The law catalogue. It lists every parametric model that
MortalityLaw can fit, with the formula and the code to pass
through law, and says where each model applies. Use it to choose a
law before fitting; there is no need to know the functional form, only its
code and the age range it is meant for. For a comprehensive review of the
mortality laws themselves, Tabeau (2001) is a good starting point.
Value
The output is of the "availableLaws" class with the following
components:
- table
Table with mortality models and codes to be used in
MortalityLaw, the model formula, the lifespan section (TYPE), the code (CODE), whether the law describesmu[x]orq[x](FIT) and whether fitting rescales the ages (SCALE_X).- legend
Table with details about the section of the mortality curve.
Details
The TYPE column says where on the lifespan the law belongs, read off
the legend in the second component of the result: a law covering the whole
range (6) is fitted over all ages, while a law for old age (5) is fitted
from the adult ages up. The FIT column says whether the law
describes a hazard, mu[x], or a death probability, q[x];
MortalityLaw and LawTable handle both. The
SCALE_X column flags the laws whose age vector is rescaled during
fitting for numerical stability, which matters when the fitted
coefficients are reused outside the fitted age range (see
LawTable).
A law is normally reached through the law argument of
MortalityLaw, but each one is also a function that can be
called by name for a plain hazard or death probability curve; those help
pages exist for reference and are kept out of the help index. A law that is
not in the catalogue can still be fitted by passing it as a function
through the custom.law argument; see the examples on the
MortalityLaw page.
A few laws carry a caveat worth knowing before choosing them, all documented in the model catalogue and their catalogue entries:
"scholey": its truncation parameter is identified only on day- or week-level data over the first year of life; on single-year ages it collapses and the fit reduces to"scholey_shifted_power", with a warning."opperman": the published middle term has a free sign; the package uses the negative branch the log-scale engine permits, which is the branch mortality data occupy."steffensen": the formula is attributed to Steffensen (1930), but the attribution is not verified against the paywalled source."HP","HP2","HP3","HP4"and"kostaki": high-parameter models; fit them withopt.method = "LF2"."gompertz_logquad"and"makeham_logquad": the sign of the quadratic term is fixed to the decelerating branch that mortality data occupy; the accelerating branch cannot be fitted."beard_makeham"and"perks": the same four-parameter Perks-Beard logistic written two ways, so they fit identical curves;"beard","kannisto"and"kannisto_makeham"are its two- and three-parameter cases. Pick one of them, not several."demoivre": the 1725 baseline, kept for completeness. Its hazard is defined only below a limiting age, so it must not be extrapolated and the fit warns every time."weibull": not defined at birth, so age 0 is reported as missing and takes no part in the fit; start the fit at age 1.
Two entries in the reference list are background for the infant laws
rather than the source of a catalogue code: Harper (1936), and de Beer and
Janssen (2016), whose infancy term is the fitted pareto_2.
References
De Moivre, A. (1725). Annuities on Lives: or, the Valuation of Annuities upon any Number of Lives. London: William Pearson.
Gompertz, B. (1825). On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies. Philosophical Transactions of the Royal Society of London, 115, 513-583.
Makeham, W. (1860). On the Law of Mortality and Construction of Annuity Tables. The Assurance Magazine and Journal of the Institute of Actuaries, 8(6), 301-310. doi:10.1017/S204616580000126X
Thiele, T. (1871). On a Mathematical Formula to express the Rate of Mortality throughout the whole of Life, tested by a Series of Observations made use of by the Danish Life Insurance Company of 1871. Journal of the Institute of Actuaries and Assurance Magazine, 16(5), 313-329. doi:10.1017/S2046167400043688
Lomax, K. S. (1954). Business Failures: Another Example of the Analysis of Failure Data. Journal of the American Statistical Association, 49(268), 847-852. doi:10.1080/01621459.1954.10501239
Vaupel, J. W. and Yashin, A. I. (1983). The Deviant Dynamics of Death in Heterogeneous Populations. IIASA Research Report RR-83-1. Laxenburg, Austria.
de Beer, J. and Janssen, F. (2016). A new parametric model to assess delay and compression of mortality. Population Health Metrics, 14(1), 46. doi:10.1186/s12963-016-0113-1
Scholey, J. (2019). The Age-Trajectory of Infant Mortality in the United States: Parametric Models and Generative Mechanisms. PAA Annual Conference, Austin.
Oppermann, L. H. F. (1870). On the graduation of life tables, with special application to the rate of mortality in infancy and childhood. The Insurance Record Minutes from a meeting in the Institute of Actuaries, 42.
