t + n]}{{_n}L_{x - n}[t, t + n]}\] where: \[\begin{aligned}
+{_n}N_{x}(t) = & \text{the population at time } t \text{ between age
+} x \text{ and } x + n \\ {_n}L_{x}[t, t + n] = & \text{the number
+of person-years lived between time } t \text{ and } t + n \text{ and
+between age } x \text{ and } x + n \end{aligned}\] When modeling actual populations migration needs to be accounted for. These migrants are also at risk of dying and giving birth in their new populations but it is unknown when exactly in the interval they migrated. A standard assumption is that half of the migrants in a given interval migrate at the beginning of the interval and the other half at the end of the interval. Example equation: \[{_1}N_{2}(2018) = \bigg[ \bigg( {_1}N_{1}(2017) + \frac{{_1}G_1[2017, 2018]}{2} \bigg) \cdot \frac{{_1}L_{2}[2017, 2018]}{{_1}L_{1}[2017, 2018]} \bigg] + \frac{{_1}G_2[2017, 2018]}{2}\] General equation: \[{_n}N_{x}(t + n) = \bigg[ \bigg( {_n}N_{x - n}(t) + \frac{{_n}G_{x - n}[t, t + n]}{2} \bigg) \cdot \frac{{_1}L_{x}[t, t + n]}{{_n}L_{x - n}[t, t + n]} \bigg] + \frac{{_n}G_{x}[t, t + n]}{2}\] where: \[\begin{aligned} {_n}N_{x}(t) = & \text{the population at time } t \text{ between age } x \text{ and } x + n \\ {_n}L_{x}[t, t + n] = & \text{the number of person-years lived between time } t \text{ and } t + n \text{ and between age } x \text{ and } x + n \\ \frac{{_n}G_{x}[t, t + n]}{2} = & \text{the net number of migrants between time } t \text{ and } t + n \text{ and between age } x \text{ and } x + n \end{aligned} \] When modeling actual populations migration needs to be accounted for.
+These migrants are also at risk of dying and giving birth in their new
+populations but it is unknown when exactly in the interval they
+migrated. A standard assumption is that half of the migrants in a given
+interval migrate at the beginning of the interval and the other half at
+the end of the interval. Example equation: \[{_1}N_{2}(2018) = \bigg[ \bigg( {_1}N_{1}(2017) +
+\frac{{_1}G_1[2017, 2018]}{2} \bigg) \cdot \frac{{_1}L_{2}[2017,
+2018]}{{_1}L_{1}[2017, 2018]} \bigg] + \frac{{_1}G_2[2017,
+2018]}{2}\] General equation: \[{_n}N_{x}(t + n) = \bigg[ \bigg( {_n}N_{x - n}(t)
++ \frac{{_n}G_{x - n}[t, t + n]}{2} \bigg) \cdot \frac{{_1}L_{x}[t, t +
+n]}{{_n}L_{x - n}[t, t + n]} \bigg] + \frac{{_n}G_{x}[t, t +
+n]}{2}\] where: \[\begin{aligned}
+{_n}N_{x}(t) = & \text{the population at time } t \text{ between age
+} x \text{ and } x + n \\ {_n}L_{x}[t, t + n] = & \text{the number
+of person-years lived between time } t \text{ and } t + n \text{ and
+between age } x \text{ and } x + n \\ \frac{{_n}G_{x}[t, t + n]}{2} =
+& \text{the net number of migrants between time } t \text{ and } t +
+n \text{ and between age } x \text{ and } x + n \end{aligned}
+\] For the youngest age group the number of births occuring in the interval needs to be calculated and then the survivorship ratios are used to calculate the proportion that survive to the next time point. Example equation: \[\begin{aligned} {_1}N_{0}(2018) =& \cdot B[2017, 2018] \cdot \frac{{_1}L_{0}[2017, 2018]}{1 \cdot l_0[2017, 2018]} \\ B[2017, 2018] =& \sum_{x=0}^{A} \frac{1}{2} \cdot {_1}F_x \cdot \frac{{_1}N^F_{x+1}(t) + {_1}N^F_x(t) \cdot \frac{{_1}L_{x+1}[t, t+1]}{{_1}L_{x}[t, t+1]}}{2} \end{aligned}\] General equation: \[\begin{aligned} {_n}N_{0}(t + n) =& \cdot B[t, t + n] \cdot \frac{{_n}L_{0}[t, t + n]}{n \cdot l_0[t, t + n]} \\ B[t, t + n] =& \sum_{x=0}^{A} \frac{n}{2} \cdot {_n}F_x[t, t + n] \cdot \frac{{_n}N^F_{x}(t) + {_n}N^F_{x - n}(t) \cdot \frac{{_n}L_{x}[t, t + n]}{{_n}L_{x - n}[t, t + n]}}{2} \end{aligned}\] where: \[\begin{aligned} {_n}N_{x}(t) =& \text{the population at time } t \text{ between age } x \text{ and } x + n \\ {_n}L_{x}[t, t + n] =& \text{the number of person-years lived between time } t \text{ and } t + n \text{ and between age } x \text{ and } x + n \\ B[t, t + n] =& \text{the number of births between time } t \text{ and } t + n \\ {_n}F_x =& \text{the age-specific fertility rate between time } t \text{ and } t + n \text{ for females between age } x \text{ and } x + n \end{aligned}\] For the youngest age group the number of births occuring in the
+interval needs to be calculated and then the survivorship ratios are
+used to calculate the proportion that survive to the next time
+point. Example equation: \[\begin{aligned} {_1}N_{0}(2018) =& \cdot
+B[2017, 2018] \cdot \frac{{_1}L_{0}[2017, 2018]}{1 \cdot l_0[2017,
+2018]} \\ B[2017, 2018] =& \sum_{x=0}^{A} \frac{1}{2} \cdot {_1}F_x
+\cdot \frac{{_1}N^F_{x+1}(t) + {_1}N^F_x(t) \cdot \frac{{_1}L_{x+1}[t,
+t+1]}{{_1}L_{x}[t, t+1]}}{2} \end{aligned}\] General equation: \[\begin{aligned} {_n}N_{0}(t + n) =& \cdot
+B[t, t + n] \cdot \frac{{_n}L_{0}[t, t + n]}{n \cdot l_0[t, t + n]} \\
+B[t, t + n] =& \sum_{x=0}^{A} \frac{n}{2} \cdot {_n}F_x[t, t + n]
+\cdot \frac{{_n}N^F_{x}(t) + {_n}N^F_{x - n}(t) \cdot \frac{{_n}L_{x}[t,
+t + n]}{{_n}L_{x - n}[t, t + n]}}{2} \end{aligned}\] where: \[\begin{aligned}
+{_n}N_{x}(t) =& \text{the population at time } t \text{ between age
+} x \text{ and } x + n \\ {_n}L_{x}[t, t + n] =& \text{the number of
+person-years lived between time } t \text{ and } t + n \text{ and
+between age } x \text{ and } x + n \\ B[t, t + n] =& \text{the
+number of births between time } t \text{ and } t + n \\ {_n}F_x =&
+\text{the age-specific fertility rate between time } t \text{ and } t +
+n \text{ for females between age } x \text{ and } x + n
+\end{aligned}\] The terminal age group at time \(t+n\) is a combination of two age groups at time \(t\), both the people that are aging into the terminal age group and those that are still alive in the terminal age group. Example equation: \[\begin{aligned} {_\infty}N_{95}(2018) =& \bigg( {_1}N_{94}(2017) \cdot \frac{{_1}L_{95}[2017, 2018]}{{_1}L_{94}[2017, 2018]} \bigg) + \bigg( {_\infty}N_{95}(2017) \cdot \frac{T_{96}[2017, 2018]}{T_{95}[2017, 2018]} \bigg) \\ \approx& \bigg( {_1}N_{94}(t) + {_\infty}N_{95}(t) \bigg) \cdot \frac{T_{95}[t, t + 1]}{T_{94}[t, t + 1]} \end{aligned}\] General equation: \[\begin{aligned} {_\infty}N_{x}(t + n) =& \bigg( {_n}N_{x - n}(t) \cdot \frac{{_n}L_{x}[t, t + n]}{{_n}L_{x - n}[t, t + n]} \bigg) + \bigg( {_\infty}N_{x}(t) \cdot \frac{T_{x + n}[t, t + 1]}{T_{x}[t, t + n]} \bigg) \\ \approx& \bigg( {_n}N_{x - n}(t) + {_\infty}N_{x}(t) \bigg) \cdot \frac{T_{x}[t, t + n]}{T_{x - n}[t, t + n]} \end{aligned}\] where: \[\begin{aligned} {_n}N_{x}(t) = & \text{the population at time } t \text{ between age } x \text{ and } x + n \\ {_n}L_{x}[t, t + n] = & \text{the number of person-years lived between time } t \text{ and } t + n \text{ and between age } x \text{ and } x + n \\ T_{x}[t, t + n] = & \text{the number of person-years lived between time } t \text{ and } t + n \text{ and above age } x \end{aligned}\] The terminal age group at time \(t+n\) is a combination of two age groups at
+time \(t\), both the people that are
+aging into the terminal age group and those that are still alive in the
+terminal age group. Example equation: \[\begin{aligned} {_\infty}N_{95}(2018) =&
+\bigg( {_1}N_{94}(2017) \cdot \frac{{_1}L_{95}[2017,
