Age-Period (AP) data
Data preparation
The package accepts several types of data formats. For AP data, users can supply a data-frame, a 3-dimensional (3D) data array, or a data matrix.
Data supplied as a data-frame
The data needs to be formatted as a data.frame with
columns name Age, Year, Deaths and
Exposures. An example is provided below.
You can access a dataset in this format for comparison by running
The first few lines of the data look like the following:
head(uk_mortalitydata, n = 20)
#> Age Year Deaths Exposures
#> 1 0 1922 74065.19 908771.5
#> 2 1 1922 23947.07 916468.6
#> 3 2 1922 10891.03 866680.1
#> 4 3 1922 4507.01 727009.9
#> 5 4 1922 2870.01 657473.8
#> 6 5 1922 2692.41 704711.4
#> 7 6 1922 2339.34 764959.2
#> 8 7 1922 2009.12 811532.0
#> 9 8 1922 1724.79 837073.8
#> 10 9 1922 1554.36 838640.6
#> 11 10 1922 1482.40 834416.1
#> 12 11 1922 1414.43 836638.8
#> 13 12 1922 1467.02 847043.2
#> 14 13 1922 1569.20 857801.8
#> 15 14 1922 1764.97 857463.9
#> 16 15 1922 1855.20 846678.5
#> 17 16 1922 2116.68 836631.3
#> 18 17 1922 2261.27 826264.3
#> 19 18 1922 2400.99 814563.2
#> 20 19 1922 2582.88 805320.9Next, we need to format the data in the format necessary for the function runBayesMoFo to work properly. To do this, we pass the dataset (separately for death and exposure) to the function preparedata_fn, which takes the arguments ages, years, and data. In the case of deaths, we pass the data using the column Age, Year, and Claim; and in the case of exposures, we use Exposure in place of Claim.
Data supplied as a 3D data array
Alternatively, users can supply the data as a 3-dimensional (3D) data
array. dxt_array_product is a 3D array containing mortality
data stratified by insurance product (see
?dxt_array_product for details), where dim one: 4 insurance
products, dim two: 83 ages, dim three: 5 years.
data("dxt_array_product")
data("Ext_array_product")
# preview of death data the 1st insurance product called "ACI"
str(dxt_array_product["ACI",,,drop = FALSE])
#> num [1, 1:83, 1:5] 0 1.01 0 0 2 ...
#> - attr(*, "dimnames")=List of 3
#> ..$ : chr "ACI"
#> ..$ : chr [1:83] "18" "19" "20" "21" ...
#> ..$ : chr [1:5] "2016" "2017" "2018" "2019" ...Similarly, users can prepare the data by either by inputting the data
as a 3-way array, or by specifying the name of the stratum to load using
the argument strat_name:
# inputting the data as a 3-way array
death <- preparedata_fn(dxt_array_product["ACI",,,drop = FALSE], ages = 35:65, years = 2016:2020)
expo <- preparedata_fn(Ext_array_product["ACI",,,drop = FALSE], ages = 35:65, years = 2016:2020)
# specifying the name of the stratum to load using `strat_name`
death <- preparedata_fn(dxt_array_product,strat_name="ACI", ages = 35:65, years = 2016:2020)
expo <- preparedata_fn(Ext_array_product,strat_name="ACI", ages = 35:65, years = 2016:2020)Data array types are less conventional but can be useful if data has been stored as it is. Preserving this data structure is useful for JAGS implementation later (for package development purposes).
Data supplied as a data matrix
Suppose the data is provided in a 2-dimensional matrix format by age and year, commonly used in the literature. For example, the following is an illustration:
# preview of death data the 1st insurance product called "ACI"
str(dxt_array_product["ACI",,,drop = TRUE])
#> num [1:83, 1:5] 0 1.01 0 0 2 ...
#> - attr(*, "dimnames")=List of 2
#> ..$ : chr [1:83] "18" "19" "20" "21" ...
#> ..$ : chr [1:5] "2016" "2017" "2018" "2019" ...To prepare data of type matrix, users need to specify the argument
data_matrix=TRUE.
Running the model
For illustrative purposes, we chose the UK mortality data.
Once the data have been prepared, they can be passed to runBayesMoFo, which is the core function in the package for estimating mortality models.
As this package is built on top of the rjags package, it
is capable of handling missing values in the death data, provided they
are coded as NA. However, if missing values are present in
the exposures data, these will be automatically replaced with a default
value of 100, and predictions will be performed using that value. If the
user wishes to perform prediction for a specific exposure value, they
can manually set the desired value of the exposure (leaving the value of
death count as NA, of course).