Wittstein, T. and D. Bumsted. (1883). The Mathematical Law of Mortality. Journal of the Institute of Actuaries and Assurance Magazine, 24(3), 153-173.
Steffensen, J. (1930). Infantile mortality from an actuarial point of view. Skandinavisk Aktuarietidskrift 13, 272-286. doi:10.1080/03461238.1930.10416902
Perks, W. (1932). On Some Experiments in the Graduation of Mortality Statistics. Journal of the Institute of Actuaries, 63(1), 12-57. doi:10.1017/S0020268100046680
Harper, F. S. (1936). An actuarial study of infant mortality. Scandinavian Actuarial Journal 1936 (3-4), 234-270. doi:10.1080/03461238.1936.10405113
Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of applied mechanics 18, 293-297. doi:10.1115/1.4010337
Beard, R. E. (1971). Some aspects of theories of mortality, cause of death analysis, forecasting and stochastic processes. Biological aspects of demography 999, 57-68.
Vaupel, J., Manton, K.G., and Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16(3): 439-454. doi:10.2307/2061224
Siler, W. (1979), A Competing-Risk Model for Animal Mortality. Ecology, 60: 750-757. doi:10.2307/1936612
Heligman, L., & Pollard, J. (1980). The age pattern of mortality. Journal of the Institute of Actuaries, 107(1), 49-80. doi:10.1017/S0020268100040257
Rogers A and Planck F (1983). MODEL: A General Program for Estimating Parametrized Model Schedules of Fertility, Mortality, Migration, and Marital and Labor Force Status Transitions. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-83-102
Martinelle S. (1987). A generalized Perks formula for old-age mortality. Stockholm, Sweden, Statistiska centralbyran, 1987. 55 p. (R&D Report, Research-Methods-Development, U/STM No. 38)
Forfar, D. O., McCutcheon, J. J. and Wilkie, A. D. (1988). On graduation by mathematical formula. Journal of the Institute of Actuaries, 115(1), 1-149.
Carriere J.F. (1992). Parametric models for life tables. Transactions of the Society of Actuaries. Vol.44
Kostaki A. (1992). A nine-parameter version of the Heligman-Pollard formula. Mathematical Population Studies. Vol. 3 277-288. doi:10.1080/08898489209525346
Thatcher AR, Kannisto V and Vaupel JW (1998). The force of mortality at ages 80 to 120. Odense Monographs on Population Aging Vol. 5, Odense University Press, 1998. 104, 20 p. Odense, Denmark
Tabeau E. (2001). A Review of Demographic Forecasting Models for Mortality. In: Tabeau E., van den Berg Jeths A., Heathcote C. (eds) Forecasting Mortality in Developed Countries. European Studies of Population, vol 9. Springer, Dordrecht. doi:10.1007/0-306-47562-6_1
Finkelstein M. (2012) Discussing the Strehler-Mildvan model of mortality Demographic Research, Vol. 26(9), 191-206. doi:10.4054/DemRes.2012.26.9
See also
MortalityLaw to fit a law; LawTable
to build a life table from fitted coefficients; availableLF
for the loss functions.
Examples
availableLaws()
#>
#> Mortality laws available in the package:
#>
#> YEAR NAME
#> 1725 De Moivre
#> 1825 Gompertz
#> <NA> Gompertz
#> <NA> Inverse-Gompertz
#> 1860 Makeham
#> <NA> Makeham
#> 1870 Opperman
#> 1871 Thiele
#> 1871 Negative-Gompertz