+2018]}{{_1}L_{94}[2017, 2018]} \bigg) + \bigg( {_\infty}N_{95}(2017)
+\cdot \frac{T_{96}[2017, 2018]}{T_{95}[2017, 2018]} \bigg)
+\\ \approx& \bigg( {_1}N_{94}(t) + {_\infty}N_{95}(t) \bigg) \cdot
+\frac{T_{95}[t, t + 1]}{T_{94}[t, t + 1]} \end{aligned}\] General equation: \[\begin{aligned} {_\infty}N_{x}(t + n) =&
+\bigg( {_n}N_{x - n}(t) \cdot \frac{{_n}L_{x}[t, t + n]}{{_n}L_{x -
+n}[t, t + n]} \bigg) + \bigg( {_\infty}N_{x}(t) \cdot \frac{T_{x + n}[t,
+t + 1]}{T_{x}[t, t + n]} \bigg) \\ \approx& \bigg( {_n}N_{x - n}(t)
++ {_\infty}N_{x}(t) \bigg) \cdot \frac{T_{x}[t, t + n]}{T_{x - n}[t, t +
+n]} \end{aligned}\] where: \[\begin{aligned}
+{_n}N_{x}(t) = & \text{the population at time } t \text{ between age
+} x \text{ and } x + n \\ {_n}L_{x}[t, t + n] = & \text{the number
+of person-years lived between time } t \text{ and } t + n \text{ and
+between age } x \text{ and } x + n \\ T_{x}[t, t + n] = & \text{the
+number of person-years lived between time } t \text{ and } t + n \text{
+and above age } x \end{aligned}\] All these tedious ccmpp equations can be written in matrix notation which helps to speed up computation and can lead to other useful relationships from matrix algebra. All these tedious ccmpp equations can be written in matrix notation
+which helps to speed up computation and can lead to other useful
+relationships from matrix algebra. \[\begin{aligned}
\mathbf{n}(t + 1) =& \mathbf{L}[t, t + 1] \cdot \mathbf{n}(t) \\
\\
\begin{bmatrix}
-n_{0, t + 1} \\ n_{1, t + 1} \\ n_{2, t + 1}\\ \vdots \\ n_{A - 2, t + 1} \\ n_{A - 1, t + 1} \\ n_{A , t + 1}
+n_{0, t + 1} \\ n_{1, t + 1} \\ n_{2, t + 1}\\ \vdots \\ n_{A - 2, t +
+1} \\ n_{A - 1, t + 1} \\ n_{A , t + 1}
\end{bmatrix}
=&
\begin{bmatrix}
-\tilde{f_{0, t}} & \tilde{f_{1, t}} & \tilde{f_{2, t}} & \dots & \tilde{f_{A - 2, t}} & \tilde{f_{A - 1, t}} & \tilde{f_{A, t}} \\
+\tilde{f_{0, t}} & \tilde{f_{1, t}} & \tilde{f_{2, t}} &
+\dots & \tilde{f_{A - 2, t}} & \tilde{f_{A - 1, t}} &
+\tilde{f_{A, t}} \\
s_{1, t} & 0 & 0 & \dots & 0 & 0 & 0 \\
0 & s_{2, t} & 0 & \dots & 0 & 0 & 0 \\
-\vdots & \ddots & \ddots & \ddots & \vdots & \vdots & \vdots \\
+\vdots & \ddots & \ddots & \ddots & \vdots & \vdots
+& \vdots \\
0 & 0 & 0 & \ddots & 0 & 0 & 0 \\
0 & 0 & 0 & \ddots & s_{A - 1, t} & 0 & 0 \\
0 & 0 & 0 & \dots & 0 & s_{A, t} & s_{A + 1, t}
\end{bmatrix}
\begin{bmatrix}
-n_{0, t} \\ n_{1, t} \\ n_{2, t} \\ \vdots \\ n_{A - 2, t} \\ n_{A - 1, t} \\ n_{A , t}
+n_{0, t} \\ n_{1, t} \\ n_{2, t} \\ \vdots \\ n_{A - 2, t} \\ n_{A - 1,
+t} \\ n_{A , t}
\end{bmatrix}
\end{aligned}\] \[\begin{aligned} \\
\text{where:} \ \ \ &\\
\mathbf{L}[t, t + 1] =& \text{Leslie matrix} \\
-\tilde{f_{a, t}} =& s_{0, t} \cdot \frac{f_{a,t} + f_{a + 1, t} \cdot s_{a + 1, t}}{2} \end{aligned}\] Preston, Samuel, Patrick Heuveline, and Michel Guillot. 2001. Demography: Measuring and Modeling Population. Wiley. Wheldon, Mark C., Adrian E. Raftery, Samuel J. Clark, and Patrick Gerland. 2013. “Reconstructing Past Populations With Uncertainty From Fragmentary Data.” Journal of the American Statistical Association 108 (501): 96–110. https://doi.org/10.1080/01621459.2012.737729. Wheldon, Mark C., Adrian E. Raftery, Samuel J. Clark, and Patrick Gerland. 2015. “Bayesian Reconstruction of Two-Sex Populations by Age: Estimating Sex Ratios at Birth and Sex Ratios of Mortality.” Journal of the Royal Statistical Society. Series A: Statistics in Society 178 (4): 977–1007. https://doi.org/10.1111/rssa.12104. Preston, Samuel, Patrick Heuveline, and Michel Guillot. 2001.