Users also have the option to perform model selection, depending on
their needs. If more than one model is provided in the
models argument, model selection is performed by default
using deviance information criterion (DIC). The argument
models can be set equal to:
- “all”, in which case, all models are fitted;
- a vector with the name of the models, in which case the models in the list are fitted;
- If the argument is left unspecified, a default set of models is fitted.
For example, the code below fit the LC,
CMB_M3, and APCI models to the data.
Note that one can also run the individual functions rather than using the function runBayesMoFo. For example,
All other functions for analysing the output (see later) would work equally. That being said, users are highly recommend to use the function runBayesMoFo even when only one model is needed. That is,
The full list of models, with the specific names, is available by
checking ?runBayesMoFo. Alternatively, one can query the
model details through the documentation within the package,
i.e. ?fit_LC.
The argument family defines the specification for the
distribution of death. A summary is as below (note that for AP data,
just suppress the subscript \(p\)):
If
family="poisson", then as proposed by Brouhns, Denuit, and Vermunt (2002), \[d_{x,t,p} \sim \text{Poisson}(E^c_{x,t,p} m_{x,t,p}) , \] where \(d_{x,t,p}\) represents the number of deaths at age \(x\) in year \(t\) of stratum \(p\), while \(E^c_{x,t,p}\) and \(m_{x,t,p}\) represents respectively the corresponding central exposed to risk and central mortality rate at age \(x\) in year \(t\) of stratum \(p\). The specification is used in conjunction with the log link function, i.e. \[\log(m_{x,t,p}) = \eta_{x,t,p} \] where \(\eta_{x,t,p}\) is the predictor that depends on the functional form of the mortality model, a full list of which is provided in Appendices A and B.Similarly, if
family="nb"(default), then a negative binomial distribution is fitted with the log link function, i.e. \[d_{x,t,p} \sim \text{Negative-Binomial}(\phi,\frac{\phi}{\phi+E^c_{x,t,p} m_{x,t,p}}) , \] \[\log(m_{x,t,p}) = \eta_{x,t,p} , \] where \(\phi\) is the overdispersion parameter. This specification of the death distribution is standard practice for incorporating overdispersion, a phenomenon commonly observed in mortality modelling. See Wong, Forster, and Smith (2023).If
\code{family="binomial"}, then \[d_{x,t,p} \sim \text{Binomial}(E^0_{x,t,p} , q_{x,t,p}) , \] where \(q_{x,t,p}\) represents the initial mortality rate at age \(x\) in year \(t\) of stratum \(p\), while \(E^0_{x,t,p}\approx E^c_{x,t,p}+\frac{1}{2}d_{x,t,p}\) is the corresponding initial exposed to risk. The binomial specification is used in conjunction with the logit link function, i.e. \[\text{logit}(q_{x,t,p})= \log(\frac{q_{x,t,p}}{1-q_{x,t,p}}) = \eta_{x,t,p}. \]
For example, the following fit the same set of models using the Poisson distribution for modelling number of death.
There are also other arguments for customising the MCMC sampling of the posterior distributions, as below:
n.adaptspecifies the number of iterations for the adaptation phase. See?rjags::adaptfor more details.n.chainspecifies the number of parallel chains for the posterior sampling under each model.n_iterspecifies the number of iterations in the posterior sampling.thinspecifies the thinning interval for the posterior sampling
Forecast
As part of the function runBayesMoFo, forecasting
can be performed by setting the argument forecast=TRUE,
with the parameter h specifying the forecast horizon. For
example, the code below fit the LC, CMB_M3,
and APCI models to the data, and forecast the models for
h=6 time points into the future.
fitmodel_forecast <- runBayesMoFo(death, expo,
models = c("LC",
"CBD_M3",
"APCI"),
forecast = TRUE,
h = 6,
n.chain = 2
)
#> Compiling model graph
#> Resolving undeclared variables
#> Allocating nodes
#> Graph information:
#> Observed stochastic nodes: 651
#> Unobserved stochastic nodes: 248
#> Total graph size: 7904
#>
#> Initializing model
#>
#> Completed: LC (1/3)
#> Compiling model graph
#> Resolving undeclared variables
#> Allocating nodes
#> Graph information:
#> Observed stochastic nodes: 651
#> Unobserved stochastic nodes: 304
#> Total graph size: 6383
#>
#> Initializing model
#>
#> NOTE: Stopping adaptation
#>
#>
#> Completed: CBD_M3 (2/3)
#> Compiling model graph
#> Resolving undeclared variables
#> Allocating nodes
#> Graph information:
#> Observed stochastic nodes: 651
#> Unobserved stochastic nodes: 302
#> Total graph size: 8266
#>
#> Initializing model
#>
#> NOTE: Stopping adaptation
#>
#>
#> Completed: APCI (3/3)Analyzing the output
After running the model, users can then query the best and worst models (in terms of DIC) among the competing models.