#> 1883 Wittstein
#> 1930 Steffensen
#> 1932 Perks
#> 1939 Weibull
#> 1954 Pareto-II
#> <NA> Inverse-Weibull
#> 1943 Van der Maen
#> 1943 Van der Maen
#> 1960 Strehler-Mildvan
#> <NA> Quadratic
#> 1971 Beard
#> 1971 Beard-Makeham
#> 1979 Gamma-Gompertz
#> 1979 Siler
#> 1980 Heligman-Pollard
#> 1980 Heligman-Pollard
#> 1980 Heligman-Pollard
#> 1980 Heligman-Pollard
#> 1983 Rogers-Planck
#> 1987 Martinelle
#> 1988 Gompertz-Makeham
#> 1988 Gompertz-Makeham
#> 1992 Carriere
#> 1992 Carriere
#> 1992 Kostaki
#> 1998 Kannisto
#> 1998 Kannisto-Makeham
#> 2019 Scholey-Shifted-Power
#> 2019 Scholey
#> MODEL TYPE
#> mu[x] = 1/[N - x] 6
#> mu[x] = A exp[Bx] 3
#> mu[x] = 1/sigma * exp[(x-M)/sigma] 3
#> mu[x] = 1/sigma * exp[-(x-M)/sigma] / (exp(exp[-(x-M)/sigma]) - 1) 2
#> mu[x] = A exp[Bx] + C 3
#> mu[x] = 1/sigma * exp[(x-M)/sigma] + C 3
#> mu[x] = A/sqrt(x+1) - B + C*sqrt(x+1) 1
#> mu[x] = A exp(-Bx) + C exp[-.5D (x-E)^2] + F exp(Gx) 6
#> mu[x] = A exp(-Bx) 1
#> q[x] = (1/B) A^-[(Bx)^N] + A^-[(M-x)^N] 6
#> mu[x] = [A + BC^x] / [BC^-x + 1 + DC^x] 6
#> mu[x] = [A + BC^x] / [1 + DC^x] 3
#> mu[x] = 1/sigma * (x/M)^(M/sigma - 1) 1
#> mu[x] = A/(x + C) 1
#> mu[x] = 1/sigma * (x/M)^[-M/sigma - 1] / [exp((x/M)^(-M/sigma)) - 1] 2
#> mu[x] = A + Bx + Cx^2 + I/[N - x] 4
#> mu[x] = A + Bx + I/[N - x] 5
#> mu[x] = A exp(Bx) exp[-(V/B)(1 - exp(-Bx))] 3
#> mu[x] = A + Bx + Cx^2 5
#> mu[x] = A exp(Bx) / [1 + KA exp(Bx)] 4
#> mu[x] = A exp(Bx) / [1 + KA exp(Bx)] + C 4
#> mu[x] = A exp(Bx) / (1 + AG/B * [exp(Bx) - 1]) 4
#> mu[x] = A exp(-Bx) + C + D exp(Ex) 6
#> q[x]/p[x] = A^[(x + B)^C] + D exp[-E log(x/F)^2] + G H^x 6
#> q[x] = A^[(x + B)^C] + D exp[-E log(x/F)^2] + GH^x / [1 + GH^x] 6
#> q[x] = A^[(x + B)^C] + D exp[-E log(x/F)^2] + GH^x / [1 + KGH^x] 6
#> q[x] = A^[(x + B)^C] + D exp[-E log(x/F)^2] + GH^(x^K) / [1 + GH^(x^K)] 6
#> q[x] = A0 + A1 exp[-Ax] + A2 exp[B(x - u) - exp(-C(x - u))] + A3 exp[Dx] 6
#> mu[x] = [A exp(Bx) + C] / [1 + D exp(Bx)] + K exp(Bx) 6
#> mu[x] = A0 + K exp[B1 x - B2 x^2] 5
#> mu[x] = K exp[B1 x - B2 x^2] 5
#> l[x] = P1 l[x](weibull) + P2 l[x](invweibull) + P3 l[x](gompertz) 6
#> l[x] = P1 l[x](weibull) + P2 l[x](invgompertz) + P3 l[x](gompertz) 6
#> q[x]/p[x] = A^[(x+B)^C] + D exp[-(E_i log(x/F_))^2] + G H^x 6
#> mu[x] = A exp(Bx) / [1 + A exp(Bx)] 5
#> mu[x] = A exp(Bx) / [1 + A exp(Bx)] + C 5
#> mu[x] = A (x + C)^-B 1
#> mu[x] = A (x + C)^-B exp(-Dx) 1
#> CODE
#> demoivre
#> gompertz
#> gompertz0
#> invgompertz
#> makeham
#> makeham0
#> opperman
#> thiele
#> neggompertz
#> wittstein
#> steffensen
#> perks
#> weibull
#> pareto_2
#> invweibull
#> vandermaen
#> vandermaen2
#> strehler_mildvan
#> quadratic
#> beard
#> beard_makeham
#> ggompertz
#> siler
#> HP
#> HP2
#> HP3
#> HP4
#> rogersplanck
#> martinelle
#> makeham_logquad
#> gompertz_logquad
#> carriere1
#> carriere2
#> kostaki
#> kannisto
#> kannisto_makeham
#> scholey_shifted_power
#> scholey
#>
#> LEGEND:
#> TYPE Coverage
#> 1 Infant mortality
#> 2 Accident hump
#> 3 Adult mortality
#> 4 Adult and/or old-age mortality
#> 5 Old-age mortality
#> 6 Full age range