Model mortality hazard, survivorship and age-specific survival probability using a mortality model

Generates an actuarial life table based on a defined mortality model.

Usage

model_survival(params, age = NULL, model, truncate = 0.01)

model_mortality(params, age = NULL, model, truncate = 0.01)

Arguments

params

Numeric vector representing the parameters of the mortality model.

age

Numeric vector representing age. The default is NULL, whereby the survival trajectory is modelled from age 0 to the age at which the survivorship of the synthetic cohort declines to a threshold defined by the truncate argument, which has a default of 0.01 (i.e. 1% of the cohort remaining alive).

model

A character string specifying the name of the mortality model to be used. Options are gompertz, gompertzmakeham, exponential, siler, weibull, and weibullmakeham. These names are not case-sensitive.

truncate

a value defining how the life table output should be truncated. The default is 0.01, indicating that the life table is truncated so that survivorship (lx) > 0.01 (i.e. the age at which 1% of the cohort remains alive).

Value

A dataframe in the form of a lifetable with columns for age (x), hazard (hx), survivorship (lx) and mortality (qx) and survival probability within interval (px).

Details

The required parameters varies depending on the mortality model. The parameters are provided as a vector.

*For gompertz and weibull, the parameters are b0, b1. *For gompertzmakeham and weibullmakeham the parameters are b0, b1 and C. *For exponential, the parameter is C. *For siler, the parameters are a0, a1, C, b0 and b1.

Note that the parameters must be provided in the order mentioned here. x represents age.

• Gompertz: \(h_x = b_0 \mathrm{e}^{b_1 x}\)

• Gompertz-Makeham: \(h_x = b_0 \mathrm{e}^{b_1 x} + c\)

• Exponential: \(h_x = c\)

• Siler: \(h_x = a_0 \mathrm{e}^{-a_1 x} + c + b_0 \mathrm{e}^{b_1 x}\)

• Weibull: \(h_x = b_0 b_1 (b_1 x)^{(b_0 - 1)}\)

• Weibull-Makeham: \(h_x = b_0 b_1 (b_1 x)^{(b_0 - 1)} + c\)

In the output, the probability of survival (px) (and death (qx)) represent the probability of individuals that enter the age interval \([x,x+1]\) survive until the end of the interval (or die before the end of the interval). It is not possible to estimate a value for this in the final row of the life table (because there is no \(x+1\) value) and therefore the input values of age (x) may need to be extended to capture this final interval.

References

Cox, D.R. & Oakes, D. (1984) Analysis of Survival Data. Chapman and Hall, London, UK.

Pinder III, J.E., Wiener, J.G. & Smith, M.H. (1978) The Weibull distribution: a method of summarizing survivorship data. Ecology, 59, 175–179.

Pletcher, S. (1999) Model fitting and hypothesis testing for age-specific mortality data. Journal of Evolutionary Biology, 12, 430–439.

Siler, W. (1979) A competing-risk model for animal mortality. Ecology, 60, 750–757.

Vaupel, J., Manton, K. & Stallard, E. (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16, 439–454.

See also

model_fecundity() to model age-specific reproductive output using various functions.

Other trajectories: model_fecundity()

Author

Owen Jones jones@biology.sdu.dk

Examples

model_mortality(params = c(b_0 = 0.1, b_1 = 0.2), model = "Gompertz")
#>     x        hx         lx        qx        px
#> 1   0 0.1000000 1.00000000 0.1051240 0.8948760
#> 2   1 0.1221403 0.89487598 0.1268617 0.8731383
#> 3   2 0.1491825 0.78135045 0.1526972 0.8473028
#> 4   3 0.1822119 0.66204041 0.1832179 0.8167821
#> 5   4 0.2225541 0.54074272 0.2190086 0.7809914
#> 6   5 0.2718282 0.42231542 0.2606027 0.7393973
#> 7   6 0.3320117 0.31225886 0.3084127 0.6915873
#> 8   7 0.4055200 0.21595427 0.3626343 0.6373657
#> 9   8 0.4953032 0.13764186 0.4231275 0.5768725
#> 10  9 0.6049647 0.07940180 0.4892807 0.5107193
#> 11 10 0.7389056 0.04055203 0.5598781 0.4401219
#> 12 11 0.9025013 0.01784784 0.6330059 0.3669941

model_mortality(
  params = c(b_0 = 0.1, b_1 = 0.2, C = 0.1),
  model = "GompertzMakeham",
  truncate = 0.1
)
#>   x        hx        lx        qx        px
#> 1 0 0.2000000 1.0000000 0.1902827 0.8097173
#> 2 1 0.2221403 0.8097173 0.2099518 0.7900482
#> 3 2 0.2491825 0.6397156 0.2333287 0.7666713
#> 4 3 0.2822119 0.4904516 0.2609450 0.7390550
#> 5 4 0.3225541 0.3624707 0.2933298 0.7066702
#> 6 5 0.3718282 0.2561472 0.3309657 0.6690343
#> 7 6 0.4320117 0.1713713 0.3742259 0.6257741
#> 8 7 0.5055200 0.1072397 0.4232876 0.5767124

model_mortality(params = c(c = 0.2), model = "Exponential", age = 0:10)
#>     x  hx        lx        qx        px
#> 1   0 0.2 1.0000000 0.1812692 0.8187308
#> 2   1 0.2 0.8187308 0.1812692 0.8187308
#> 3   2 0.2 0.6703200 0.1812692 0.8187308
#> 4   3 0.2 0.5488116 0.1812692 0.8187308
#> 5   4 0.2 0.4493290 0.1812692 0.8187308
#> 6   5 0.2 0.3678794 0.1812692 0.8187308
#> 7   6 0.2 0.3011942 0.1812692 0.8187308
#> 8   7 0.2 0.2465970 0.1812692 0.8187308
#> 9   8 0.2 0.2018965 0.1812692 0.8187308
#> 10  9 0.2 0.1652989 0.1812692 0.8187308
#> 11 10 0.2 0.1353353        NA        NA

