Calculates reproduction indices from a mortAAR life table

For population studies it is of vital importance to estimate growth or decline of a population. For archaeological datasets this is rarely attempted, probably because the data quality seems to scanty. Nevertheless, the calculation of such measures seems worth the try, at least it should give an impression if the resulting values are unrealistic high or low.

Usage

lt.reproduction(
  life_table,
  fertility_rate = "BA_log",
  growth_rate = "fertility",
  gen_len = 20
)

Arguments

life_table: an object of class mortaar_life_table.
fertility_rate: string or numeric. Either fertility rate according to McFadden & Oxenham 2018a if infants are represented well or Taylor et al. or fertility rate according to data by McFadden & Oxenham 2018a for P(5-19) index after Bocquet-Appel 2002. Options: 'McFO' (McFadden/Oxenham), 'TOMc' (Taylor et al.), 'BA_linear' (linear fit), 'BA_power' (power fit) or 'BA_log' (logistic fit). Default: BA_log'. Additionally, the user can specify an arbitrary number in lieu of the fertility rate.
growth_rate: string or numeric. Either derived directly from the fertility calculations or from regression analysis by either McFadden & Oxenham 2018b (\(10.06 * D0--14/D) -- 1.61\)) or Bocquet-Appel and Masset (\((1.484 * (log10(200 * d5--14/d20 * d60/d20))**0.03 - 1.485)\)). Options: 'fertility', 'MBA', McFO'. Additionally, the user can specify an arbitrary number in lieu of the growth rate.
gen_len: numeric. Length of generation for determining the rate of doubling the population. Default: 20.

Value

A data.frame with basic reproduction indices:

m: Mortality rate (= natality rate n).

\(0.127 * d5--14/d20 + 0.016\)
dep: Dependency ratio.

\(DR = (sum(D0--14) + sum(D60+)) / sum(D15--59) \)
TFR: Total fertility rate.
GRR: Gross reproduction rate.

\(GRR = TFR * 0.488\)
NRR: Net reproduction rate.

\(NRR = sum(GRR * age specific fertility * age specific survival / 10000)\)
r: Intrinsic growth rate in per cent per year.
Dt: Doubling time in years.

\(Dt = 100 * ln(2) / r\)
D30_D5: Ratio D30+/D5+ after Buikstra et al.
BR: Birth rate from ratio D30+/D5+.

\(BR = -11.493 * D30_D5 + 12.712\)
DR: Death rate from ratio D30+/D5+.

\(DR = -5.287 * D30_D5 + 6.179\)

Details

There are different approaches to calculate reproduction rates (e. g. Henneberg 1978). We largely follow the methodology by Hassan 1981. Typically, a Total fertility rate (TFR) of 6-8 is assumed for prehistoric populations (Ascadi/Nemeskeri 1970; Henneberg 1978; Hassan 1981). Recently, McFadden and Oxenham 2018a have published a formula to estimate the Total fertility rate from archaeological data, provided that infants are represented fully in the archaeological record.
Unfortunately, this will not be the case for most archaeological datasets. Therefore, we used the data published by McFadden and Oxenham to apply it to the P(5-19)-index after Bocquet-Appel 2002. We approximated the ratio by three different methods of fitting (linear, logistic, power) and recommend logistic fitting, but the others are available as well.
We have also added the option to use the formula by Taylor et al. 2023 that uses the ratio of adults aged 15–49 in relation to those aged 15 or older.
The Gross reproduction rate (GRR) is calculated by multiplying the TFR with the ratio of female newborns, assumed to be a constant of 48.8 of all children (Hassan 1981, 136). The Net reproduction rate is arrived at by summing the product of the GRR, the age specific fertility rate as defined by Hassan (1981, 137 tab. 8.7) and the age specific survival taken from the life table and dividing the result by 10000.
The Rate of natural increase or Intrinsic growth rate r (growth in per cent per year) can be computed from the fertility following Hassan (1981, 140). Alternative ways to calculate the intrinsic growth rate derive from Bocquet-Appel and Masset (1977) and recently from McFadden and Oxenham 2018b. The latter present a regression analysis based on the index D0–15/D also used for fertility calculations (see above) in connections with modern data.
Whatever is chosen as base for the growth rate calculations is used for computing the doubling time in years, assuming exponential steady growth.
Also calculated is the mortality rate m after Bocquet-Appel and Masset (1977) in per cent of a given population. Furthermore, the ratio of dependent individuals is reported that is usually (but probably erroneously for archaic societies (Grupe et al. 2015, 423) assumed to apply to those aged below 15 or 60 and above.
Finally, Buikstra et al. 1986 have made the interesting observation that the relation of those individuals aged 30 years and above to those aged 5 years and above is very closely related to the birth rate and also closely (but less significantly) related to the death rate. Therefore, these indices are calculated as well.

References

acsadi_history_1970mortAAR

masset_bocquet_1977mortAAR

bocquet_appel_2002mortAAR

buikstra_et_al_1986mortAAR

grupe_et_al_2015mortAAR

hassan_1981mortAAR

henneberg_1976mortAAR

mcfadden_oxenham_2018amortAAR

mcfadden_oxenham_2018bmortAAR

taylor_et_al_2023mortAAR

Examples

schleswig <- life.table(schleswig_ma[c("a", "Dx")])
lt.reproduction(schleswig)
#>    method  value              description
#> 1       m   4.39                Mortality
#> 2     dep 105.83         Dependency ratio
#> 3     TFR   7.01     Total fertility rate
#> 4     GRR   3.42  Gross reproduction rate
#> 5     NRR   1.69    Net reproduction rate
#> 6       r   2.63 Rate of natural increase
#> 7      Dt  26.39   Doubling time in years
#> 8  D30_D5   0.57           ratio D30+/D5+
#> 9      BR   6.20               Birth rate
#> 10     DR   3.18               Death rate

odagsen <- life.table(list(
  "corpus mandibulae" = odagsen_cm[c("a", "Dx")],
  "margo orbitalis" = odagsen_mo[c("a", "Dx")]
))
lt.reproduction(odagsen)
#> $`corpus mandibulae`
#>    method value              description
#> 1       m  3.37                Mortality
#> 2     dep 62.32         Dependency ratio
#> 3     TFR  6.46     Total fertility rate
#> 4     GRR  3.15  Gross reproduction rate
#> 5     NRR  1.82    Net reproduction rate
#> 6       r  2.98 Rate of natural increase
#> 7      Dt 23.24   Doubling time in years
#> 8  D30_D5  0.69           ratio D30+/D5+
#> 9      BR  4.74               Birth rate
#> 10     DR  2.51               Death rate
#> 
#> $`margo orbitalis`
#>    method value              description
#> 1       m  2.42                Mortality
#> 2     dep 34.05         Dependency ratio
#> 3     TFR  5.83     Total fertility rate
#> 4     GRR  2.84  Gross reproduction rate
#> 5     NRR  1.45    Net reproduction rate
#> 6       r  1.87 Rate of natural increase
#> 7      Dt 37.09   Doubling time in years
#> 8  D30_D5  0.69           ratio D30+/D5+
#> 9      BR  4.77               Birth rate
#> 10     DR  2.52               Death rate
#>