Mortality Rates are usually published following an abridged description, i.e., by age groups 0, [1, 5[, [5, 10[, [10, 15[ and so on. For some applications, a detailed (single) ages description is required. Despite the huge number of the proposed methods in the literature, there is a limited number of methods ensuring a high performance at lower and higher ages in the same time. For example, the 6-terms Lagrange interpolation function is well adapted to mortality interpolation at lower ages (with unequal intervals) but is not adapted to higher ages. On the other hand, the Karup-King method allows a good performance at higher ages but not adapted to lower ages. Interested readers can refer to the book of Shryock, Siegel and Associates (1993) for a detailed overview of the two cited methodsThe package Q2q allows combining both the two methods to allow interpolating mortality rates at all ages. First, it starts by implementing each method separately, then the resulted curves are joined based on a 5-age averaged error between the two curves.
You can install the released version of Q2q from CRAN with:
The package Q2q provides two functions getqxt() and getqx().
The getqxt() functions allows interpolating age specific mortality rates (ASMRs) starting from an abridged mortality surface. This later should provides the five ages mortality rates, usually noted as (nQ{x,t}) in the literature, with (x) representing the age and (t) the year and (n) the length of the age interval which is set to be (5) except for age (0) and the age group (1-5).
The general formulation of the function is getqxt(Qxt, nag, t) with Qxt is the matrix of the five ages mortality rates with no header and identification column. The number of rows in the matrix should correspond to the number of age groups refereed as nag in the function. t corresponds to the number of years contained in the mortality matrix, and it should be equal to the number of columns in Qxt.
The function results principally in qxt which represents the matrix of the age-specific mortality rates for age (x) and year (t). This matrix had resulted from the junction of two initial matrices qxtl and qxtk obtained with the Lagrange and the Karup-King methods respectively. These two matrices are also provided as function returns. additionally, for each year (t), the junction age is provided in a vector $jUnct_ages. The functions also returns the survivorship matrix lxt and the matrix of theoretical deaths qxt.
The getqx() function is a special case of getqxt() with t=1.
Lets consider the Algerian mortality surface from 1977 to 2014, for males, by 5 ages groups.
LT <- read.table("https://raw.githubusercontent.com/Farid-FLICI/farid-flici.github.io/y/completed_Mortality_Rates_Males_nQx.txt", dec=".", sep="\t", quote="" )
LT <- data.matrix(LT)Lets consider a case of an annual life table (Algeria, males, year 1995).
Ax <- LT[ 2:18 , 18 ]
Ax
#> [1] 0.057600000 0.008595076 0.005458048 0.004412398 0.006377689 0.009247171
#> [7] 0.010432682 0.010874390 0.013575676 0.017743403 0.022464104 0.034233594
#> [13] 0.052863976 0.060080280 0.114340818 0.221651890 0.313934576We plot the mortality curve.
plot(x=c(0,1, seq(5, (length(Ax)-2)*5, by=5)), y=log(Ax), type="b", xlab="Age", ylab="log(nQx)", main="Figure 1. Mortality Rates (Algeria, males, 1995)" )
We interpolate [q_{x,t}] using getqx()
library(Q2q)
#>
#> Attaching package: 'Q2q'
#> The following object is masked _by_ '.GlobalEnv':
#>
#> getqx
interpolated_curve <-getqx(Qx=Ax, nag=17)
str(interpolated_curve)
#> List of 6
#> $ qx : num [1:80, 1] 0.0576 0.00272 0.00231 0.00195 0.00165 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:80] "0" "1" "2" "3" ...
#> .. ..$ : NULL
#> $ lx : num [1:81, 1] 10000 9424 9398 9377 9358 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:81] "0" "1" "2" "3" ...
#> .. ..$ : NULL
#> $ dx : num [1:80, 1] 576 25.7 21.7 18.3 15.4 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:80] "0" "1" "2" "3" ...
#> .. ..$ : NULL
#> $ qxk : num [1:80, 1] 0.0218 0.0164 0.01217 0.00927 0.00779 ...
#> $ qxl : num [1:70, 1] 0.0576 0.00272 0.00231 0.00195 0.00165 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:70] "0" "1" "2" "3" ...