+Demography: Measuring and Modeling Population. Wiley. Wheldon, Mark C., Adrian E. Raftery, Samuel J. Clark, and Patrick
+Gerland. 2013. “Reconstructing Past Populations With Uncertainty From
+Fragmentary Data.” Journal of the American Statistical Association 108
+(501): 96–110. https://doi.org/10.1080/01621459.2012.737729. Wheldon, Mark C., Adrian E. Raftery, Samuel J. Clark, and Patrick
+Gerland. 2015. “Bayesian Reconstruction of Two-Sex Populations by Age:
+Estimating Sex Ratios at Birth and Sex Ratios of Mortality.” Journal of
+the Royal Statistical Society. Series A: Statistics in Society 178 (4):
+977–1007. https://doi.org/10.1111/rssa.12104. Site built with pkgdown 2.0.6. Site built with pkgdown 2.0.7.Demographic Balancing Equation
CCMPP
-demCore::ccmpp function and the second actually gives a more detailed explanation of CCMPP.demCore::ccmpp function and the second actually gives a
+more detailed explanation of CCMPP.demCore::ccmpp function
-demCore package implements ccmpp() and includes some example inputs for Thailand from Wheldon et al 2015.demCore::ccmpp function:demCore package implements ccmpp() and
+includes some example inputs for Thailand from Wheldon et al 2015.demCore::ccmpp
+function:
@@ -129,9 +144,16 @@ demCore::ccmpp functionnet-migration (or immigration and emigration separately)
-
-
library(demCore)
@@ -186,7 +208,9 @@ demCore::ccmpp function ages_mortality = seq(0, 85, 5),
ages_asfr = seq(15, 45, 5)
)
ccmpp can be run to produce a full time series of projected populations between 1960 and 2000 in five year age groups up to 80+.ccmpp can be run to produce
+a full time series of projected populations between 1960 and 2000 in
+five year age groups up to 80+.
thailand_population <- ccmpp(
inputs = demCore::thailand_initial_estimates,
@@ -206,77 +230,167 @@ demCore::ccmpp function
CCMPP Details
-Lexis Diagrams
-LexisPlotR which is highly recommend.LexisPlotR which is highly recommend.
CCMPP for most age groups with out migration
-
CCMPP for most age groups with migration
-CCMPP for the youngest age group with out migration
-
CCMPP for the terminal age group with out migration
-
Leslie Matrix
-demCore::leslie_matrix is a function to produce the leslie matrix from input vectors of the survivorship ratio, asfr, and the sex-ratio at birth. Below is the Thailand example leslie matrix for the rates between 1960 and 1965.demCore::leslie_matrix is a function to produce the
+leslie matrix from input vectors of the survivorship ratio, asfr, and
+the sex-ratio at birth. Below is the Thailand example leslie matrix for
+the rates between 1960 and 1965.
leslie <- demCore::leslie_matrix(
survival = demCore::thailand_initial_estimates$survival[year_start == 1960 &
@@ -348,9 +462,17 @@ Leslie Matrix
References
-References
-
All vignettes
Introduction to life tables
-Katie Paulson
+Katie +Paulson
March 2020
@@ -93,11 +94,41 @@March 2020
What is a life table?
-A life table is a table that includes information to describe the dying out of a birth cohort. This can also be a synthetic birth cohort, in which case we refer to it as a period life table.
-Life tables are one of the most important devices in demography – they have been used since the 1600s! They can also be useful for other fields, because they are generalizable to other discrete “time to event” data.
-Typically, life tables have one row per age group, with columns representing life table metrics, also known as parameters. The life table parameters used in this package are \(_nm_x\), \(_na_x\), \(_nq_x\), \(_np_x\), \(l_x\), \(_nd_x\), \(e_x\), \(_nL_x\), and \(T_x\). In this notation, \(x\) refers to age, and the metrics apply to either the age \(x\) directly or to the interval between ages \(x\) and \(x+n\) where \(n\) indicates the length of the interval, typically in years. We often shorthand by removing the “n” from the notation, with interval length implied.
-Here’s an example of a period life table, for males in Austria in 1992, which is saved in the package as austria_1992_lt (source: Preston 2001):
