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Commit 804d0b9e authored by Reinhold Kainhofer's avatar Reinhold Kainhofer
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Finish Usage vignette

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R/deathProbabilities.R
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......@@ -45,6 +45,8 @@ setMethod("deathProbabilities","valuationTable.trendProjection",
finalqx = exp(-object@trend * damping) * qx;
} else {
# dampingFunction interpolates between the two trends:
# The damping functions does NOT give yearly weights,
# but rather cumulative weights from the base year until the observation year!
weights = sapply(YOB + 0:(length(qx)-1), object@dampingFunction);
finalqx = qx * exp(
-(object@trend * (1 - weights) + object@trend2 * weights) *
......
vignettes/using-the-valuationTables-package.Rmd
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......@@ -41,18 +41,18 @@ Provided types of valuation tables are:
: Death probabilities of a given base year are projected into the future
using age-specific trends $\lambda_x$. The death probability of an $x$-year old in year
`baseYear + n` is calculated as:
$q_x^{(baseYear+n)} = q_x^{(baseYear)} \cdot e^{-n\cdot\lambda_x}$
$$q_x^{(baseYear+n)} = q_x^{(baseYear)} \cdot e^{-n\cdot\lambda_x}$$
: Consequently, the death probabilities for a person born in year `YOB` can be calculated as
$q_x^{YOB} = q_x^{(base)} \cdot e^{-(YOB+x-baseYear)\cdot \lambda_x}$
$$q_x^{YOB} = q_x^{(base)} \cdot e^{-(YOB+x-baseYear)\cdot \lambda_x}$$
* Cohort life table approximation using age shift
: Class `valuationTable.ageShift`
: Death probabilities for cohort $YOB$ are obtained by using death probabilities
for cohort $X$ and modifying the technical age with a birth-year dependent shift:
: $q_x^{YOB} = q_{x+shift(YOB)}^{(base)}$
* Observed life table
: Class `valuationTable.observed`
: Death probabilities observed during several years. The probabilities are
stored as a matrix with observation year and age as dimensions.
$$q_x^{YOB} = q_{x+shift(YOB)}^{(base)}$$
<!-- * Observed life table -->
<!-- : Class `valuationTable.observed` -->
<!-- : Death probabilities observed during several years. The probabilities are -->
<!-- stored as a matrix with observation year and age as dimensions. -->
* Mixed life table
: Class `valuationTable.mixed`
: Arithmetic mean of two life tables with given weights. This approach is
......@@ -214,19 +214,141 @@ deathProbabilities(lt)
```
### Observed life tables
The observations for the given years
<!-- ### Observed life tables -->
<!-- The observations for the given years -->
TODO
<!-- TODO -->
### Cohort life tables with trend projection
TODO
A cohort life table with trend projection needs the following parameters:
* The base table $q_x^{(base)}$ (death probabilities) for the given base period as a vector
* Age-specific trend factors $\lambda_x$ as a vector
* The base year (numeric)
*
```{r}
atPlus2 = valuationTable.trendProjection(
name = "Austrian Census Males 2011, 2% yearly trend",
baseYear = 2011,
deathProbs = deathProbabilities(mort.AT.census.2011.male),
ages = ages(mort.AT.census.2011.male),
trend = rep(0.02, length(ages(mort.AT.census.2011.male)))
)
```
Some life tables do not assume a constant age-specific trend over time, but rather
assume that the currently observed high mortality improvements are just a
temporary effect, so the current trend is in effect only for some time and
then reduces to some kind of long-term trend.
There are two conceptual approaches: One is to use a trend dampening function
that is simply applied to the starting trend. So, while the initial trend might
be 3\%, i.e. the projection will use `(ObservationYear-BaseYear) * OriginalYear`,
over time it will assume the value
`dampeningFunction(ObservationYear-BaseYear) * OriginalTrend`. The dampening
function in this case gives the cumulated trend effect from the base year until
the observation year.