#> .. ..$ : NULL
#> $ junct_age: num 30plot the interpolated mortality curve using plot()
plot(x=c(0:79), y=log(interpolated_curve$qx), type="b", xlab="Age", ylab="log(qx)", main="Figure 2. Interpolated Mortality Rates (Algeria, males, 1995")
plot(x=c(5:79), y=log(interpolated_curve$qxk[6:80]), type="p", cex=0.75, pch=1,xlab="Age", ylab="log(qx)", main="Figure 3. Interpolated Mortality Rates (Algeria, males, 1995) - Junction age", xlim=c(0,80))
lines(x=c(0:69), y=log(interpolated_curve$qxl), lwd=2)
lines(x=rep(interpolated_curve$junct_age, 2), y=c(min(log(interpolated_curve$qxl)),max(log(interpolated_curve$qxk))), lty=2, lwd=1.25, col="red")
Lets consider the following mortality surface with of a dimension 17 by 38, the Algerian mortality surface from 1977 to 2014 for males.
Bxt <- LT[2:18, 2:39]
head(Bxt[ ,1:5 ])
#> V2 V3 V4 V5 V6
#> [1,] 0.12771000 0.11457000 0.10983285 0.107010000 0.101360000
#> [2,] 0.05712550 0.04550331 0.05024614 0.053628820 0.050487403
#> [3,] 0.01737471 0.02004402 0.01430249 0.010897732 0.010348424
#> [4,] 0.01218803 0.01563632 0.01006031 0.006878649 0.006382928
#> [5,] 0.01319020 0.01601963 0.01230083 0.010250912 0.009546511
#> [6,] 0.01533404 0.01735250 0.01615231 0.016405813 0.015426454
tail(Bxt[ ,1:5 ])
#> V2 V3 V4 V5 V6
#> [12,] 0.05676323 0.04748539 0.05589881 0.06149508 0.05773371
#> [13,] 0.07900273 0.08034157 0.08121839 0.08712466 0.08291744
#> [14,] 0.11802724 0.11413719 0.11408799 0.11920273 0.11195253
#> [15,] 0.17857622 0.15288799 0.16690673 0.16547548 0.16143084
#> [16,] 0.27407753 0.23679338 0.22764444 0.21208642 0.19996918
#> [17,] 0.40216832 0.35013558 0.34325999 0.33074950 0.30638017The surface of log(nQxt) can be plotted using
persp(x=c(0,1, seq(5,75, by=5)), y=c(1977:2014), z=log(Bxt), theta=-10, expand = 0.8 , phi=25, xlab="age", ylab="year", zlab="log (nQxt)", main="Figure 4. Five ages mortality surface, Algeria, males, 1977-2014")
Then lets deduce the single ages mortality surface using getqxt()
interpolated_surface <- getqxt(Qxt=Bxt, nag=17, t=38)
str(interpolated_surface)
#> List of 6
#> $ qxt : num [1:80, 1:38] 0.12771 0.02187 0.01629 0.01181 0.00837 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:80] "0" "1" "2" "3" ...
#> .. ..$ : NULL
#> $ lxt : num [1:81, 1:38] 10000 8723 8532 8393 8294 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:81] "0" "1" "2" "3" ...
#> .. ..$ : NULL
#> $ dxt : num [1:80, 1:38] 1277.1 190.8 139 99.1 69.4 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:80] "0" "1" "2" "3" ...
#> .. ..$ : NULL
#> $ qxtk : num [1:80, 1:38] 0.0587 0.046 0.0353 0.0276 0.0236 ...
#> $ qxtl : num [1:70, 1:38] 0.12771 0.02187 0.01629 0.01181 0.00837 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:70] "0" "1" "2" "3" ...
#> .. ..$ : NULL
#> $ junct_ages: num [1, 1:38] 16 15 26 62 46 45 28 30 30 29 ...persp(x=c(0:79), y=c(1977:2014), z=log(interpolated_surface$qxt), theta=-10, expand = 0.8 , phi=25, xlab="age", ylab="year", zlab="log (qxt)", main="Figure 5. Detailled ages mortality surface, Algeria, males, 1977-2014")
Shryock, H. S., Siegel, J. S. and Associates (1993). Interpolation: Selected General Methods. In: D. J. Bogue, E. E. Arriaga, D. L. Anderton and G. W. Rumsey (eds), Readings in Population Research Methodology. Vol.1: Basic Tools. Social Development Center/ UN Population Fund, Chicago, pp. 5-48-5-72.