A table showing the DIC of all models fitted can be returned too.
One can retrieve the fitted results for the best and worst performing
models, both of which are of type fit_result.
The function plot_param_fn plots all the fitted
parameters of the model specified, using posterior samples generated. If
more than one models were specified previously when running
runBayesMoFo, then only the best model will be
illustrated.
As evident in the plot, all fitted and forecasted parameters will be
included, with solid lines indicating the medians and dashed lines
representing the credible intervals generated from the posterior
samples. By default, the intervals are constructed based on 95%
credibility, but can be changed using the argument
pred_int. For instance, for 80% credible intervals,
The argument legends argument can be used to suppress
the legends for better visualisation (e.g. if visibility is blocked by
the legend boxes).
The function plot_rates_fn plots the fitted death rates
of the model specified for specific ages and years, using posterior
samples generated. Again, if more than one models were specified
previously when running runBayesMoFo, then only the
best model will be illustrated. By default, the (log) death rates will
be plotted against age for the first nine years.
As before, both fitted and forecasted death rates will be included,
with solid lines indicating the medians and dashed lines representing
the credible intervals (95% by default but can be changed using the
argument pred_int) generated from the posterior samples.
Also, observed crude death rates will also be included as coloured
dots.
plot_rates_fn(fitmodel_forecast)
#> Warning in plot_rates_fn(fitmodel_forecast): Too many years selected, only
#> printing the first 9 years.
For better visualisation, one may customise the argument
plot_years to plot only selected years. Note that if more
than nine years have been specified, then only the first nine years will
be plotted.
The argument plot_type allows users to plot death rates
against year instead to better visualise temporal variations in death
rates. The argument plot_ages can be used accordingly to
specify which ages to plot.
The function summary_fn produces a summary of the model
results, including posterior means, standard deviations, medians, lower
and upper quantiles based on the credibility specified using
pred_int.
To obtain the posterior means and standard deviations of all death rates,
#posterior means
summary_fitmodel$rates_summary$mean
#posterior standard deviations
summary_fitmodel$rates_summary$stdTo obtain the posterior medians and lower/upper quantiles of all death rates,
#posterior medians
summary_fitmodel$rates_pn$median
#lower quantiles
summary_fitmodel$rates_pn$lower
#upper quantiles
summary_fitmodel$rates_pn$upperCorrespondingly, for model parameters,
Convergence diagnostics
Users can assess if convergence has been attained by the MCMC
posterior sampling procedure. The functions diag_rates_fn
produces (by default) trace plots, density plots, as well as effective
sample sizes of the posterior samples of death rates under the best
model. For example,
The effective sample sizes can be viewed as:
diagnostics_rates_result$ESS
#> q[1,2,12] q[1,2,16] q[1,2,17] q[1,16,12] q[1,16,16] q[1,16,17] q[1,17,12]
#> 135.41161 82.48078 43.02568 128.19355 45.68017 47.54397 129.67851
#> q[1,17,16] q[1,17,17]
#> 65.93268 94.86059The arguments plot_strata, plot_ages,
plot_years can be used to specify which death rates to
examine, as follows.
#> $ESS
#> q[1,6,17] q[1,6,21] q[1,6,25] q[1,16,17] q[1,16,21] q[1,16,25] q[1,26,17]
#> 29.29179 32.68255 40.86880 47.54397 40.91059 94.21674 31.55983
#> q[1,26,21] q[1,26,25]
#> 32.84719 237.42925The arguments trace and density can be used
to specify only plotting one of them or none according to personal
preferences.
#for only trace plots
converge_diag_rates_fn(fitmodel_forecast, plot_ages = c(35,45,55), plot_years = c(2016,2020,2024), trace = TRUE, density = FALSE)Auto-correlation plots can also displayed to check if posterior samples are too correlated.
#for only acf plots
converge_diag_rates_fn(fitmodel_forecast, plot_ages = c(35,45,55), plot_years = c(2016,2020,2024), trace = FALSE, density = FALSE, acf_plot = TRUE)
#> $ESS
#> q[1,6,17] q[1,6,21] q[1,6,25] q[1,16,17] q[1,16,21] q[1,16,25] q[1,26,17]
#> 29.29179 32.68255 40.86880 47.54397 40.91059 94.21674 31.55983
#> q[1,26,21] q[1,26,25]
#> 32.84719 237.42925Similarly, convergence can be assessed for fitted parameters using
the function converge_diag_rates_fn.