To implement this trend reduction with the ValuationTables package, simply pass
a one-argument function as the `dampingFunction` slot to the class, the argument
will be the number of years from the base year (NOT the calendar year!):
```{r}
atPlus2.damp = valuationTable.trendProjection(
name = "Austrian M '11, 2% yearly, damping until 2111",
baseYear = 2011,
deathProbs = deathProbabilities(mort.AT.census.2011.male),
ages = ages(mort.AT.census.2011.male),
trend = rep(0.02, length(ages(mort.AT.census.2011.male))),
# damping function: 2011: full effect, linear reduction until yearly trend=0 in 2111:
# 2011: 100%, 2012: 99%, 2013: 98% => For 2013 we have a cumulative trend
# of 297% instead of 300% for three full yearly trends!
dampingFunction = function(n) { n - n*(n+1)/2/100 }
)
plot(mort.AT.census.2011.male, atPlus2, atPlus2.damp, YOB=2011, legend.position=c(0.8,0.75))
```
The other approach is to assume that instead of the initial trend, after some
time a second trend (slot trend2) takes over. In this case, the `dampingFunction`
slot is again a one-argument function that now gives the weight of the first trend, while `1-dampingFunction(year)` will give the weight of the second trend. As the weights
will be applied for the whole period from the base- to the observation year, the weights
need to be cumulated and normalized.
The argument
in this case is the actual calendar year (not the year since the base year like it was in the one-trend case above!)
```{r}
atPlus2.damp2 = valuationTable.trendProjection(
name = "Austrian M '11, 2% yearly, 1% long-term",
baseYear = 2011,
deathProbs = deathProbabilities(mort.AT.census.2011.male),
ages = ages(mort.AT.census.2011.male),
trend = rep(0.02, length(ages(mort.AT.census.2011.male))),
trend2 = rep(0.01, length(ages(mort.AT.census.2011.male))),
# damping function interpolates between the two trends:
# until 2021 trend 1, from 2031 trend 2, linearly beteen
dampingFunction = function(year) {
if (year <= 2021) 1
else if (year>2031) 14.5/(year-2011)
else 1 - (year-2021)*(year-2021+1)/20/(year-2011)
}
)
plot(mort.AT.census.2011.male, atPlus2, atPlus2.damp, atPlus2.damp2, YOB=2011, legend.position=c(0.8,0.75))
```
### Cohort life tables with age-shift
TODO
Age-shifted cohort life tables are an approximation to full cohort life tables.
Full cohort life tables apply a trend or improvment factors to the death
probabilities of a base year to obtail death probabilities for a given birth year.
Age-shifting rather modifies the age of the corresponding person and uses the
same, unmodified base table for all cohorts. Basically, it works like this:
> A 60-year old born in 1950 has the same death probability as a 50-year old
> born in 1900, so instead of looking at the cohort 1950, we can look at the
> cohort 1900 and for a person born 1950 we treat him as if he were 10 years
> younger.
So, an age-shifted cohort life table just needs the base table and for each
birth year the amount the age is modified.
```{r}
baseTableShift = getCohortTable(atPlus2, YOB=2011);
baseTableShift@name = "Base table of the shift (YOB 2011)"
atShifted = valuationTable.ageShift(
name = "Approximation with age shift",
baseYear = 2011,
deathProbs = deathProbabilities(baseTableShift),
ages = ages(baseTableShift),
ageShifts = data.frame(
shifts = c(
rep( 0, 3),
rep(-1, 3),
rep(-2, 3),
rep(-3, 3),
rep(-4, 3),
rep(-5, 3),
rep(-6, 3)
),
row.names = 2011:2031
)
)
ageShift(atShifted, YOB=2021)
plot(baseTableShift, atPlus2, atShifted, YOB=2021, legend.position=c(0.8,0.75))
```
As one can see, for ages above 40 years, the table with 2% yearly trend and the
corresponding age-shifted table have roughly the same mortalities. Below 40 years,
the two are very different, so this approximation through age-shifting should
really be used with extreme care!
## Modifying life table objects
......
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