model_mortality(
  params = c(a_0 = 0.1, a_1 = 0.2, C = 0.1, b_0 = 0.1, b_1 = 0.2),
  model = "Siler",
  age = 0:10
)
#>    x        hx         lx        qx        px
#> 1  0 0.3000000 1.00000000 0.2606669 0.7393331
#> 2  1 0.3040134 0.73933313 0.2666366 0.7333634
#> 3  2 0.3162145 0.54219987 0.2786665 0.7213335
#> 4  3 0.3370930 0.39110690 0.2969238 0.7030762
#> 5  4 0.3674870 0.27497795 0.3216226 0.6783774
#> 6  5 0.4086161 0.18653882 0.3529771 0.6470229
#> 7  6 0.4621311 0.12069488 0.3911330 0.6088670
#> 8  7 0.5301797 0.07348713 0.4360763 0.5639237
#> 9  8 0.6154929 0.04144114 0.4875201 0.5124799
#> 10 9 0.7214946 0.02123775 0.5447766 0.4552234

model_mortality(
  params = c(b_0 = 1.4, b_1 = 0.18),
  model = "Weibull"
)
#>     x        hx         lx         qx        px
#> 1   0 0.0000000 1.00000000 0.06148553 0.9385145
#> 2   1 0.1269140 0.93851447 0.13686918 0.8631308
#> 3   2 0.1674640 0.81006076 0.16657160 0.8334284
#> 4   3 0.1969509 0.67512765 0.18857273 0.8114273
#> 5   4 0.2209701 0.54781698 0.20648690 0.7935131
#> 6   5 0.2416003 0.43469995 0.22177479 0.7782252
#> 7   6 0.2598783 0.33829446 0.23520126 0.7647987
#> 8   7 0.2764068 0.25872718 0.24722531 0.7527747
#> 9   8 0.2915718 0.19476327 0.25814734 0.7418527
#> 10  9 0.3056374 0.14448565 0.26817620 0.7318238
#> 11 10 0.3187936 0.10573804 0.27746379 0.7225362
#> 12 11 0.3311819 0.07639956 0.28612450 0.7138755
#> 13 12 0.3429115 0.05453977 0.29424691 0.7057531
#> 14 13 0.3540682 0.03849161 0.30190116 0.6980988
#> 15 14 0.3647210 0.02687095 0.30914389 0.6908561
#> 16 15 0.3749264 0.01856396 0.31602155 0.6839785
#> 17 16 0.3847313 0.01269735 0.32257279 0.6774272

model_mortality(
  params = c(b_0 = 1.1, b_1 = 0.05, c = 0.2),
  model = "WeibullMakeham"
)
#>     x        hx         lx        qx        px
#> 1   0 0.2000000 1.00000000 0.1977871 0.8022129
#> 2   1 0.2407624 0.80221294 0.2151206 0.7848794
#> 3   2 0.2436881 0.62964040 0.2169760 0.7830240
#> 4   3 0.2454959 0.49302353 0.2182027 0.7817973
#> 5   4 0.2468237 0.38544447 0.2191342 0.7808658
#> 6   5 0.2478803 0.30098040 0.2198903 0.7801097
#> 7   6 0.2487612 0.23479772 0.2205292 0.7794708
#> 8   7 0.2495187 0.18301798 0.2210836 0.7789164
#> 9   8 0.2501844 0.14255570 0.2215743 0.7784257
#> 10  9 0.2507790 0.11096903 0.2220149 0.7779851
#> 11 10 0.2513168 0.08633225 0.2224152 0.7775848
#> 12 11 0.2518083 0.06713065 0.2227822 0.7772178
#> 13 12 0.2522610 0.05217514 0.2231213 0.7768787
#> 14 13 0.2526810 0.04053375 0.2234366 0.7765634
#> 15 14 0.2530729 0.03147703 0.2237313 0.7762687
#> 16 15 0.2534403 0.02443463 0.2240082 0.7759918
#> 17 16 0.2537863 0.01896108 0.2242693 0.7757307
#> 18 17 0.2541134 0.01470869 0.2245164 0.7754836
#> 19 18 0.2544236 0.01140635 0.2247511 0.7752489

model_survival(params = c(b_0 = 0.1, b_1 = 0.2), model = "Gompertz")
#>     x        hx         lx        qx        px
#> 1   0 0.1000000 1.00000000 0.1051240 0.8948760
#> 2   1 0.1221403 0.89487598 0.1268617 0.8731383
#> 3   2 0.1491825 0.78135045 0.1526972 0.8473028
#> 4   3 0.1822119 0.66204041 0.1832179 0.8167821
#> 5   4 0.2225541 0.54074272 0.2190086 0.7809914
#> 6   5 0.2718282 0.42231542 0.2606027 0.7393973
#> 7   6 0.3320117 0.31225886 0.3084127 0.6915873
#> 8   7 0.4055200 0.21595427 0.3626343 0.6373657
#> 9   8 0.4953032 0.13764186 0.4231275 0.5768725
#> 10  9 0.6049647 0.07940180 0.4892807 0.5107193
#> 11 10 0.7389056 0.04055203 0.5598781 0.4401219
#> 12 11 0.9025013 0.01784784 0.6330059 0.3669941