#> NOTE: Only showing three randomly selected alpha.
#> NOTE: Only showing three randomly selected beta.
#> NOTE: Only showing three randomly selected kappa.
#> NOTE: Only showing three randomly selected gamma.The effective sample sizes can be viewed as:
diagnostics_param_result$ESS
#> rho rho_gamma sigma2_kappa sigma2_gamma phi alpha[1,31]
#> 969.64746 20.61668 500.96781 13.73325 569.60223 1398.90826
#> alpha[1,16] alpha[1,15] beta[1,22] beta[1,14] beta[1,7] kappa[1,22]
#> 993.10171 531.46272 11.51433 21.21420 9.20101 1707.96677
#> kappa[1,18] kappa[1,13] gamma[1,28] gamma[1,4] gamma[1,43]
#> 121.54441 210.70046 13.81269 28.38499 21.55733By default, the function examines a selection of the parameters. But
the arguments plot_params can be used to specify which set
of parameters to examine, as follows.
converge_diag_param_fn(fitmodel_forecast, plot_params = c("kappa","gamma","rho","phi","sigma2_kappa"))
#> NOTE: Only showing three randomly selected kappa.
#> NOTE: Only showing three randomly selected gamma.
#> $ESS
#> rho rho_gamma phi sigma2_kappa kappa[1,6] kappa[1,1]
#> 969.64746 20.61668 569.60223 500.96781 105.06140 608.95448
#> kappa[1,22] gamma[1,19] gamma[1,1] gamma[1,31]
#> 1707.96677 37.41380 13.86666 12.44684To check the names of parameters available for examining:
fitmodel_forecast$result$best$param
#> [1] "alpha" "beta" "kappa" "gamma" "rho"
#> [6] "sigma2_kappa" "rho_gamma" "sigma2_gamma" "phi"For the rate-related parameters such as alpha, beta, kappa, gamma etc., only three of the randomly selected subset will be examined when specified. If a particular parameter is to be assessed, users need to specify clearly the indices of the parameters to be examined. For instance, the following will assess the beta parameters for the first stratum and the third age, as well as kappa for the first stratum and the fourth year.
#> $ESS
#> kappa[1,4] gamma[1,2]
#> 168.574037 9.357196To check the full list of parameters available for examining:
colnames(fitmodel_forecast$result$best$post_sample[[1]])[!startsWith(colnames(fitmodel_forecast$result$best$post_sample[[1]]),"q[")]
#> [1] "alpha[1,1]" "alpha[1,2]" "alpha[1,3]" "alpha[1,4]" "alpha[1,5]"
#> [6] "alpha[1,6]" "alpha[1,7]" "alpha[1,8]" "alpha[1,9]" "alpha[1,10]"
#> [11] "alpha[1,11]" "alpha[1,12]" "alpha[1,13]" "alpha[1,14]" "alpha[1,15]"
#> [16] "alpha[1,16]" "alpha[1,17]" "alpha[1,18]" "alpha[1,19]" "alpha[1,20]"
#> [21] "alpha[1,21]" "alpha[1,22]" "alpha[1,23]" "alpha[1,24]" "alpha[1,25]"
#> [26] "alpha[1,26]" "alpha[1,27]" "alpha[1,28]" "alpha[1,29]" "alpha[1,30]"
#> [31] "alpha[1,31]" "beta[1,1]" "beta[1,2]" "beta[1,3]" "beta[1,4]"
#> [36] "beta[1,5]" "beta[1,6]" "beta[1,7]" "beta[1,8]" "beta[1,9]"
#> [41] "beta[1,10]" "beta[1,11]" "beta[1,12]" "beta[1,13]" "beta[1,14]"
#> [46] "beta[1,15]" "beta[1,16]" "beta[1,17]" "beta[1,18]" "beta[1,19]"
#> [51] "beta[1,20]" "beta[1,21]" "beta[1,22]" "beta[1,23]" "beta[1,24]"
#> [56] "beta[1,25]" "beta[1,26]" "beta[1,27]" "beta[1,28]" "beta[1,29]"
#> [61] "beta[1,30]" "beta[1,31]" "gamma[1,1]" "gamma[1,2]" "gamma[1,3]"
#> [66] "gamma[1,4]" "gamma[1,5]" "gamma[1,6]" "gamma[1,7]" "gamma[1,8]"
#> [71] "gamma[1,9]" "gamma[1,10]" "gamma[1,11]" "gamma[1,12]" "gamma[1,13]"
#> [76] "gamma[1,14]" "gamma[1,15]" "gamma[1,16]" "gamma[1,17]" "gamma[1,18]"
#> [81] "gamma[1,19]" "gamma[1,20]" "gamma[1,21]" "gamma[1,22]" "gamma[1,23]"
#> [86] "gamma[1,24]" "gamma[1,25]" "gamma[1,26]" "gamma[1,27]" "gamma[1,28]"
#> [91] "gamma[1,29]" "gamma[1,30]" "gamma[1,31]" "gamma[1,32]" "gamma[1,33]"
#> [96] "gamma[1,34]" "gamma[1,35]" "gamma[1,36]" "gamma[1,37]" "gamma[1,38]"
#> [101] "gamma[1,39]" "gamma[1,40]" "gamma[1,41]" "gamma[1,42]" "gamma[1,43]"
#> [106] "gamma[1,44]" "gamma[1,45]" "gamma[1,46]" "gamma[1,47]" "gamma[1,48]"
#> [111] "gamma[1,49]" "gamma[1,50]" "gamma[1,51]" "gamma[1,52]" "gamma[1,53]"
#> [116] "gamma[1,54]" "gamma[1,55]" "gamma[1,56]" "gamma[1,57]" "kappa[1,1]"
#> [121] "kappa[1,2]" "kappa[1,3]" "kappa[1,4]" "kappa[1,5]" "kappa[1,6]"
#> [126] "kappa[1,7]" "kappa[1,8]" "kappa[1,9]" "kappa[1,10]" "kappa[1,11]"
#> [131] "kappa[1,12]" "kappa[1,13]" "kappa[1,14]" "kappa[1,15]" "kappa[1,16]"
#> [136] "kappa[1,17]" "kappa[1,18]" "kappa[1,19]" "kappa[1,20]" "kappa[1,21]"
#> [141] "kappa[1,22]" "kappa[1,23]" "kappa[1,24]" "kappa[1,25]" "kappa[1,26]"
#> [146] "kappa[1,27]" "phi" "rho" "rho_gamma" "sigma2_gamma"
#> [151] "sigma2_kappa"The arguments trace and density can be used
to specify only plotting one of them or none according to personal
preferences.
#for only trace plots
converge_diag_param_fn(fitmodel_forecast, plot_params = c("kappa[1,4]","gamma[1,2]"), trace = TRUE, density = FALSE)Auto-correlation plots can also displayed to check if posterior samples are correlated.
#for only acf plots
converge_diag_param_fn(fitmodel_forecast, plot_params = c("kappa[1,4]","gamma[1,2]"), trace = FALSE, density = FALSE, acf_plot = TRUE)
#> $ESS
#> kappa[1,4] gamma[1,2]
#> 168.574037 9.357196Several other commonly used MCMC convergence diagnostics, such as
Gelman’s R statistic (Gelman and Rubin
(1992)), Geweke’s Z scores (Geweke
(1991)), Heidel’s Stationarity and Half-width tests (see Heidelberger and Welch (1981) and Heidelberger and Welch (1983) for more details),
can be computed and illustrated using the function
converge_diag_fn.
converge_diag_result<-converge_diag_fn(fitmodel_forecast, plot_gelman = TRUE, plot_geweke = TRUE)
#> Note: no convergence issues identified.
#Gelman's R
head(converge_diag_result$gelman_diag$psrf)
#> Point est. Upper C.I.
#> q[1,1,1] 1.007521 1.022611
#> q[1,2,1] 1.122386 1.438192
#> q[1,3,1] 1.677692 2.917502
#> q[1,4,1] 1.003516 1.003721
#> q[1,5,1] 1.269089 1.861320
#> q[1,6,1] 1.543866 2.665951
#Geweke's Z
head(converge_diag_result$geweke_diag$z)
#> q[1,1,1] q[1,2,1] q[1,3,1] q[1,4,1] q[1,5,1] q[1,6,1]
#> 3.606717 1.164595 -5.137911 -2.043750 -1.228851 -5.200268
#Heidel's Stationarity and Half-width tests
head(converge_diag_result$heidel_diag)
#> stest start pvalue htest mean halfwidth
#> q[1,1,1] 1 1 0.18887881 1 0.0007217432 4.272638e-06
#> q[1,2,1] 1 1 0.85854603 1 0.0007715985 5.760703e-06
#> q[1,3,1] 1 801 0.12233304 1 0.0008153714 5.008481e-06
#> q[1,4,1] 1 1 0.50954807 1 0.0008232145 6.828502e-06
#> q[1,5,1] 1 601 0.07154669 1 0.0008998373 4.979771e-06
#> q[1,6,1] 1 601 0.23798990 1 0.0009619807 8.592257e-06
